Under consideration for puSl_cation i_ J. Fluid Mech. Statistical ensemble of large eddy simulations By D A N IELE C A RAT It, M IC H A EL M . R O G ER S2 AND ALAN A.W RAY 2 2Association Euratom-Etat Beige, Universit_ Libre de Bruxelles, CP 231, 1050 Bruxelles, Belgium. 2NASA Ames Research Center, Moffett Field, CA 94035, [IS.A. (1Received : ) A stat_ _sem b]e of laxge eddy s_ ul_tJons :isrun sinukzaneous_ fDr the sam e _w. The hfDzra atJon provided by dqe di erent ]arge-sca]e vebc/ty e]ds Jsused in an ensem b]e-averaged vein ofthe dynam JcmodeL This pl_duces bcalm odelpaxam eners that on_ depend on the stati_ properties of the ow. A n inportDnt plImperty of the ensem b]e-ave_aged dynam i:proceduz_ Jsthat Jtdoes not require any spatial aver- agog and can thus be used h ful)yiqhGm ogeneous ows. A ]so,d_e 6ns_n b]e ofLE S's provides stat/st_ of the _xge-sza]e vebcJLy that can be used fDrbuik_hg new models fDr the subgrJd_aca]e stresstensor. The ensem b]e-ave_aged dynam _ pzooedure has been inp_m ent_d with varbusm odels fDr three _s: dec_yilg Js3tr_p_ D/zbul_nce, _9l_ /sotr_pJc tu_bu]enae, and the tine_devebp_g plane wake. It/sfDund that the resl]_sare aln ost hdepesdent ofthe num ber ofLE S's _ the stati_l ens_n b]e p_mvJded that the enssu b]e cnntahs at ]east16 ree_t_ns. i. Introduction Thenumberofdegreesoffreedom needed tochazact_r/zeavebc_y e]d u, thatcorre- spondstD a tu_bul_It Gw Jsknc_n to hcL_ease as Re _/4 (R_!istheReyno]dsnumber) _2 dlree din ens_nal tunbu]ent systmm s.D Jrectnum em-Jcals_ u%_tJons _3N S)ofthe N av_r{ Stokes equat_ns gove_nhg the evol/tJon ofsuch syst_n sare thus ]inited tom oderate_ sn a/lReynolfls numbers. There Jsthus an ht_mest _ deve_hg techniques h whJd% on_ a fraction of the tDt_inumber old--of fre_dcm Jsactual_ sinulat_d. Am ong these tedlnrlues, Large Eddy S1nu]atJon _ES) and Reyno]ds Averaged Navi_x{Stokes _RANS) s_ u]atJon have attzact_d mud% interest _ the past f_w decades. InLES, the numberofd_of freedcma Jsreduced byus_g a spat/al _g: 0_)= /dyG(x-y)u_), (ii) whemeC isthe ]£erkemnelandH, istheLES e]d.lnRANS,anensemb]eave_g_ngJs usedtDde netheRANS e]dUi: 2 D.Ca_mt_.MM.R_,andA.A.Wray Inbothcasetsh,eequationfDsr_ ior fDrU_ cont_ an unknown _ t_nn that requires m odellhg. The purpose of the appzoad_ devebped here is_ cam bine conc_0ts frnm the two m ethods toproduce a statJstJcmlver_d3n ofLES. T he p_sent approad% ism ot/vatsd by the factthat, inboth LE S and RAN S,models fDrthe degrees offreedcm thathavebeen elininatBd areinspJrsd fr_n sta_theor_ oftufou]6nce. ItJsthus inplb_]y assure ed that the ]fetingand ensemb]e averaging can both be regarded as projgctg_e operati]ns that assocJat_ a hum bet of di erent ve_bcJty r_lg_tJons w Jth a shg]e LE S orRAN S e]d.T here is,how ever,an inportant di erenoe between enssttb]e averaging and spat/al ker_g. The ensemb]e averaging operation ze- duces the number ofdegIEes offreedam by somuch that aln ostno usefnlinfDnn atJon on the ucD_atbns 6"ui:--u_ - (}ican be deduced from the knowledge of[/{ abne. On the otherhand, InLES the statistizsofthe unresolved sca]eszz' : u_ _7_mustbe cbs_ly related _ the statistizsof the reso]ued scales {, shoe there isno clear sc_Je separation between them .Hence, know]edgeofthestatistizalpropertJesoftheLES e]dsui should be heJpful in devebping LE