Separated Turbulent Flow Simulations Using a Reynolds Stress Model and Unstructured Meshes

Transcription

1 AIAA AIAA 43rd Aerospace Scences Meeng & Exhb Reno, NV, Janary, 5 Separaed rblen Flo Smlaons Usng a Reynolds Sress Model and Unsrcred Meshes Emre Alpman and Lyle N. Long Deparmen of Aerospace Engneerng he Pennsylana Sae Unersy Unersy Par, PA, 168, U.S.A. Nmercal smlaon of hree-dmensonal, separaed, hgh Reynolds nmber rblen flos s performed sng a second-order accrae cell-cenered fne olme mehod on nsrcred meshes. he compaons nclde an applcaon of Reynolds Sress Modelng RSM), hch consss of coplng he Reynolds ranspor eqaons h he Fare aeraged Naer-Soes Eqaons. he reslng sysem of 1 copled, non-lnear paral dfferenal eqaons s soled sng PUMA_RSM hch s an n-hose an nsrcred grd compaonal fld dynamcs code ren n ANSI-C. Compaons are performed on nsrcred meshes composed of erahedral cells. In order o redce he CPU me and memory reqremens, parallel processng s appled h he MPI Message Passng Inerface) commncaon sandard. he reslng parallel code s rn on Beolf clsers. Resls for hgh Reynolds nmber flo arond a 6:1 prolae spherod and a sphere are presened and compared h expermenal resls. In he prolae spherod case predcons of mean pressre and crcmferenal locaons of cross flo separaon pons are n good agreemen h expermen. he locaons of prmary and secondary separaon pons are comped h an error of roghly hree degrees. Mean pressre and sn frcon predcons for sphere solons are also n good agreemen h he measremens. he comped separaon locaon s ery close o he measred one. he dsrbon of rblen sresses shos ha he rblen flo arond a sphere s hghly ansoropc and sppors he noon ha sng ansoropc rblence models are necessary for hreedmensonal separaed flos. I. Inrodcon eparaed rblen flo arond hree-dmensonal geomeres s one of he mos challengng problems n S aerospace. For example, one of he common problems n helcoper fselage aerodynamcs s separaed rblen flo. Flo separaes from he hb or fselage and hen mpnges on he al roor, empennage and conrol srfaces. Separaed flo arond a body resls n many phenomena n aerodynamcs, sch as drag ncrease, lf loss, nseady flcaons, ec. herefore, accrae rblence predcon s a ey o ndersandng and predcng separaed rblen flos arond aerodynamc deces 1. he presence of hree-dmensonaly and crare nrodces changes n he rblence srcre, hs naldang many of he rblence models dely sed for smple and mldly complex shear layers. herefore, becomes exremely mporan o employ more physcs n prodng sable closre models for adeqae predcon of hese complex flos. Many of he closre models n he lerare e.g. -ε, -ω, Spalar-Allmaras, Baldn-Lomax) are based on he Bossnesq approxmaon 3 : µ j j µ 3 δ 1) hch assmes ha he prncpal axes of he Reynolds sress ensor are concden h hose of he mean sran rae a all pons n a rblen flo. For complex hree-dmensonal flos h sdden changes n mean sran rae he Gradae Research Asssan, exa15@ps.ed Professor, lnl@ps.ed Copyrgh by Emre Alpman. Pblshed by he Amercan Inse of Aeronacs and Asronacs, Inc., h permsson. 1 Amercan Inse of Aeronacs and Asronacs

