EFFECT OF P-NORMS ON THE ACCURACY ORDER OF NUMERICAL SOLUTION ERRORS IN CFD

Transcription

1 rocdigs of NIT 00 opyright 00 by ABM 3 th Brazilia ogrss of Thrmal Scics ad girig Dcmbr 05-0, 00, brladia, MG, Brazil T O -NORMS ON TH ARAY ORDR O NMRIAL SOLTION RRORS IN D arlos Hriqu Marchi, marchi@ufpr.br dral ivrsity of araá (R), Mchaical girig Dpartamt, uritiba, R, Brazil. Márcio Adré Martis, madr@uictro.br ivrsidad stadual do tro-ost (NINTRO), Dpartamto d Matmática, Guarapuava R. Abstract. Th objctiv of this work is to valuat th us of svral p-orm typs i th vrificatio of umrical solutios i omputatioal luid Dyamics (D). Thortical aspcts of umrical rrors ad vctor orms ar discussd, ad rsults of umrical xprimts ar prstd. Th o-dimsioal advctio-diffusio quatio is usd as xampl, which is solvd by th fiit volum mthod usig schms of first, scod ad third-ordr accuracy. Th study compasss sv typs of mtrics, ivolvig l, l ad l -orms; fift variabls of itrst; ad fift uiform grids. Th aalytical dductios about th quivalcs amog orms ar corroboratd by th umrical rsults. With rgard to th variabls of itrst studid, it was foud that th ordrs varid accordig to th typ of orm mployd. W foud prmac, dgratio or lvatio of th ordr of th umrical schm usd hr. Amog th mtrics ivstigatd, th os that maitaid th ordr of th umrical schm wh applid to th umrical rror wr: ma l -orm, l -orm of ma squar odal rrors ad th l -orm. Kywords: vrificatio, fiit volum, umrical rror, D.. INTRODTION I th currt litratur it is commo to us vctor orms i umrical vrificatio procdurs i which, basically, th umrical rror ivolvd ad its ordr of accuracy ar stimatd. Ths orms ar usd bcaus thy charactriz mtrics that allow for th aalysis of th ordr of accuracy of th rror of a giv umrical solutio. Simos ad Str (003), alcão t al. (006), Myrs t al. (007), Mathou t al. (008), ad Ju t al. (009) ar xampls of works that adopt this approach. Dtrmiig th ordr of accuracy is importat, abov all, from th followig stadpoits: us of rror stimators for cass of ukow aalytical solutios,.g., GI (Grid ovrgc Idx) (Roach, 998) ad Richardso (Marchi ad Silva, 00); for cofirmatio of th thortical ordr of accuracy of th umrical modl mployd; or to dtrmi th practical ordr of accuracy wh th thortical ordr is ukow. I th ivstigatio of this ordr for a fixd umbr of variabls ad discrtizatio itrval, th choic of th orm to b mployd may lad to diffrt rsults, which i tur may lad to icorrct itrprtatios. Studis that dal with ths ffcts ar currtly ot availabl i th litratur. Th objctiv of this work is to valuat th us of svral typs of orms i th vrificatio of umrical solutios i D. W itd to dmostrat that, i th prst cotxt, vctor orms ar ot quivalt, ad to idtify th mtrics that maitai th thortical ordr of accuracy of th umrical modl adoptd. To this d, som thortical aspcts of umrical rrors ad vctor orms ar discussd, ad th rsults of umrical xprimts ar prstd. As th modl problm, w cosidr th o-dimsioal advctio-diffusio quatio solvd by th fiit volum mthod with schms of first, scod ad third ordrs of accuracy. Th o-dimsioal approach is motivatd by th possibility of grid rfimt up to a ordr of millios of ods, which allows for vrificatio of asymptotic bhaviors. It is also assumd that th o-dimsioal rsults ar applicabl to two ad thr dimsios. I this study w cosidr sv typs of mtrics obtaid by usig l, l ad l -orms, fift variabls of itrst, ad fift uiform grids.. NMRIAL VRIIATION IN D Th umrical solutio of a problm whos mathmatical modl is a quatio or a st of diffrtial quatios ca b gralizd by (Garby ad icard, 008) ( ) ( ), x S x x R, g x R () whr rprsts th dpdt variabl or st of variabls, x is th idpdt variabl or st of variabls, is th calculatig domai, ad is its boudary. osidrig ( V, ) ad ( W, ) as two ormd vctor spacs of V W fiit dimsios, h : V W corrspods to th oprator that rprsts th applicatio of a umrical mthod applicatio of V i W. To xmplify: a procss of discrtizatio by fiit volums i grids M ( h ) paramtrizd by

