Large-Eddy Simulation of a Circular Cylinder on Unstructured Grids

Transcription

1 AIAA -98 Large-Edd mlan f a Crclar Clnder n Unsrcred Grds Rber P. Hansen Uned aes Mlar Academ Wes Pn, NY Lle N. Lng The Pennslana ae Uners Uners Park, PA 4 h AIAA Aersace cences Meeng and Ehb Janar / Ren, NV

2 LARGE-EDDY IMULATION O A CIRCULAR CYLINDER ON UNTRUCTURED GRID R. P. Hansen * L. N. Lng Dearmen f Cl and Mechancal Engneerng Dearmen f Aersace Engneerng Uned aes Mlar Academ The Pennslana ae Uners Wes Pn, Ne Yrk 1996 Uners Park, PA 168 Rber.Hansen@sma.ed lnl@s.ed ABTRACT A cmanal sd f nsead, searaed, l-renlds nmber fld fl s made sng a secnd-rder accrae, cell-cenered fne-lme mehd n nsrcred grds. The hree-dmensnal smlans nclde largeedd smlan rblence mdelng sng a fed-ceffcen magrnsk sbgrd-scale mdel. The fl ab a hree-dmensnal crclar clnder a Renlds nmber 39 and Mach nmber. as sded sng a sandard Naer-kes frmlan h and h he sbgrd-scale mdelng. A hrd case as als esed sng a mdfed frm f he Naer-kes eqans h sbgrd-scale mdelng. Cmarsns are made h eermen and seleced hree-dmensnal nmercal calclans erfrmed n srcred grds. The freqenc f he nsead re sheddng as n gd agreemen h eermen, heer, he nmercal dssan nheren n he nd fl calclans affeced he deals f he me-aeraged flfeld. The resls shed an ncreased base ressre, shrened recrclan znes n he clnder ake, and ecesse drag cmared eermen fr all cases. Inrdcn Whle rblems nlng fl er cmle gemeres hae alread been aemed sng nsrcred grd mehds 1,,3, he abl f nsrcred mehds accrael smlae fl nlng largescale nseadness and rblence has n been clearl demnsraed. The allre f accrael slng cmle rblems sng he er effcen cell r elemen dsrbn ffered b nsrcred grds maes cnned research n hs area. The effeceness f nsrcred grd slns fr fl er sandard gemeres sch as cbes and clnders has n been ssemacall dcmened fr bh he large-scale mean reres and rblen reres. The nsead fl er a sqare clnder 4 and cbe 5 sng nsrcred grds has been sded as ell as -dmensnal slns fr crclar clnders 6 sng nsrcred mehds. Nmers sdes hae rded crcal nsghs n he adanages and dsadanages f ars algrhms aled hreedmensnal clnders sng srcred grds 7,8,9,1. The sd f he hree-dmensnal crclar clnder sng nsrcred grd mehds, heer, s eher scarce r nn-esen. Ths aer resens resls fr a smlan f fl er a hree-dmensnal clnder sng an nsrcred, cell-cenered, fne-lme mehd. The hscs f fl ab a crclar clnder s far mre cmle han s relael smle shae mgh sgges. Dese beng ne f he ms eermenall esed gemeres n fld mechancs, hree-dmensnal nmercal sdes f he crclar clnder hae nl been aemed n he las decade. Parallel cman, as ell as mre erfl ecr cmans, has made greaer grd resln ssble, hch n rn has made he se f LE feasble fr rblems reqrng a large nmber f grd ns. Drec Nmercal mlan f fl er a crclar clnder a a Renlds nmber f 3 has een been accmlshed 11. Mch hgher Renlds nmbers hae been acheed sng Deached - Edd mlan. Tran e al. 1 hae cmed resls fr he hree-dmensnal clnder Renlds nmber eeral eermens rde a er cmlee daa se fr fl er a crclar clnder a Renlds