S m odels. T he advantage of studying a statJstiml ensem - b]e of LE S's Jsthe abil_y to extract sta_ infDnn atJen fDrbuilding m ode/s fDr the unresoIIed scales.This willbe expbred inSectbn 3. Thede nitJon ofequJu-al_tand indep6_ldentLES eJds isnotn_ obvJousand _nou]d probably depend on dqe m otg/ati)n _r the sinu]atign. W e propose in Secti]n 2 scrneconditigns underwhJch two sinu]atJans ofa tu_bu]6_t ow w illbe supposed tD be independent and equ_t. InSect/Dn 3,wew ill/now that the know ]edgeofan ensem b]e ofLE S'syields a good fram e_ork fDrdevebpmg a bcalversbn ofthe dynam Jcprocedure inwhiln m odelparam et_rsare cam put_d ushg statisticalquant/tJ6_. The applbatJon of thisapproach to JsotropJc turbu]anoe Jspz_s_nt_d inSectJon 4.App]Jcati]n to the wake ow Jspresented h Se_ti3n 5.InthJs lastcase, itJs_nown that the know]edge ofan ensemb]e of realisations can be used to devebp new models that explhit/y inoorpoma_e averaged quantities m ade avaJ]ab]e dlrough the enssm b]e. 2. StatJstJcalensemb]e ofLES's T he equatbn fDr]azgeeddy sinu]atbn (LE S)isobt2mhed by applying a spatial itertm theNavJ_r{Stokesequat_bns. TheLES equat_n thusdescrgDes thee-vo]ut=bn ofa ltered ve]oc/L-y eli _, which explhgz]y depends on the sm a/lsc2Jes through the subgrg] Stre_STij : f_{tlj -- t_ _1): _)_E_+ _./fij_ : -i_,_+ _4,Vzun_- #3_u. (21) Stadtsxalensebm]eof_ eddsyinu]atbns 3 ForsireplbJtyw,eon_oons]d9e1rccp,nr_]e owsh, whi:hp,thepressudreividebdy thedens_yis,detenbnedbyd_ehccmpr_il_-y condg_oTnh.eunknow_nensor Tu appears h the equation fDr d_e ]azge-sca]e vebcJty ui but ]tdepends on the s_ a/l-smm]e ve]oc/by e]d.The purpose ofthis study isto expbrE the advantages ofsgnukaneously nnlning several sta_ equ_t and i%dependent LES's fDr the sam e ow .In pract_, we thus replace the equation (2A) by the _Dlk_ i%g set ofequatJons fDrR large- scale vebc_y e]ds 3[ : OtiS[+ Bju/u, = -0z_ r + - r:%j, (22) wherer = I,...,R. Itis worth ment]oning that the use ofan 6_ssm b]e ofLE S's isnot per sem uch m ore expens_e than the use ofa shg]e rea/_t/3n. To show this,_t us eonsdHer a stationary LES and denote by t,_the tine of the transit period betwe6n the beginning of the sin u]atbn and the tine at which the tumbula_oe becomes _lly deve_ed. Let us also denote by t_the time 00eyond _t)requk_d toconverge the sta_. T hen, the CPU time required forobtaining converged statJstScsw 991a shg]e LES istl+ _ .With an enssa b]e of reel_satJons, statisticsar_ accum ulat_d over both the ensemb]e and tgn e.Tbus, fmr equkra]6nt sam pie, the ensem b]e only needs tobe advanced intgneby the am ount t._/R. The totalC PU cost fDr the ensemb]e isthus R (it+ t_/R ),w hx_n am ounts _ an overhead of {R - l)_,tovera shg]e r_a]/s_ti3n. Ifthe ratiobetween the transi_t phase and the tine needed to conv_z_le statistizsissm a]l then the additional cost w illbe m odera_e. In the exam pies trsatsd bek_, thisadd]tJonal cost/s total_ negl_0]e. Moreover, ifthe LE S isnot stationary and ifthere isno dJrenTJon ofhcln ogene/ty, the enssm b]e-averaged applT]ach isprEsum ably the only way to obtain statiC. 