2 rocdigs of NIT 00 opyright 00 by ABM 3 th Brazilia ogrss of Thrmal Scics ad girig Dcmbr 05-0, 00, brladia, MG, Brazil h 0. I practic, o has a approximatio for th solutio h i grids M ( h ) gratd by a computatioal cod (umrical algorithm) applid at poits S h : S : h h () Th smallr h is, th mor rfid th discrtizatio of th calculatig domai ad thus, it is xpctd that th approximat umrical solutio is mor accurat. Howvr, this is ot always th cas, so a structurd procss of umrical vrificatio is dd. Ths procsss basically cosist i obtaiig a stimat for th umrical rror () ivolvd, ad its moitorig through th us of a giv mtric. Amog othr aspcts, th aim is to vrify whthr 0 for h 0 (3) V. Numrical rror ad its Ordr of Accuracy Th umrical rror ( ) ca b dfid as th diffrc btw th xact aalytical solutio ( ) of a variabl of itrst ad its umrical solutio ( ), i.., ( ) (4) whr ca b causd by four sourcs of rror (Marchi ad Silva, 00): trucatio, itratio, roud-off ad programmig. Wh th othr sourcs ar abst or vry mior i rlatio to trucatio rrors, ca also b calld a discrtizatio rror. By aalogy to th gral quatio of trucatio rror, th discrtizatio rror of a umrical solutio is giv by (rzigr ad ric, 00; Roach, 998; Marchi ad Silva, 00) c h c h c h (5) L whr th cofficits c j, j,,3... ar ral umbrs that ar fuctios of th dpdt variabl (of th problm) ad its drivativs, but ar assumd to b idpdt of th siz (h) of th cotrol volums cosidrd i th discrtizatio procss. By dfiitio, th tru ordrs ( V ) of th rror ar th xpots of h i q. (5). Ths ordrs ar ral umbrs that follow th rlatio: L 3... Th smallst xpot, L, is calld th asymptotic ordr. Wh h 0, th first parcl (q. (5)) is th pricipal compot of th discrtizatio rror, i.., it domiats th total valu of (Marchi ad Silva, 00). L is oft tratd i th litratur as rror ordr or accuracy ordr ad is dotd by. Rsults ad discussios about umrical vrificatio procdurs ar ormally ctrd o this ordr. Roy (005), alcão t al. (006) ad Mathou t al. (008) ar xampls of works that follow this mthodology. Thr ar currtly a cosidrabl umbr of mthods to stimat discrtizatio rrors. Ths mthods ca b classifid ito a priori ad a postriori mthods, but i gral both cosidr th domiat trm of th gral xprssio of th discrtizatio rror (q. (5)). I othr words, thy cosidr ch (6) a rror of ordr for h 0. Sic it is impossibl to adopt this limit i practic, h 0 is cosidrd th procss of rfimt of M ( h ). sually, th ffctiv ( ) ad appart ( ) ordrs ar admittd, which corrspod, rspctivly, to th local slop for th rror curv ad its stimat vrsus h i logarithmic scal graphs (Marchi ad Silva, 00). Thrfor, thy ar mployd as approximatios for. osidrig th umrical solutios ad, for i two grids, fi ( M ( h )) ad coars ( M ( h )), rspctivly, th algbraic xprssio for is dtrmid by log ( ) / ( ) (7) log( q)