nmber 39. Nrberg 13 sded he effec f freesream rblence n he measred fl, and rdes srface ressre daa as ell as freqenc f sheddng fr a range f Renlds nmbers. Krhaall e al. 14 and Lrenc and hh 15 measred me-aeraged elces and Renlds sresses n he clnder near-ake a Renlds nmber 39. Ong and Wallace 16 rde eermenal daa a he same Renlds nmber a sns frher dnsream f he clnder, bend he recrclan zne. These mre recen, ler Renlds nmber eermens hae rded cmanal * Asssan Prfessr, LTC, Uned aes Arm Prfessr, Asscae ell, AIAA. Crgh b Hansen and Lng. Pblshed b he, Inc. h ermssn

3 researchers h a es case ha s easl hn he caabl f crren cmng er. A he Renlds nmber f 39 he aached bndar laer n he clnder srface s lamnar, he searaed shear laers are n he earl sages f ransn, and he ake s fll rblen. 17 A hs Renlds nmber, he bndar laer s hck engh be easl resled b he cmanal grd. The eermenal ff-bd daa (ressre, elces, Renlds sresses) are als clse engh he clnder s ha an ecesse nmber f cmanal cells n he clnder ake can be aded. Dese he fac ha sme ar f he flfeld arnd a clnder a Renlds nmber 39 s n ransn, a cndn ha s bsl mre dffcl cme, he erall adanages f sng hs Renlds nmber are jdged egh an dffcles cased b he eakl ransnal shear laers. The cmanal resls f Breer 7, Beadan 8, and Mal 18 fr Renlds nmber 39 hae added grea nsgh n he adanages and dsadanages f ars nmercal schemes aalable n srcred grds. Breer 7 sed he magrnsk LE mdel h fe dfferen fl calclan schemes n a srcred grd. He shed ha he e f nmercal scheme had a large mac n feares f he me-aeraged flfeld and arameers sch as drag. Cenral dfference arman schemes rdced mre accrae resls, hle he schemes emlng ndng ended shren he recrclan bbble and redc ecesse drag cmared eermenal resls. Beadan 7 sed a fne dfference arach h ffh- and seenh-rder nd based schemes dscreze he cnece erms. Hs smlans fr hree cases -- n rblence mdel, magrnsk LE mdel, and dnamc LE mdel -- all er-redc he lengh f he recrclan bbble, h he dnamc mdel fallng hn nl ercen f he eermenal ale. Oher characerscs f he recrclan bbble, sch as he lcan and ale f he mamm sreamse elc alng he ake cenerlne, ere less accrae. Beadan als ealaed he ne-dmensnal rblence secra a seeral dnsream lcans fr bh he ffh- and seenhrder accrae smlans. Whle he man effr f hs research s ealae a er dffcl es case sng nsrcred grds, mremens n he sln accrac ere aemed b sng a nel slng mehd. eeral researchers hae sed he echnqe f eressng he Naer-kes eqans n erms f sead and nsead cmnens. Ha e al. 19 sed a decmsn f nsananes arables smlar ha dne n he resen research fr nesgang he eln f a re n a hreedmensnal bndar laer. Ha e al. decmsed each arable n a me-ndeenden base fl and a me-deenden dsrbance. B sbsng hs decmsn n he Naer-kes eqans and sbracng he base fl, he frmed ha he called he dsrbance eqans The slng arach sed here, called he nnlnear dsrbance eqans (NLDE), as frs rsed b Mrrs e al.,1, cme slns hgh-seed crclar and recanglar je nse rblems. The NLDE hae snce been sccessfll aled a are f scs and nscd rblems. 