2 AIAA AIAA 43rd Aerospace Scences Meeng & Exhb Reno, NV, Janary, 5 Bossnesq hypohess can be serosly n error becase he Reynolds sresses ll adjs o sch changes a a rae nrelaed o mean flo processes and me scales 3. he Bossnesq approxmaon also yelds he necessy of defnng lengh and me scales of rblence for eddy scosy calclaons. In zero and one eqaon models, hese scales are defned n an adhoc fashon for dfferen flos. he models are soropc n nare and are only ald for o-dmensonal smple shear flos. herefore, o-eqaon models conse he mnmm leel of closre ha s physcally accepable 3. Alhogh o-eqaon models can be speror o zero and one eqaon models, ho modfcaon, hey sll fal o capre many of he feares assocaed h complex flos. Whle hese models can be modfed o mproe her predce accracy, he modfcaons are largely adhoc and canno be easly generalzed 3. Moreoer, n he case of srong separaon, een he modfed o-eqaon models ere shon o fal o predc flo physcs de o her soropc nare 4. herefore, ansoropc models sch as fll Reynolds sress ranspor models RSM) 5, are necessary for accrae predcon of hree-dmensonal separaed flos. Alhogh RSM does no represen he nseady nare of rblence as does dynamc models sch as Large Eddy Smlaon LES) 6, s ery effece n compng he me aeraged qanes and reqres mch less CPU me han LES. he presen or employs RSM for smlaon of separaed rblen flos arond hree-dmensonal bodes. he mehod consss of coplng he ranspor eqaons for Reynolds sresses and rblence dsspaon rae 5 h he Fare aeraged Naer-Soes eqaons 3. Fare aeragng maes possble o predc he araons n mean densy ho explcly modelng he pars relaed o densy araons. he proposed model has been sed and aldaed for aros hree-dmensonal hgh Reynolds nmber flos 7-9. Alhogh he Reynolds sress ranspor eqaons are ren for me aeraged qanes, he mehod can sll be sed for nseady flo smlaons hen he characersc freqences of he phenomenon are sffcenly lo compared o he characersc freqences of rblence 1. II. Mehodology he reslng sysem of 1 copled non-lnear paral dfferenal eqaons s nmercally soled sng he comper code PUMA_RSM, hch s an n-hose compaonal fld dynamcs code ren n ANSI C. PUMA_RSM s a modfed erson of he comper code PUMA hch has been sed and aldaed by Long e al. for nmercal solon of nmeros problems he eqaons are dscrezed by a second order fne olme mehod and soled sng an explc for-sage Rnge-Ka ype negraon echnqe. Alhogh explc mehods sffer from lmaons on he maxmm alloable me sep sze, hey reqre a mnmm nmber of arhmec operaons per eraon sep and are easer o parallelze compared o mplc mehods 4. Solons arond a 6:1 prolae spherod and a sphere are obaned sng nsrcred meshes composed of 5.1 mllon and 3.8 mllon erahedral cells, respecely. he cells are clsered n he cny of he sold bondary for sffcen bondary layer resolon. Here he non-dmensonal dsance y of he frs cells s on he order of one for boh cases. Unsrcred grds are relaely easy o consrc arond complex hree-dmensonal bodes and hey offer ery effcen cell dsrbons for accrae solons of complex real lfe problems. he seconal es of he compaonal meshes generaed for he problems and a close-p e of he cells near he sold bondary are dsplayed n Fgres 1 and. Md Plane Grd 6:1 Prolae Spherod Fgre 1 Seconal and close-p es of he cells arond a 6:1 prolae spherod. Amercan Inse of Aeronacs and Asronacs