2 i. Statist_lly equ_t and _de6_ndent LES's The know ]edge of an enssab]eofLES'scan enlybeusEfmliftheLES e]ds{[ area/l hdependent. Yet allthese _]ds have to correspond to the sam e experim entalsituation ifsome meaningful statisticsare to be e_tracted _ the ensemb]e. W e therE_e must de ne what w illbe cons_de_ sta_ equkra36nt but independent LE S e_]ds.A l- though a proof ofex_ce and un_ueness of sol/tJons fDrthe N av_er{Stokes equations isnot yet avai]ab]e, _ apzact_poi%tofv]_w a ow descrgoed by theNavJ6_r{Stokes equat/]ns orby an LES equation Jsassure ed tobe ful_ de ned by the knowledge of i. The doraa_] _ inw hJ_h the ow iscr_nsi_er_]. 2. The condkSons on the boundary _ of this dcrna_ u_(@T),t.): b_(t)where the functions b_(_)are gJu_n. 4 D.CazatiM.M.Rcgers,aAnd.A.Wray 3.The]nJtJaleondkJons ui(x,0) : u(/0<) Vx 6 D. How ever,inasinu]atJon ofa t]]rbu]Ent ow onl] the dam ain and the boundary conditions axe r_orous]z xed. Indeed,besause of the lack of sens/tJul-y to in_ eondit_ns in turbu]6nce, di er_nt sinu]atbns w kh di erent inJt/aloonditbns _haring sam e properties are considered to chara_ the sam e ow. Thus, the requ/r_n ent that the in95al oond/r_bns are known issmm ewhat l__]axed fDr tulbulmt ows and the point (3)isthus re_]aced by a weaker consist: 3'.The inJtig_lGond95on ui(x_0): ul/(x;wl) isgenerated ushg random num betaswl and satis esa certain number ofoonst_ts: P, [ul/]: Ps, s : i,...S. For exam p]e,iqhomogeneous tu_bulmce, the most important consist w illbe on the spectrum E (k)ofu{_: k /a ],<, :E(k) (22) where 0 is the Fourier transfDnn oful/ and isthe solh] angle in thewavenumber dak : k2d dk. For channel ow, one could impose the p]ane_veraged pro ]eofboth the ve_bc_-y U _) and the Reyno]ds stress R u _ ): (u"_ = U (rJ)6i,l (2.4) l /2,z ({U','--[]_)6i,l)2)x,z : Ri] [_;) (25) where :c,!]and z are l_o_ the bngkudina]_ the wallnonn a]_and the tran_ d_ns and (.•)r,z reprEs_qts the average Jnplmnes parallel to the wail W e w illnot dg_-uss indetm/l the m him alconstzailts that must be inposed on the in_ oondkJons in order to have a reasonable sJmu]at_n. In fact,this m inimal set of constra_hts will probably d_oend on the type of ow as well as on the quantities that are measured in the sinu]atJon. H ere we only suppose that these constrahts do exist inorder to gJue a precise de nitbn ofequ_tLES's: i) TwoLES'sarestatist_cally equiva]entifthedom ain ofthe ow and the boundary condO:ions are the sam e and ifthe initialoond_dons s_ti_ the sam e set of constlmg_ts. Carrying an ensemble of equkral_t LES's can be cc_ putatJonal_ e ectJue only if the di erent m 6rnbers 91 the set of LES's are 91dep6ndent. Here again, the de nitJon of ind_pend6nt LES's m x/ht d_0end on the ow as well as on the quantit_ that are measured in the sinulatJon. A long the sam e ]he as fDr the de nkJon of sta_ equJva_t LES 's,we propose the fDlk_ hg de n95bn : ii) Two LES 'saxestatistically 91dependent iftheirhJtialconditJons are generated w _[h uncoated random numbers u!l. Stadst_alensem bl_of ]azgeeddy sin ulltbns 5 W e reinark that fDr a statbnary ow, sud% equiraJ6nt and hdependent ]n_ c_n- d/t_bns can be obtained by m/nnilg a sd%g]e LES and recordilg s_velDc/ty e]ds ssparatEd by at ]sastone l_/ge_dy turnover t_ e when tuz]0ulanoe isful}Fdeve/Dped. 22. U nk_rsalm odelpazam et_r _ LES Clas_ closure strat_gJes in LES amount to m ode]lhg Tu in t_nms of the resokred vebc_y eli: _,j: Cm[3_t, ], (2.6) where isthe ]t_rw_dth.The_nsorm[3 issupposed tocharacter_ the dependence on both the k_rw]dth and