3,4,5,6,7 Hansen e al. 8 fnd he freqenc f nsead sheddng er a -dmensnal clnder be mre accrae sng he NLDE frmlan cmared a sandard Naer-kes sln. 3 Gernng Eqans and Nmercal Imlemenan Gernng Eqans The saall flered, me-deenden, Naer-kes eqans are sed. In negral frm he are: V Q dv nˆ d The cnsered ecr arable s gen b: (1) Q () E here he erbar ndcaes a saall flered qan and he lde () denes are (dens eghng) aeragng. The fld dens s gen b, he Caresan elc cmnens are,, and, and E s he energ er n mass. The fl ecr cnsss f: here r ˆ ˆj kˆ z z ( ) Θ E z q z (3)

4 4 ( ) Θ z z z q E ( ) Θ z z zz z z zz zz z z z z z q E z q Θ The sbgd-scale (G) sresses and hea fles aear as j and q, resecel, n he abe. Usng he magrnsk mdel, he G sresses are defned b: ( ) j j s j j C µ µ (4) here C s s he magrnsk cnsan and s assgned a ale f.1 n hs sd. The resled defrman rae ensr s defned b: j j j 1 (5) The mdel fr he G hea fles s: T R q 1 Pr µ γ γ (6) here Pr s he rblen Prandl nmber and s cmmnl chsen n he range f.3.5. The majr shrcmng f he fed-ceffcen magrnsk mdel s s behar clse he all. In he bndar laer, he large scale eddes arach he sze f he small scales. The cal grd s carse n he bndar laer, casng he magrnsk mdel creae ecesse dssan. One mehd f crrecng hs s dam he rblen scs cnrbn near he all. Use f a an Dres damng fncn h LE rdes he reqred redcn n edd scs near he all. The magrnsk ceffcen n eqan 4 s relaced b C s * defned as: ( ) 3 / * 1 A s s e C C (7) here A s a cnsan eqal 5., and s he dsance, n scs all ns, frm he all he feld n here edd scs s be ealaed. The scs all n s defned b: d ν (8) here d s he dsance frm a cell cener he neares face ha s n a scs srface, s he shear sress a he all, s he fld dens a he all, and ν s he ceffcen f dnamc scs. Calclan f n a arned, nsrcred grd reqres arclar care and s dscssed n he secn n nmercal mlemenan. Use f he nnlnear dsrbance eqans (NLDE) reqres he resled-scale arables be decmsed n a mean qan ha s ndeenden f me and a me-deenden errban. r eamle: ) (, () ) (, ) (, () ) (, (9) The decmsn f each arable s sbsed n he flered eqans, and he erms cnanng rel mean fl qanes are laced n he rgh-hand sde, s ha eqan 1 ma n be ren as: V R nd nd Q dv ˆ ˆ (1) here nd dv Q R V ˆ e Q Θ ) ( ) ( ) ( z e e e

5 Θ Θ ( ( z ) ) z Θ ( ) ( z ( ) ) q z ( ) q The remanng cmnens f and are smlarl eanded. The arable Q n eqan 1 s a ne ecr f cnsered arables cnssng f nl neracns beeen he errbans hemseles r he errbans and he mean fl. cnans all f he G sresses and hea fles, and Q and are he same as defned b eqans and 3 ece all erms are n resled mean qanes. Deendng n h he mean fl s baned, he ale f R ma r ma n be zer 8. When R s a cnsan s calclaed nl nce a he begnnng f a cman. A sead, nfrm fl s chsen as he mean fl n hs sd, redcng R zer. Nmercal Imlemenan The gernng eqans are dscrezed sng a secnd-rder accrae, cell-cenered fne-lme mehd h ndng fr sabl. The cde sed n hs research s LOU (LO Unrcred), rgnall ren b Lng 9 n Cnnecn Machne ORTRAN (CM). Wenberg and Lng 3 laer mdfed hs cde se adae meshng. The resen research nclded dang f he LOU cde ORTRAN9 and he