3 AIAA AIAA 43rd Aerospace Scences Meeng & Exhb Reno, NV, Janary, 5 Fgre Seconal and close-p es of he cells arond a sphere. Alhogh, n a compaonal sense, RSM coss mch less han LES and DNS; he solons sll reqre consderable comper resorces. he RSM eqaons hae been me aeraged, so hey are no me dependen. LES and DNS solons reqre hosands or mllons of me seps. RSM represens a 3-D problem, hle LES and DNS are 4-D problems space and me). B RSM sll reqres mllons of grd cells and parallel processng s appled o redce he CPU me and memory reqremens. he MPI Message Passng Inerface) 5 commncaon sandard s sed for hs prpose and he code s rn on a Beolf clser. Or COs Effece Compng Array COCOA3) s a clser of off-he-shelf PCs conneced a fas eherne. hese PCs rn RedHa Lnx h MPI 5. COCOA3 conans 6.4 GHz. dal Inel Xeon processors, each hang GB of RAM and dal 1 Mbps fas Eherne cards. III. Mahemacal Formlaon he negral form of he goernng eqaons RSM) can be ren as: Ω q dω F ds F ds F ds Q dω = ) S S S Ω here q s he ecor conanng he conserae arables, F s he conece flx ensor, F s he scos flx ensor, F s he rblen flx ensor and Q s he ecor conanng rblen sorce erms. he conserae arable ecor q can be ren as: [ E ε ] q = 3) zz here s he mean densy,,, are mean elocy componens n he x, y and z-drecons, E s he oal energy per n mass, are he Reynolds sresses and ε s he rblen dsspaon rae. In he eqaons. ) denoes Fare aeragng 3, ). denoes non-eghed aeragng and. ) denoes an aerage qany ha s neher a Fare aerage nor a non-eghed aerage 8. Here oal energy s dfferen han he Fare aeraged oal energy hch conans rblen nec energy. p V E = γ 1) 4) 3 Amercan Inse of Aeronacs and Asronacs

4 AIAA AIAA 43rd Aerospace Scences Meeng & Exhb Reno, NV, Janary, 5 Amercan Inse of Aeronacs and Asronacs 4 In Eqaon 4) p s he mean sac pressre. he deals of he flx ensors and he sorce ecor are as follos: A. Conece Flx ensor = ε ε ε H H H p p p F zz zz zz 5) here H s he oal enhalpy per n mass. p E H = 6) B. Vscos Flx ensor = F V z zz y x zz q q q 7) here he mean scos sresses and lamnar hea ransfer raes are approxmaed as follos 6 : j j x x x δ µ µ 3 8)

5 AIAA AIAA 43rd Aerospace Scences Meeng & Exhb Reno, NV, Janary, 5 q K 9) C. rblen Flx ensor here ) d and ) zz q q q x y zz z d ) d ) d ) 1 3 F = d ) d ) d ) 1) 1 3 d ) d ) d ) zz 1 zz zz d ) d ) d ) 1 d ) d ) d ) 1 d ) d ) d ) ) ) ) 1 3 d d d ε 1 d ε are dffson erms for Reynolds sresses and rblen dsspaon erm, respecely. Here he dffson of Reynolds sresses s modeled sng he rple correlaons model of Hanjalc and Lander 7. ε ε 3 d ) n jn = Cs m ε m jm n m nm ν m n 11) A smlar expresson s ren for he rblen dsspaon rae: d ) ε n = Cε m ε ε ε ν m n 1) Where C =. 11, C =. 18. In he aboe eqaons, s he rblen nec energy. s ε 1 = ) 13) he rblen hea ransfer raes are approxmaed sng a smple graden model 8. q µ C Pr p 14) here = C µ Re 15) µ µ 5 Amercan Inse of Aeronacs and Asronacs

6 AIAA AIAA 43rd Aerospace Scences Meeng & Exhb Reno, NV, Janary, 5 Re = µ ε 16).5 C µ =.9 exp 17) 1. Re ) Here C s he specfc hea a consan pressre, p Pr s he rblen Prandl nmber. D. rblen Flx ensor P ε P ε φ Q = 18) P ε φ P ε φ zz zz zz P ε φ P ε φ P ε φ S here P s he exac prodcon erm de o mean sran raes 5 : P = j 19) j he dsspaon ensor ε s approxmaed sng local soropy assmpon 3 ε = ε δ ) 3 he φ erm n Eqaon 18) s he pressre redsrbon erm. hs erm s he mos mporan em n he closre becase conrols boh he separaon and reaachmen processes 7-1. he erm s spl no slo and rapd pars 3 and her correspondng all echo erms hch arse from he reflecon of pressre flcaons from he rgd all 8. he slo and he rapd erms are modeled as follos 9 : φ, 1 C1 ε δ 3 1) 1 φ C P P δ, mm 3 ) he models for slo 3 and rapd 9 all echo erms are dsplayed n he follong eqaons: 6 Amercan Inse of Aeronacs and Asronacs