the speci c rea/_tbn ofthe large-sc_de ow ul•On the other hand, we w illassum e that the paz_m etherC depends on]_,on the type of ow and on the itershape and _noulflnot depend on any parthu]ar l_l_tJon ofthe lalge-sca]eve]oc/ty e]d._Inthe fDl_ hg, we w illl__f_rto thisass/m ed property as the unkre_d/ty of the model param et6_s h LE S:For a g_ gegm etry and fDr a gk_n R eynolls num her, the m edelpa/am et__rsshou]dbe the sam e_ia/lequk_t LE S 's.This concept ofunk_i-y does not inp_ that the model p_ ehe_s are constant _ space and tin e. C ]ear_, C = C Cx,t)can be a e]d quant_y that needs tobe adapted both _ space and tin e to the bcalcond_tJons of the ow. However, h our approad_ the var/at_bns off.' al_ not supposed to take their oz/2h h poss_]e uctuatbns _ the ]arge-s3a]e ow. Rather, C isexpected to depend on/_zon the averaged propert_ of the ow, and in that sense it shares m any propert_ w i_ RAN S quanta. T he assum pt_on that the model pazam ehers are un_zersal has a dk_zt in uence on the _nn ulatSon ofm odels inan ensemb]e of statist_ equ_t LES's. Inthe equa- tbns (22), these m ode]s should have the fDl_ hg structl_re: where C /snow hdependent ofthe rea/_tbn _de_ 'r. Itmust be not_, however, that the devebpm ent ofthe dynam]c procedure h soree ways chall_nges th/s v_ewpoint. In the dynam _cprocedure, hfDn_ at_on fram the snall scales of_ isused _r estimathg the model par-_ epees. This procedure isknown tD produoe h_h_ uctuat_g m odelpazam et_rs.Sud_ a property issam etlmesregarded as a proof of the capabil_-y of dqe dynam izprocedure ha produoe model pazam eters that account fDrthe]ocalcondJr_onsofthe ow.Howev_r, these uctuatSons_ ("ar_respon- sable _3r hstabil_es, and sam e avezaghg prooedures are used ha avoid this di cu]ty. Of ,:ollrse, ]Ilore sot>histicated mof|e]s wit]* more than one tern) have also been proposed, hut the specific roles of the model parameters and of the model tensors m,_ remain fl*esame, 6 D .CazatJ_M .M .Rogers,andA.A.Wray W e propose in Section 3 an approach that reconcJl_ the dynam Jcprocedure w J_h the concept ofa unJuersmlm odelparam eter. In thissense, _:isfaJrl_di erent from other procedures inwhich the concept ofa unJuersg_Iparam eterhas not been adopted, such as dle bcaldynam i:pzocedure devebped by G hosalet al (1995),the Lagrang]sn dynam procedure proposed by M 6neveau et aLL (1996), or the tine l%gghg procedure proposed by Pi]n el]iand L_ (1995). 23. New m ode/]_g concepts Theknow]edgeofan ens_nb]eofLES e]ds opens new po_=do_ inthemode]]hg of the T_. Indeed,it ]snow conoekrab]e to introduce an exp]Jc_itdependence on ensem b]e- averaged quantiti_ into the mode]s fDrT_']. 2J i. M odelhmsel on the uch_athg strain tensor T he rstm odelw epropose isbased on the uctuatJng part ofthe rate-of-stra_h tenser: (- >) T_]: -2_,.,%j-<_{j- -2,,,_0s, (2.8) where u_ isthe eddy viscos_y. Th]s _3nnu]atJon has sam e nice properties. The averaged totaldg_Jpat_n ]Sgiven by j ,r ,r and consequently the tnJou]6nt d_JpatJon or_91ates on_ _om the uctuat/ng part of the strain t_nser. The m ean part contr]bu tinson_ tothe m o]6cufard_s_atbn. T hisprop- erty ensures that the m odelw illnotproduce d_s_atbn ina ]am inar region. Inaddit_bn, whi_ thismodel isd_patJue on average _orov]ded the eddy vi_x)sdty ispositJue), hdi- vJdual_tJons can