addn f Message Passng Inerface (MPI) lbrar sbrne calls all arallel eran. Large-edd smlan, n he frm f a cnsan ceffcen magrnsk mdel, as als added fr rblence mdelng. The ndng echnqe emls Re s fl dfference slng 31. A sandard 4 h rder Rnge-Ka negran s sed elcl adance he sln n me. Unsrcred grds cmsed f erahedral cells ere generaed sng he Grdl/VGRIDns 3.3 sfare 3,33. The grd as decmsed n sbdmans r znes sng he METI 4. arnng sfare 34. eeral les ere als ren fr LOU handle a are f re- and s-rcessng asks. Parallel eran f he cde sng MPI reqres he echange f all cnsered arables beeen he cells n eher sde f a face ha defnes a znal bndar. Ths echange ccrs seeral mes er me se. Addnal message assng reqremens are ncrred hen calclang he edd scs damng gen b eqan 7. Eqan 8 mles an asscan beeen each cell cener and he neares sld, scs face. Each cell denfes he lcan f and dsance frm s neares all face nce as he cde nalzes. In a arned, nsrcred grd he cell and s neares sld srface ma n be ar f he same zne. Ths reqres nfrman, secfcall he shear sress and he dens, a he all be sen frm he face cener he cell here edd scs s be ealaed. Ths addnal message assng akes lace each me se. 5 mlans mlans and Resls A crclar clnder n a crssfl a Renlds nmber 39 has been smlaed a Mach.. The clnder s fr dameers n san. A smmer bndar cndn s laced a he end alls, creang he effec f an nscd all. gre 1 shs he srface grd f he clnder. Armael 8 cells defne he clnder srface n he sanse() drecn, r seen cells er dameer. The farfeld bndares are laced 1 dameers frm he clnder srface ece fr he dnsream e lane, hch s dameers frm he clnder. The cmlee grd cnsss f 38,8 erahedral cells. All me-aerages are cmed sng en ccles f sheddng. anse aeragng s als erfrmed sng 13 ns acrss he san. All cmans ere dne n he NAA Ames Research Cener 51-rcessr Orgn ssem (LOMAX clser) sng 64 rcessrs. Each ccle f sheddng reqred armael.8 hrs f all me, r ab.5 secnds er eran er cell. gre 1. rface Mesh f he Crclar Clnder Three cases are cmared he baselne LOU cde h LE rblence mdelng, he baselne Y Z X

6 Table 1. Ke Parameers fr he Three-Dmensnal Clnder Cman C D C b L r /D Baselne LOU LOU / LE NLDE / LE Breer 7, Case B OK.63 Breer 7, Case D OK 1.37 Eermen /U. -. LOU - / LE LOU - /magrnsk Mdel LOU - NLDE /LE Beadan, 5h Order Breer, Case B Eermen, Krhaall e al. Eermen, Ong and Wallace -.4 LOU cde h LE rblence mdelng, and he NLDE frmlan h LE mdelng. A smmar f ke me-aeraged arameers s gen n Table 1. The cmed ceffcen f drag, C D, s sbsanall hgher n all cases han he eermenal ale. The base ressre ceffcen, C b, s defned as he ceffcen f ressre a he n n he clnder srface frhes dnsream. The magnde f C b s als larger han he eermenal ale fr all cases. The rhal nmber s slghl abe he eermenal ale fr he baselne cde, b falls hn he eermenal range fr he cases h LE. The case B frm Breer 7 s sed fr cmarsn becase ha case sed an nd arman fr fl calclans. The rhal nmber fr hs case as n rded, b Breer saed as hn he range f eermen. The lengh f he recrclan bbble, L r, s nn-dmensnalzed b he clnder