7 AIAA AIAA 43rd Aerospace Scences Meeng & Exhb Reno, NV, Janary, 5 ε 3 3 φ C n n n n n n f m m j j δ, 1 1 3) 3 3 φ C n n n n n n f φ δ φ φ, m, m, j j, 4) here n s he srface normal. Near all dampng fncon f s aen as.4 3 / / εyn, y n beng he normal dsance o he all. he coeffcens C 1, C, C 1, C are opmzed by Lander and Shma 8. Fnally he sorce erm S s gen by he follong relaon 3 : here * ε s he modfed dsspaon rae hch goes o zero a he sold all 8. * ε ε ε S = C C 5) ε1 ε j IV. Nmercal Mehod he goernng eqaons are nmercally soled sng an nsrcred grd fne olme echnqe, hch employs a Lax-Fredrch scheme 31 for conece flxes and space cenered scheme for scos and rblen flxes. he reslng sem-dscree eqaons are nmercally negraed sng a for sage Rnge-Ka negraon echnqe h local me seppng o enhance he conergence o seady sae. he solon of he eqaons reqres approprae me seps lmed by he sably bond of he scheme. he local me sep s comped hrogh sably analyses for he nscd conecon, scos dffson, rblen dffson and rblen dsspaon. A he sold bondares a no slp condon s appled here all he elocy componens, as ell as he Reynolds sresses, are se o zero. he rblen dsspaon rae a he all s comped sng Eqaon 6). ε = ν n 6) Where n s he dsance normal o he all. he sold bondares are also assmed o be adabac. Alhogh me aeraged qanes are soled for, he goernng eqaons are assmed o be me dependen for erae prposes. herefore, hey reqre nal condons. In hs sdy he freesream condons are sed for he nal elocy and densy. Pressre s comped sng he eqaon of sae. Freesream rblence s assmed o be soropc. For he prolae spherod solons he characersc freesream elocy of rblence s assmed o be.3% of he mean speed 3 hle hs ale s aen as.45% for he sphere resls. he rblen dsspaon rae s hen comped assmng ha he freesream lengh scale for large eddes s.1 m 1. V. Resls and Dscsson Resls for hgh Reynolds nmber flo arond a 6:1 prolae spherod and a sphere are presened here A. Resls for a 6:1 Prolae Spherod For he nmercal solon he freesream Reynolds nmber, freesream Mach nmber and flo angle of aac are aen o be 6.5x1 6,.13 and 3 degrees, respecely. he solons reqred 13.4 GB of memory and oo nearly 8 days on 3 COCOA3 processors. Fgre 3 shos he pressre dsrbon of he mdplane and comparson h expermenal daa 33. he predcons sho good agreemen h measremens excep a he exreme af end of he spherod. he man reason for hs dscrepancy s he nmercal dsspaon of he orcal flofeld. Vorces moe aay from he body o coarser grd regon as hey rael along he longdnal drecon. 7 Amercan Inse of Aeronacs and Asronacs

8 AIAA AIAA 43rd Aerospace Scences Meeng & Exhb Reno, NV, Janary, 5-1 Mdplane Cp Dsrbon Re = 6.5x1 6, M =.13, α = 3 deg -.5 Cp.5 1 Expermen Upper Srface) Expermen Loer Srface) RSM Loer Srface) RSM Upper Srface) x/l Fgre 4 Mdplane Cp dsrbon of a 6:1 prolae spherod. Re = 6.5x1 6, M =.13, α = 3. hree-dmensonal separaon s boh dffcl o model and poorly ndersood. Unle o-dmensonal separaon, hree-dmensonal separaon s rarely assocaed h he anshng of he all shear sress 3. he flo arond a prolae spherod separaes on he lee sde of he body and forms a srface ha rolls no a orex. hs, sn frcon lnes on eher sde of he separaon conerge asympocally oard he separaon lne 34. A hgh angles of aac secondary separaon mgh occr a a hgher crcmferenal angle. Sch behaor can be easly obsered from Fgre 5 here he orcy conors a dfferen axal saons are dsplayed along h he srface sn frcon lnes. I s clear from Fgre 5 ha he sn frcon lnes asympocally conerge oard a separaon lne. Vorex formaon s also eden a he af porons of he body js afer hs separaon lne. A he pper lee sde of he body a second separaon lne s also obsered. Predced prmary and secondary separaon locaons a he axal saon x/l =.738 are ablaed and compared h measremens 33 n able 1. Fgre 5 Vorcy conors a dfferen axal locaons h srface sn frcon lnes. Re = 6.5x1 6, M =.13, α = 3. 8 Amercan Inse of Aeronacs and Asronacs