have negative d_JpatJon, thus r_or_enthg the 91verse transfers ofenergy flrrn the s_ allunres3]ued sma]es to the large ones 03ackscatter) CLeith 1990; M ason & Thcm son 1992; C aratiet al 1995a). It]s general_ b_ that bac_hscatt6r ordinates fr_n uctuatJon phencrnena h the subgrJd soa]es,and representation ofth_ e ect through uctuatJons 91 the stra_h t_nsor isthus very reasonable. Resuksus91g th]smodel_Dr thewake ow al_gk_] _ se_tJonsbei3w. Ithas_dy be_n used 91 the channel ow, where the plane ofhom ogene/ty Jsused to compute the average (Schum ann 1975).How ever,the ensem b]e ofLE S'sa]k_ sthe use ofsuch models even 91 _IIF hhcrn ogeneous ows. O fcourse, many othermode]sm _htbe Cons_er_d a_bng the sam e lhes, and the uctuatJng stra_ rate isnot the on_ quant/ty that Could enter the m ode]_ In thispaper, w e w ill_ our 91vest_atJons to the model (2.8)91 the study of the wake ow. However, we m entJon hereafh_r another posmb]e use of the kn_]6dgeofan ens_nb]eofLES's91 _hecaseofan_otropJc ows. Stads t_ensem b]eof l_ge eddy sireu]atbns 7 2_32. Anisotr_pJc m ode/ A n_tzopJc e ects are aln ostuni_lI] observed in tuJnul_nce. How ever,anJsotrmpy us]a/l] ork/inat_s from cornpl_x intE_acthnsbetween ow dkectJon, sol_dboundarg_and external constzahts lakep_ess]_ gzad_nt or g]obalrotat]on. ItJs thus quJ_e di cult to predict a pr_ri the main oonsequences of thisanJsotropy. In the context of sta_ averaged LES, we have access at any instant tn mean quant_ that will dJsplmy the an/sotzopJc structure ofthe tu_bul_noe eve_ fDrfi]l]_i"uhornogeneous ows.A m ode/that wou]d dkectl] take advantage of the enssm b]e ofLES's could be: r,__ _r_ik'y;iS_l, (210) where the factor N plmys the ro]eofan eddy v_cos_ but th__ough an anJsotropi: relation between the subgrkl scale_and the str_th t_nsor.T he t_nsor 7U _hou]d be ameasure ofthe anJsotropy. Itcrx/]d be constructed f_]m the vebc/ty uclmatJons: (2i1) Thism odel reduces to the _leddy viscosLty model _)r isotropiz turbul_nce (%u = 6u ).The s_n of the d_patbn depends on]},on the s_n of}L shce the product ofr_) and dle stra_h tensor isggen by 7ijSij = t_SijTik"/jlSkl = it (5'0"7ik)2. (212) Moreover, if there is no tuzbuhnce in one dkectbn (6u_ = C), the mode/has the properW that the components Ti_, = raj = 0. Th_s is an exp_ property that Js m issed by i_tmp_c eddy viscosity models. 3. C oup]Jng the dynam Jc procedure and the ensemb]e ofLES's 3i. C lass _mldynam _ proo_ures T he dynam]c proosdur_ isbased on an JdentJZy (Germ ano 1992) that relat_stheunknown stress generated by di erent kars: L,j + r_3 - T_j = 0, (3 1) wheretl}j : _i uj--ui_) isthe subgrkl s_9_]est:llm_generat3sdby the sucx:_applbation tothe ve_bcJty e]d oft_vo _ that axe respectg_]_ denot_] by - and by ^.The Leonaxd tensor Jsgi_n by Lij = ui uj - ui _j •Itdepends only on '_,so tglatitdoesnot requk_ any m odellhg. This i:lentJty (3l) isof course only valid fmr the _xact and unknown subgr]d scal9 stresses.W hen mode]s are used, _,3 _ Om[2 _] and T,] _ Om T._ ],the di er_nce E,:): Li3 + Cm_--_j- ornT between the r_ht hand s_e and the ]eft=hand s_e 8 D.Cazat/M_.M.Rogers,aAnd.A.Wray of(3 l)can be cons_ered as a m essure of the perfmnn anoe ofthe m odet The dynam prooedure uses thism easure h order topl_x_lDe the m odelparam et_r(_by m him ]zing E_j .W hen a hcrnogeneous