dameer as L r /D n he las clmn f Table 1. There s a drec crrelan beeen he drag ceffcen and he lengh f he recrclan bbble he shrer he recrclan zne he hgher he drag ceffcen. gre shs he me-aeraged sreamse elc alng he ake cenerlne fr all cases. Als /D gre. Tme-Aeraged Velc Alng he Wake Cenerlne shn are eermenal resls and seleced resls frm Beadan 8 and Breer 7. The mean fl s n he se -drecn n all fgres and he dsance alng he ake s nn-dmensnalzed b he clnder dameer. In all he resen cases, he recrclang zne s sgnfcanl shrer han shld be. The cases sng LE are er smlar he Breer case B ha ses a magrnsk sbgrd-scale mdel and an nd scheme fr he fl calclans. The addn f LE he baselne cde has he desrable effec f crrecng he mnmm elc n he recrclang zne a ale cnssen h eermen. The lengh f he recrclang zne s, heer, slghl shrened hen LE s added. Use f he NLDE frmlan shrens he recrclang zne een frher and redces he mnmm elc n he bbble a ale bel he eermenal ale. gre 3 shs he me-aeraged, san-aeraged sreamlnes fr he NLDE mdel. Ths case has he shres me-aeraged recrclan zne, b s herse qalael he same as he nn-nlde cases. gre 4 shs he me-aeraged sreamlnes fr a hree-dmensnal nmercal cman frm Breer 7 6

7 (case D3) hch sed a cenral dfference arman scheme h a dnamc sbgrd-scale LE rblence mdel. All arameers f hs smlan are er clse he eermenal ranges, and he me-aeraged sreamlnes are als acceed be an accrae reresenan f he eermenal daa. Cmarsn f gre 3 and gre 4 sh he me-aeraged recrclan zne f he resen sd be less han half he lengh f he Breer case D3, cnssen h he daa n Table 1. Addnall, he smaller, cnerrang rces seen n gre 3 are absen frm gre 4. b sng cenral dfference armans. The nheren nmercal dssan n Re s ndng scheme, sed n he resen research, aears be ecesse, casng he ake be dmnaed b he larges scales. Ths resls n a ake ressre ha s l and a drag ha s hgh. 1 z/d The drag and lf ceffcen me-hsres fr armael s and ne-half sheddng ccles are shn n gre 7 and gre 8. Whle he me /D gre 3. Tme-Aeraged reamlnes fr he NLDE Case gre 5. Is-rface f Absle Vrc gre 6 shs he nsananes rc srcre n he cener-lane erendclar he clnder as baned frm he baselne LOU cde. The srcre f he rc n he ake s er smlar ha baned frm a rel -dmensnal smlan. The ake s dmnaed b large cheren srcres. Mch smaller scale mn shld be sermsed n he erdc re sheddng. The erszed large-scale mn als aears resnsble fr ndcng he cner-rang smaller re adjacen he clnder srface. gre 4. Tme-Aeraged reamlnes frm Breer (reference 7) Case D3 gre 5 reeals he srng hree-dmensnal fl ha resls frm erfrmng he clnder cman n hree dmensns. hn s an s-srface f absle rc. ch hree-dmensnal as absen hen smlar cmans ere erfrmed drng he resen research sng grds nl dameers n san. The reasn fr he r erfrmance f all cases n he areas f drag, base ressre ceffcen, and lengh f he recrclang zne aears be relaed he se f ndng. Breer 7 as able ban sgnfcanl beer resls han he B case sed here fr cmarsn 7 gre 6. Insananes Vrc Magnde (darker shadng ndcaes hgher rc magnde) C D LOU LOU /LE NLDE /LE U /D gre 7. Drag Ceffcen Tme Hsr