9 AIAA AIAA 43rd Aerospace Scences Meeng & Exhb Reno, NV, Janary, 5 able 1. Prmary and Secondary Separaon Locaons, x/l =.738 Re = 6.5x1 6, M =.13, α = 3. RSM Solon Measremen Crcmferenal Locaon of Prmary Separaon [degrees] Crcmferenal Locaon of Secondary Separaon [degrees] he predcons are ery close o he expermenal daa. he dscrepancy s approxmaely hree degrees for boh cases. Anoher ndcaon of hree-dmensonal separaon s he n he cny of he separaon pon near he all 34. Fgre 6 shos he longdnal elocy conors a axal saon x/l =.738. o localzed mnmm regons darer regons) can be obsered from hs fgre. hese regons are close o he measred prmary and secondary separaon pons. Fgre 6 Longdnal elocy conors, x/l =.738. Re = 6.5x1 6, M =.13, α = 3. For a more dealed analyss, he crcmferenal dsrbon of longdnal elocy, predced a a dsance of.14 cm from he all, s ploed n Fgre 7. In he fgre local mnma occr a crcmferenal locaons of 11 and 151 degrees hch are ery close o he measred prmary and secondary separaon pons. Fgre 8 shos he laeral sn frcon dsrbon a x/l =.738 and comparson h he measremens 33. Predcons sho qalae agreemen h expermen. he sn frcon coeffcen s dffcl o measre and expermenal daa alays conan some nceranes 34. In addon o hs, accrae sn frcon coeffcen compaon n nsrcred grd h seed cells s also ery dffcl. B qalae agreemen s sll good and encoragng. he prolae spherod resls clearly shoed ha RSM can be effecely sed for he predcons of hreedmensonal separaed flos here he models based on Bossnesq approxmaon old mos lely fal de o her soropc nare. In addon, LES solons old reqre hosands or mllons of meseps. he RSM nmercal solons sccessflly represened he physcs of he problem. he prmary and een he secondary separaon locaon ere predced accraely h approxmaely hree degrees of dscrepancy from he expermen. 9 Amercan Inse of Aeronacs and Asronacs

10 AIAA AIAA 43rd Aerospace Scences Meeng & Exhb Reno, NV, Janary, 5 Longdnal Velocy Dsrbon a x/l =.738 a a dsance of.14 cm from he bondary) Re = 6.5x1 6, M =.13, α = 3 deg U/U nf φ [deg] Fgre 7 Crcmferenal dsrbon of longdnal elocy a.14 cm aay from he all, x/l =.738. Re = 6.5x1 6, M =.13, α = 3. C fla Laeral Sn Frcon Comparson a x/l =.738 Re = 6.5x1 6, M =.13, α = 3 deg φ [deg] Expermen RSM Solon Fgre 8 Sn frcon dsrbon, x/l =.738. Re = 6.5x1 6, M =.13, α = 3. B. Resls for a Sphere For he sphere solon, he freesream Reynolds nmber and freesream Mach nmber are aen o be 1.14x1 6 and.1763, respecely. he solon reqred 9.7 GB of memory and oo nearly 7 days on 3 COCOA3 processors. Fgre 9 shos he crcmferenal pressre dsrbon and comparson h expermenal daa 35. he resls are n ery good agreemen h he measremens h slgh dscrepances beeen 9 and 1 degrees. Sch dscrepances ere also obsered by he LES solons n Ref. 6. As n he prolae spherod case, flo separaon s analed by plong sn frcon lnes and he crcmferenal sn frcon dsrbon. Fgre 1 shos he sn frcon lnes along h he elocy conors and Fgre 11 shos he crcmferenal sn frcon dsrbon and comparson h he measremens 35. I s clear from Fgre 1 ha he sn frcon lnes sop a a crcmferenal angle of approxmaely 1 degrees, hch ndcaes he separaon locaon. Bondary layer hcenng beyond hs pon s also eden from he elocy dsrbon conors. he predced separaon locaon s ery close o he expermenal ale 35. hs s also sppored by Fgre 11 here he comped sn frcon coeffcen s n ery good agreemen h he expermen 35 h some dscrepances n he cny of 9 degrees. hese dfferences are de o lamnar-rblen ranson hch s no modeled by he crren nmercal mehod. 1 Amercan Inse of Aeronacs and Asronacs