d/r_ctbn ex/sts in the prob]sn ,the estimatJon fDrC isgJ_an by (Gena ano et al 1991;Lii_ 1992;Ghosmletal 1995): (? _ (L,jA/Iij}h (32) <M,jM,j>,, ' where ]_[0 : ;n_Tj -- _TtT and the average <'-">h is supposed to be taken over the ho- rnogeneous dkectbn (s). Obvinusly, this approach is _ to spec/algecm _ with hcmogeneous dkectfion (s). Ccm plex gecm etr_s recfuxe an a/bexnati_ trea_ ent in whJch a bcalde nitJc_ of the parameter C can be proposed. This is the case in the bca_l dy- nam Jcprocedure devebped by Ghosalet aL (1995) aswellas in the LagxangJan dynam _c proceduIe proposed by M eneve_u et al (1996). Inboth cases, the m odelparam et3er is dk_ct_ re_]arEalto the ]alxle-scale eli Ki dlrough the t_nsor L,; and _][u-Itw illthus vary fzmm one realisation to another, even ifdne underlying LE S's are supposed to be equiralmt. As _dy m entbned inSectbn 22, the dynam r procedure thus produoes m odelpamam etE_sthat are not unJuersa]_ In the early stages of itsdeve]opment, the fact that d_em odelparam _ are dk_-T-]y re3ated to the specic r_lg_tJon ofthe c_ was consklered advantageous because thisa]im_ed the model tobe m ore adap tatJue.Hcw ever, this property proved to be prob]sa atJc because itgenerahes highly variable m odelpa- ram ete_sthat cause num er_al ilstab_. Smm e ofthese practizalprob]6rn shave bea_ resolved inthe afmren entJoned ]ocaland Lagrangiml versbns ofthe dynam]c prooedure. 32. Ensem bA_av_aged dynam _cpxcx_dure The enssm ble-averaged dynam Jcprocedure (EA D P )wepropose here isconoep mal_ very c]ose to the vollm e_averaged orplane-averaged vermbns (32)of the dynam i:procedure. The only di er_nce cornes from the nahare ofthe average, wh_h isnow an ensemb]e aver- age over the set ofLES's.C ons_erg_g thatR LES's (22) ai_ cornputed sireu]taneous_, the m odelpaxam et_r isnow g_en by: C _ (L'j3Ii3) (33) (M,jMu) ' where (...) now z_presents the ensemble average.The expressbn (33) is only va/kt ifthe parameter (7 is sbw ly dependent on space and c&n be t_en out of the t_st ]t_r _. Such an assam ptizn is not very restri-tk_, however, shce the ensemb]e averaging is l_ely to smooth out the rapid v-oziatSzaas in the e]d. In the ne_t S_n, _vw ill be seen that the m ode/coe c_nt does hdeed beccm e an oother and s_ oother as the ensemble size Statist_mlensem b]_of ]_mIeeddy sireul_tbns 9 isiqcreesed. The fDnn u]at_n (33 )guarantees _hat the model param et_rsare un_ers_l shce they depend only on d%e statJst_propert_ of the 19xg_ ve3ocky e]ds. Ina sense, the ensemb]e ofLE S'sco_onds toan arti c/aldk_mJ:bn ofhQm ogenedty, whxh alvays exists hdependent of the ecrup]_x_y of the cw .Inthe unexp_ cases inwhlh the m odel ooe c_t reinails _gni cant_ varigb]e in space _r ]azge ensem b]e sizes,the EA D P coul_ be coupled w 9/_other approaches l_e the bcaldynam _ procedure deve_ed by G hos_let al (1995), the Lagm_ngJan dynam _cprocedure proposed by M en- eveau etal (1996), or the tine ]agg_]g procedure proposed by P_m el]land Lh (1995). The ceuplhg of the EADP wg_h any of these methods woul_ l_ad to a neglkl_o]e cost shoe itwould be used only once fDr the whole enssm bl=. In smm e cases, the fact that the m odel pazam eter cannot be adapted to the specic ree]/s_tJon of the l_rge scale ow m _ht be cmns_ered as a drawback of the EAD P. In pa_llr, a greater adaptabil_y