8 aeraged ale f drag s hgh, he me arans are cnssen h eermen and her nmercal sdes. The eak--eak flcans are als smaller fr he hree-dmensnal smlans cmared he -dmensnal case, alhgh he hree-dmensnal NLDE case alls larger drag flcans han he her. cmnen hen sng a dnamc LE mdel h cenral dfference armans. a) LOU LOU /LE NLDE /LE C L.5 b) U /D gre 8. Lf Ceffcen Tme Hsr A cmarsn f he Renlds sresses h eermenal daa as als made. All Renlds sress cmnens are nrmalzed b U. gre 9 cmares he sreamse nrmal Renlds sress ( ) f he dfferen cases. The near ake aears er smlar fr all cases. Onl frher dnsream are sme dfferences bserable. gre 1 shs he sreamse Renlds nrmal sress cmnen ( ) a he lcan /D 1.54 behnd he clnder. The sreamse cmnens cmed b he baselne cde and he baselne cde h LE nder-redc he eak Renlds sresses b as mch as 1%. The NLDE ersn redcs he eaks hn ne ercen, b he lcans f he eaks are slghl dslaced frm he eermenal lcans. I shld be ned ha here s a degree f nceran f he eac eak f he eermenal daa based n he fac ha a nmber f cres cld f he daa rdng dfferen eaks. Percenage dfferences sed n hs dscssn are relae he larges ales led. gre 11 shs he Renlds shear sress cmnen fr he NLDE case n he z lane. gre 1 shs he shear sress ale a /D 1.54 fr all cases. All he cases er-redc he shear Renlds sress b as mch as 36%. The baselne LOU cde sng LE aears be slghl mre accrae han he her cases, b s sll 7% er he eermenal ale. The laeral nrmal Renlds sress cmnen s shn n gres 13 and 14. All calclans sgnfcanl er-redc he cmnen b a leas 1%. The calclans f Mal 18 als sgnfcanl er-redc he magnde f hs cmnen. Breer 7 sccessfll cares hs The sanse ( ) Renlds nrmal sress s shn n gre 15 n he lane erendclar he clnder as. gre 16 shs he sanse sress a 8 c) gre 9. reamse Renlds Nrmal ress n he z Plane: (a) LOU, (b) LOU /LE, (c) NLDE /LE; ( 15 cnrs frm.5.4 ) < > andard LE NLDE /LE Eermenal z/d gre 1. reamse Renlds Nrmal ress a /D 1.54 (eermenal daa frm reference 15)

9 /D 1.54 fr he NLDE case agans her nmercal smlans. The resls f Beadan 8 are sed as he benchmark becase eermenal daa s n aalable fr hs cmnen. The eak sress ale frm Beadan s 35% hgher han he NLDE case r Mals 18 calclan. gre 16 ndcaes a redced leel f hree-dmensnal n he near-ake fr he resen sd. The redced sanse sresses cnrbe he hgher ales f he n-lane and cmnens. Ths recse cndn has been shn b Mal and Balachandar 35 lead a hgher ale f base ressre ceffcen and hgher drag. Bh Beadan 8 and Mal 18 sed grds h armael 15 cells er dameer alng he san f he clnder, r rghl ce he sanse dens sed n hs sd. Dese he dfferences n grd dens beeen he resen sd and Mal, h sed a secnd-rder fne dfference mehd, he eak ale fr sanse Renlds sress s he same. Ths ndcaes ha he seenh-rder scheme sed b Beadan accns fr he beer accrac f hs resls, and n he grd dens. gre 13. Laeral Renlds Nrmal ress n he z Plane NLDE (15 cnrs frm.9.81) < > andard LE NLDE /LE Eermenal z/d gre 14. Laeral Renlds ress a /D 1.54 (eermenal daa frm reference 15) gre 11. Renlds hear ress n he z Plane NLDE (15 cnrs frm ) gre 15. anse Renlds Nrmal ress n he z Plane NLDE (15 cnrs frm.1.1)..1 < > andard LE NLDE /LE Eermenal < > NLDE /LE Mal Beadan z/d gre 1. Renlds hear ress a /D 1.54 (eermenal daa frm reference 15) z/d gre 16. anse Renlds Nrmal ress a /D 1.54 (Mal Rn II frm reference 18, Beadan fh-rder frm reference 8) 9