11 AIAA AIAA 43rd Aerospace Scences Meeng & Exhb Reno, NV, Janary, Crcmferenal Pressre Dsrbon of a Sphere Re = 1.14x Cp Expermen RSM φ [deg] Fgre 9 Crcmferenal Cp dsrbon of a 6:1 prolae spherod. Re = 1.14x1 6, M = Fgre 1 Velocy dsrbon h srface sn frcon lnes Re = 1.14x1 6, M = Cf*sqrRe) Mdplane Sn Frcon Coeffcen Dsrbon Re = 1.14x1 6, M = Expermen. RSM Solon φ [deg] Fgre 11 Crcmferenal sn frcon dsrbon. Re = 1.14x1 6, M = Amercan Inse of Aeronacs and Asronacs

12 AIAA AIAA 43rd Aerospace Scences Meeng & Exhb Reno, NV, Janary, 5 aneda 36, hrogh hs flo salzaons sggesed a mechansm for orex sheddng n he ae of a sphere. He obsered ha a orex shee separang from he sphere forms a par of srong sreamse orces. Sch a behaor s obsered n Fgre 1 hch shos a me aeraged so-orcy srface. hs srcre as also obsered n he LES resls by Jndal e al 6. Fgre 1 Iso-orcy srface n he ae of a sphere Re = 1.14x1 6, M = In order o anale he rblen srcre of he flo, normalzed and dsrbons are ploed and dsplayed n Fgres 13 and 14. Normalzaon s preformed by ddng he qany by mean densy and rblen nec energy. In soropc rblence, normalzed and ae he ales of /3 and respecely. he large deaons from hese ales mean non-soropy s presen n he rblence srcre. I s clear from hese fgres ha he flo s hghly ansoropc. he fgres clearly sppor he noon of employng ansoropc rblence models, sch as RSM, for accrae predcon of hree-dmensonal separaed flos. Fgre 13 Normalzed conors Re = 1.14x1 6, M = Amercan Inse of Aeronacs and Asronacs