m _ht be des_]e fDr very _tenn itt_nt cws wkh, _Dr in_ce, bca/ized turbulent spots appearhg insole a ]am inar sea _ enningsonet a]_ 1987). Indeed,wh_% the tuJou]ent spots app6mr random lyina stati_]ly hcmogeneous doraa_h, the m odelparam etar pr_d_J3sd by the EADP isqu_nstant and isweakly a _ by the turbul_zt spots. In fact,the EADP imp]]c_ assam es that the model tEnsgr 7rhj_[] should, abne, take care of the _rbul_=nce act/u_y. W e _mg/ize however that,because the pe_ct m odelis not ava/lmb]e,the assureptJon ofa un_ers_iparam et_r m _ht be sireet/n es inappropriate, dependhg on both the nature of the ow and the model adopted _r _n{j_[]. However, itmust be notsd that the same di culty woul_ be encountered fDr mode]s that use pazam et_rs that are chosen a pr93ri as well as fDr dynam]c m ode/s that are based on voJam e or plane averaging. M or_over, in the test cas_s presented inthe folk_ hg sectigns, the pr_dJct_ons of the EA D P are encouraging. M ore localized approaches (Ghosalet a_ 1995; M eneveau etal 1996; PJmm elli& Li/ 1995) _r whkh the assureptJon of a unJm_salm odelparam et_r _ not adopted woul_ probably respond m ore strongly to int_m 9d_nt ows. Itmust be noted, however, that dnese generalized dynam _cprocedures are usual_ cramb_ed w _h m ode3s that are based on sta_approaches. In these cases, use of the I_ dynam _cm ode/in a sense contrad_Ts the underlying sta_assum ptJonsused tobui]d the eddy v_cos_y m ode]_ FinaI_, w ereinazk that the m odelparam et_r]sthe on_ coupling between the di er_nt LES's. The di erence between DNS, LES, and an ensemb]e ofLES's coup]ed through theEADP isiUustzated _ gure l.Asa oonssquence, theEADP ]sp_ suited fmr d_r_utsd proeesshg on parallel cam put_rs. T he most natazal inplan entat%]n of thisprocedure am ounts to nnaning each m _n ber of the enssm bl_ ofLE S on a s_0arat_ 10 D. Cazati M .M .Rogers,andA .A .Wray a:DNS b:LES i : i *C=_nlllutc_t k...!_.y.y-7..t_:!....' . _','_n_put?" t c:Ensemble LES FIGURE I. The dif[erences between DNS, LES, and ensenlb[e of LES's using tile EADP are illustrated. In DNS (top), only the right ]rand side of tile Navier-Stokes equatious is ,we(led for _tdvancing the velocity field in time. In tr,'u:litional LES (middle), an additional modelling term is needed In the EADP (bottom), one substep, _'o,nmon for all the LES"'s.', is added for computing tim model parameters used in em'h of the simulations. This is the only point where information is required from the other fields. node. Cmm m unJcatJon between the di er_t processes is]in_ to the cornputatJon of the model coe cJent.O therw iseeach e]d _i[isadvanced h time _dependently of the others. T hJs property _hou]d guarantBe very good scalabil_y iflarge ensureb]e sizesaxe expbred. 4. Tests in isotropJc turbulence 4i. D ecmyiqg t3/r]:u]e_me The EA D P descr_3ed inthe previous section was tes_d h decay_g JsotropJc tu_bu]enoe _r 323 LE S's.T he t_nsor n_[._ was chosen tooorre_0ond to the Sm agorinsky m ode]= r,rj _ -2Cb 77,r 1-/,;5r,j (41) A serg_ of num erizal experka ents has det_mn bed (Caxati et al. 1996) hcxq b_rge the ensemb]e of sin ukane_)us LE S's must be (i.e. how :b__e/{ shoutfi be). T he _ used to detrain he them him alsJze of the ensemb]e were fDcx_sed on i. The spatialvarJabil_y ofC. 2. The percentage ofnegatJue (7.