10 CONCLUION The accrac f he hree-dmensnal clnder drag, base ressre, sheddng freqenc, and lengh f he recrclan zne ere all degraded b alcan f he NLDE mehd. The nl benef frm sng he NLDE frm as a mdes mremen n he eak nrmal Renlds sress n he sreamse drecn. The alcan f Large Edd mlan sng a fed-ceffcen magrnsk mdel had a nceable effec n ke arameers f he hree-dmensnal clnder ess. Relae he baselne LOU cde, he rhal nmber as mre accrae and he mnmm elc n he recrclang zne as nceabl mred hen LE as sed. The erall r qal fr all hree-dmensnal clnder cases s arbed rmarl he ndng sed n he fne-lme frmlan. The se f ndng nrdces ndesrable leels f arfcal dssan. Grd refnemens sdes shld be cndced, heer, searae he effecs f grd resln and dssan cased b ndng n he accrac f hese resls. The arfcal dssan nheren n ndng s necessar, an een, fr nmercal sabl. Oher researchers 7,18 hae shn ha cenral dfference arman schemes are serr n srcred grds. The cenral dfference schemes, heer, reqre he elc addn f arfcal dssan fr nmercal sabl. Elc arfcal dssan s n a crren caabl n hree-dmensnal nsrcred grds, lmng nsrcred mehds he se f ndng. Whle he srcred, hgher-rder, nded mehd sed b Beadan 8 erfrmed beer han he mehd sed n he resen research, deendable hgher rder mehds fr nsrcred fne-lme grds hae n been deeled. The lmae effec f ecesse dssan eerenced n he hree-dmensnal clnder cases s he dmnan f large-scale rces n he clnder near-ake, leadng a ressre ha s l and a drag redcn ha s ecesse. References 1 Mneck, R., Alcan f an Unsrcred Grd Naer-kes ler a Generc Helcer Bd - Cmarsn f Unsrcred Grd Resls h rcred Grd Resls and Eermenal Resls, NAA TM rnk, N. and Przadeh,. Terahedral ne-vlme lns he Naer-kes Eqans n Cmle Cnfgrans. NAA / TM , December Grsmer, M., rang, W., Tmar, R., and Wzeman,. Cbal: A Parallel, Imlc, Unsrcred Eler/Naer- kes ler. Adances n Engneerng fare, Vl. 9, N. 3-6, Camarr,. and ale, M.V. Tards he Large- Edd mlan f Cmle Engneerng ls h Unsrcred Grds, INRIA Rer 3844, December hah, K. and erzger, J. A ld Mechancans Ve f Wnd Engneerng: Large Edd mlan f l Pas a Cbc Obsacle. Jrnal f Wnd Engneerng and Indsral Aerdnamcs, Vl. 67 and 68,. 11-4, elam, R.P. ne Elemen Mdelng f l Arnd a Crclar Clnder sng LE, Jrnal f Wnd Engneerng and Indsral Aerdnamcs, Vl. 67 and 68, Breer, M. Large Edd mlan f he b-crcal l Pas a Crclar Clnder: Nmercal and Mdelng Asecs. Inernanal Jrnal fr Nmercal Mehds n lds, Vl. 8, N. 9, , December Beadan, P., and Mn, P., Nmercal Eermens n he l Pas a Clnder a b-crcal Renlds Nmber, rer N. T-6, Dearmen f Mechancal Engneerng, anfrd Uners, Mal,. Cman f Three-Dmensnal ls Pas a Crclar Clnder f L Asec Ra, Phscs f lds 13, 1. 1 Krachenck, A., and Mn, P., Nmercal des f l Oer a Crclar Clnder a Re D 39, Phscs f lds 1,. 11 Mal, R. and Balachandar,. Generan and Eln f Vrcal rcres n Blff Bd Wakes. 34 h Aersace and cences Meeng and Ehb. AIAA Paer 96-1, Tran, A., hr, M., reles, M., and alar, P., Deached-Edd mlans Pas a Crclar Clnder, l Trblence and Cmbsn 63: , Nrberg, C., Effecs f Renlds Nmber and L- Inens ree-ream Trblence n he l Arnd a Crclar Clnder. Pblcan 87/, Dearmen f Aled Thermscences and ld Mechancs, Chalmers Uners f Technlg, Ghenbrg, eden,