13 AIAA AIAA 43rd Aerospace Scences Meeng & Exhb Reno, NV, Janary, 5 Fgre 14 Normalzed conors Re = 1.14x1 6, M = VI. Conclson Hgh Reynolds nmber separaed rblen flos arond a 6:1 prolae spherod and a sphere ere predced sng RSM. he model consss of a nmercal solon of he Fare aeraged Naer-Soes eqaons copled h he ranspor eqaons for Reynolds sresses and he rblen dsspaon rae. he prolae spherod solon as performed a a Reynolds nmber of 6.5x1 6, Mach nmber of.13 and flo angle of aac of 3 degrees. Resls ere encoragng and shoed good agreemen h expermens. he behaor of he srface sn frcon lnes and crossflo separaon pon a x/l =.738 ere conssen h expermenal daa. he predced locaons of he prmary and secondary separaon ere hn hree-degrees of he measremens. he sphere solon as performed a a Reynolds nmber of 1.14x1 6, and a Mach nmber of he sphere resls ere also n good agreemen h he measremens. Slgh dscrepances ere obsered beeen he comped and measred sn frcon coeffcens, hch are manly de o he lamnar-rblen ranson. Analyss of he normalzed rblen sresses reealed ha he rblen srcre of he flo as hghly ansoropc. hs sppored he necessy of employng ansoropc rblence models for hree-dmensonal separaed flos. Resls clearly shoed ha RSM can handle hree-dmensonal separaed flos here mos of he eddy scosy models old be nsffcen. Snce he model s no flo dependen can be employed for oher geomeres ncldng complex shapes) and flo condons. References 1 Solez, F. J., Parallel Mehods for Compng Unseady Separaed Flos Arond Complex Geomeres, Ph.D. hess, Aerospace Engneerng Deparmen, Penn Sae Unersy, Sae College, PA,. Lashmnarayana, B., rblence Modelng for Complex Shear Flos, AIAA Jornal, Vol.4, No. 1, 1986, pp Wlcox, D. C., rblence Modelng for CFD, DCW Indsres Inc, ISBN X. 4 Sloms, J. F., Gors, J. J., Mller R. W., Maron. A., Nmercal Smlaon of Crclaon Conrol Arfols as Affeced by rblence Models, AIAA 4 h Aerospace Scences Meeng & Exhb, AIAA -851, Reno, NV, Janary. 5 Lander, B. E., Reece, G. J., Rod W., Progress n he Deelopmen of a Reynolds-Sress Closre, Jornal of Fld Mechancs, Vol. 68, No. 3, pp , Jndal, S., Long. L. N., Plassmann, P. E., Sezer-Uzol, N., Large Eddy Smlaons on a Sphere Usng Unsrcred Grds, AIAA Fld Dynamcs Conference and Exhb, AIAA 4-8, Porland, OR, 4. 7 Gerolymos, G. A., Valle, I., Wall-Normal-Free Reynolds Sress Closre for hree-dmensonal Compressble Separaed Flos, AIAA Jornal, Vol. 39, No. 1, 1, pp Gerolymos, G. A., Valle, I., Wall-Normal-Free Reynolds Sress Model for Roang Flos Appled o rbomachnery, AIAA Jornal, Vol. 4, No.,, pp Amercan Inse of Aeronacs and Asronacs