11 14 Krhaall, A., hh, C., and Lrenc, L. The Near Wake f a Crclar Clnder a.3<m <.6: a PIV d. 3 nd Aersace cences Meeng and Ehb, AIAA Paer , Lrenc, L., and hh, C., Characerscs f he Plane Trblen Near Wake f a Crclar Clnder, A Parcle Image Velcmer d. Prae Cmmncan, 1993 (daa aken frm reference 18) 16 Ong, L. and Wallace, J. The Velc eld f he Trblen Ver Near Wake f a Crclar Clnder. Eermens n lds, Vl., , Zdrakch, M., l Arnd Crclar Clnders, Ofrd Uners Press, Ofrd, Mal, R. Large-edd mlan f l Pas a Crclar Clnder. Annal Research Brefs, Cener fr Trblence Research, anfrd Uners, Ha, X., Jsln, R.D. and Pmell, U. Large Edd mlan f Bndar Laer Transn n e Wngs, ICAE/LaRC Wrksh n Transn, Trblence, and Cmbsn, Hamn, VA, Jne Jl 7, Mrrs, P.J., Lng, L.N., Bangalre, A., Wang, Q., A Parallel Three-Dmensnal Cmanal Aeracscs Mehd Usng Nnlnear Dsrbance Eqans, J. f Cmanal Phscs, Vl. 133, Mrrs, P.J., Lng, L.N., Lckard, D.P., Wang, Q., Nmercal Predcn f Hgh eed Je Nse, AIAA Paer , Mrrs, P.J., Lng, L.N., chedegger, T.E., Wang, Q., Pln, A.R., Hgh eed Je Nse mlans, AIAA Paer 98-9, L, J., Lng, L.N., Md, A., Hgher Order Accrae lns f h Arake l elds sng Parallel Cmers, AGARD Cnference, Amserdam, Lng, L.N., A Nn-Cnserae Nnlnear lfeld lng Mehd, AIAA Paer -1998,. 7 ezer-uzl, N. and Lng, L.N., Hgh Accrac Wake and Vre mlan Usng a Hbrd CD/DV Mehd, AIAA Paer -9,. 8 Hansen, R.P., Lng, L.N., and Mrrs, P.J., Unsead, Lamnar l mlans Usng he Nnlnear Dsrbance Eqans, 6 h AIAA/CEA Aeracscs Cnference, AIAA Paer -1981, Ma, Jne,. 9 Lng, L.N., Parallel Algrhms fr Gas Dnamcs. In Parallel Cmanal ld Dnamcs 9, Elseer cence Pblshers, Amserdam, , Wenberg, Z. and Lng, L.N., A Massel Parallel ln f he Three-Dmensnal Naer-kes Eqans n Unsrcred Grds, AIAA Paer 94-76, Re, P. Armae Remann lers, Parameer Vecrs, and Dfference chemes. Jrnal f Cmanal Phscs, Vl. 43, , amareh, J. Grdl: A rface Mdelng and Grd Generan Tl. Prceedng f he Wrksh n rface Mdelng, Grd Generan, and Relaed Isses n CD lns, NAA CP-391, Ma Przadeh,. Vscs Unsrcred Three- Dmensnal Grds b he Adancng-Laers Mehd, AIAA Paer , Kars, G. and Kmar, V. METI A fare Package fr Parnng Unsrcred Grahs, Parnng Meshes, and Cmng ll-redcng Orderng f arse Marces, Versn 4.. Uners f Mnnesa, Dearmen f Cmer cence/arm HPC Research Cener, Mnneals, MN 55455, eember Mal, R. and Balachandar,. Effec f Three- Dmensnal n he Lf and Drag f Nmnall T- Dmensnal Clnders. Phscs f lds 7 (8), Ags 1995, L, J., Lng, L.N., Drec Aeracsc and Aerdnamc mlan f Ml-Hle Engne Lners, AIAA Paer 98-33, Chczesk, T, Mrrs, P.J., and Lng, L.N., Large- Edd mlan f Wall Bnded hear l Usng he NLDE, AIAA Paer -7,. 11