14 AIAA AIAA 43rd Aerospace Scences Meeng & Exhb Reno, NV, Janary, 5 9 Chassang, J. C., Gerolymos, G. A., Valle, Effcen and Robs Reynolds-Sress Model Compaon of hree-dmensonal Compressble Flos, AIAA Jornal, Vol. 41, No. 5, 3, pp Chassang, J. C., Gerolymos, G. A., Valle, Reynolds-Sress Model Dal-me Seppng Compaon of Unseady hree-dmensonal Flos, AIAA Jornal, Vol. 41, No. 1, 3, pp Mod, A., Long, L. N., Sezer-Uzol, N., Plassmann P., Scalable Compaonal Seerng Sysem for Vsalzaon of Large-Scale CFD Smlaons, Jornal of Arcraf o be pblshed). 1 Solez, F., Long, L., N., Morrs, P. J., Sharma, A., Landng Gear Aerodynamc Nose Predcon sng Unsrcred Grds, Inernaonal. Jornal of Aeroacoscs, Vol. 1, No., 13 Mod, A., Unseady Separaed Flo Smlaons Usng a Clser of Wor Saons, M.S. hess, Aerospace Engneerng Deparmen, Penn Sae Unersy, Sae College, PA, Mod, A., Long, L. N., Unseady Separaed Flo Smlaons sng a Clser of Worsaons, 38 h AIAA Aerospace Scences Meeng & Exhb, AIAA -7, Reno, NV, Janary,. 15 Hansen, R. P., Long, L. N., Large Eddy Smlaon of a Crclar Cylnder on Unsrcred Grds, 4 h AIAA Aerospace Scences Meeng & Exhb AIAA -98, Reno, NV, Janary. 16 Alpman, E., Long, L. N., and Kohmann, B. D., Undersandng Dced Roor Anorqe and Dreconal Conrol Characerscs, Par I: Seady Sae Smlaon, Jornal of Arcraf, Vol. 41, No. 5, 4, pp Alpman, E., Long, L. N., and Kohmann, B. D., Undersandng Dced Roor Anorqe and Dreconal Conrol Characerscs, Par I: Unseady Smlaon, Jornal of Arcraf o be pblshed) 18 Mod, A., Long, L. N., Plassmann, P. E., Real-me Vsalzaon of Wae-Vorex Smlaons sng Compaonal Seerng and Beolf Clsers, VECPAR, Porgal, Jne. 19 Long, L. N., Mod, A., rblen Flo and Aeroacoscs Smlaons sng a Clsers of Worsaons, NCSA Lnx Reolon Conference, IL, Jne, 1. Sharma, A., Long, L. N., Arae Smlaons on an LPD 17 Shp, 15 h AIAA CFD Conference, AIAA 1-589, Anahem, CA, Jne 1. 1 Long, L. N. Solez F., Sharma, A., Aerodynamc Nose Predcon sng Parallel Mehods on Unsrcred Grds, 7 h AIAA/CEAS Aeroacoscs Conference, AIAA 1-196, Maasrch, Neherlands, May 1. Schezer. F., Compaonal Smlaon of Flo arond Helcoper, M.S. hess, Aerospace Engneerng Deparmen, Penn Sae Unersy, Sae College, PA, May, Hansen R., Separaed rblen Flo, Ph.D. hess, Mechancal Engneerng Deparmen, Penn Sae Unersy, Sae College, PA, Ags, 1. 4 Alpman, E., Nmercal Smlaon of Roary ng Flofelds n Parallel Compers, M.S. hess, Aerospace Engneerng Deparmen, Mddle Eas echncal Unersy, Anara, rey, Jly 1. 5 Pacheco, P. S., Parallel Programmng h MPI, Morgan Kafmann Pblshers Inc., Vandromme, D., Ha Mnh H., Abo Coplng of rblence Closre Models h Aeraged Naer-Soes Eqaons, Jornal of Compaonal Physcs, Vol. 65, 1986, pp Hanjalc, K., Lander B. E., A Reynolds Sress Model of rblence and Is Applcaons o hn Shear Flos, Jornal of fld Mechancs, Vol. 5, 197, pp Lander, B. E., Shma. N., o-momen Closre for he Near-Wall Sblayer: Deelopmen and Applcaon, AIAA Jornal, Vol. 7, No , pp Gbson, M. M., Lander B. E., Grond Effecs on Pressre Flcaons n he Amospherc Bondary Layer, Jornal of Fld Mechancs, Vol. 86, 1978, pp Shr, C. C., A Prelmnary Nmercal Sdy of Amospherc rblen Flos n he Idealzed Planeary Bondary Layer, Jornal of Amospherc Scences, Vol. 3, 1973, pp Hrsch C., Nmercal Compaon of Inernal and Exernal Flos, Vol. : Compaonal Mehods for Inscd and Vscos Flos, John Wley & Sons, Chesnaas, C. J., Smpson R.L., Dealed nesgaon of he hree-dmensonal Separaon Abo a 6:1 Prolae Spherod, Jornal of Fld Mechancs, Vol. 3, No. 6, pp , Jne Krepln, H. P., Volmers H., Meer H. U., Shear Sress Measremens on an Inclned Prolae Spherod n he DFVLR 3 m x 3 m Lo Speed Wnd nnel, Gongen Daa Repor, DFVLR Rep, IB -84 A 33, Wezel,. G., Smpson R. L., Chesnaas, C. J., Measremen of hree-dmensonal Crossflo Separaon, AIAA Jornal, Vol. 36, No. 4, 1998, pp Achenbach, E., Expermens on he Flo pas Spheres a ery Hgh Reynolds Nmbers, Jornal of Fld Mechancs, Vol. 54, No. 3, 197, pp aneda, S., Expermanal Inesgaons of he Wae Behnd a Sphere a Lo Reynolds Nmbers, Jornal of Physcs Japan, Vol. 11, No. 1, 1956, pp Amercan Inse of Aeronacs and Asronacs