3 rocdigs of NIT 00 opyright 00 by ABM 3 th Brazilia ogrss of Thrmal Scics ad girig Dcmbr 05-0, 00, brladia, MG, Brazil whr ( ) ad ( ) corrspod to th rrors for ad, ad q h / h is th grid rfimt ratio. Accordig to th dfiitio of, ca b obtaid oly wh th aalytical solutio for is dtrmid. osidrig M ( h ) (fi), M ( h ) (coars) ad M ( h S ) (suprcoars) grids, with a costat rfimt ratio of q h / h hs / h, ad thir rspctiv umrical solutios, ad S, th appart ordr ( ) ca b obtaid by mas of (Marchi ad Silva, 00) log ( S ) /( ) (8) log( q). Global rror by Vctor Norm ach poit i of M ( h ) has a discrtizatio rror associatd to it a local rror, which is usually tratd as a odal rror. Howvr, it is ormal to attmpt to quatify th rror at all th M ( h ) poits to obtai th global rror. Th stimat of th global discrtizatio rror is, amog othrs, a itm i which th procss of umrical vrificatio ca b summarizd (Roy, 005). or poits, lmts or calculatio volums, th global discrtizatio rror ( g ) is dtrmid by a xprssio that ivolvs all th odal rrors, i.., g cih i (9) whr rprsts th mathmatical oprator that stablishd this rlatioship btw th odal valus, i,...,. I umrical vrificatio procdurs of D, is obtaid, i most cass, through vctor orm. I gral, it is commo procdur to us l, l ad l -orms. Howvr, o justificatios ar availabl that poit to quivalcs amog ths orms, or dcisio-makig critria with rgard to th adoptio of a mtric charactrizd by a giv orm. As a xampl, i th umrical vrificatio of a lamiar flow problm dscribd by Navir-Stoks quatios, arptr t al. (005) cosidr th us of l ad l -orms to dtrmi g. At th d, thy admit that th rsults lad to qualitativly similar coclusios ad mtio th quivalc btw th orms. Howvr, thy do ot dscrib th ordrs of accuracy ivolvd. po aalyzig ths rsults, o fids that th citd ordrs ar distict. By dfiitio, from a aalytical stadpoit, two orms r ad s i a vctor spac V R, dotd by ad, r s ar calld quivalt if ral costats k ad k R xist, such that (Golub ad Va Loa, 996) k v v k v (0) r s r whr v V is ay vctor of dimsio. Th orms (l -orm) ar xampls of aalytical quivalc: (l -orm), (l -orm, or uclida orm) ad v v v () v v v () v v v (3) Howvr, i D umrical vrificatio procdurs, th applicatio of ths orms may lad to diffrt itrprtatios about th accuracy of th umrical rsults obtaid, du to chags i th ordr of accuracy. It ca thrfor b statd that, i this cotxt, th quivalc btw th ordrs of l, l ad l -orms is ot vrifid. 3. T O VTOR NORMS ON TH ARAY ORDR I this sctio, w ivstigat th ffct rsultig from th us of th l, l ad l -orms o th ordr of th umrical mthod utilizd. Th sctio bgis with a prstatio of th problm modl adoptd, followd by a dscriptio of th umrical ad aalytical rsults obtaid with th us of sv mtrics l -orm ad its ma, l -orm, ma l -orm

4 rocdigs of NIT 00 opyright 00 by ABM 3 th Brazilia ogrss of Thrmal Scics ad girig Dcmbr 05-0, 00, brladia, MG, Brazil obtaid i two distict ways, l -orm ad its ma o th umrical rror ivolvd ( ) ad o th dpdt variabl ( ) of th problm. 3. Modl roblm osidrig th mathmatical modl of cosrvatio of thrmal rgy with stady o-dimsioal flow, icomprssibl fluid, without gratio of hat ad viscous dissipatio, ad with costat proprtis ad vlocitis i a cotiuous mdium, o has th advctio-diffusio quatio: d d (4) dx dx whr = clt umbr, is th dpdt variabl of th problm (tmpratur) ad x is th idpdt variabl (spatial coordiat). Th lgth of th calculatig domai (D) cosidrd was th itrval [0,]. Th boudary coditios applid (Dirichlt coditios) wr: (0) 0 (). Thus, th aalytical solutio for q. (4) is ( x) x (5) Th umrical solutios to this problm wr obtaid usig th fiit volum mthod (Vrstg ad Malalaskra, 007), with first, scod ad third-ordr umrical approximatios. Th TDMA mthod was usd to solv th systm of quatios rsultig from th procss of discrtizatio (rzigr ad ric, 00). Th computatioal cod was dvlopd usig th ortra Itl 9. applicatio with quadrupl prcisio. Th calculatios wr prformd i 5 distict grids, with a rfimt ratio of q 3. Amog ths grids, th coarsst had 5, ad th fist grid had 3,94,845 calculatig volums, whr h D. Th variabls of itrst, i.., th variabls for which th solutios wr obtaid ad th ordrs aalyzd, wr: (a) [ ( )], th odal rror for th umrical solutio of at th ctral poit of th grid; (b), th global rror, dtrmid usig l -orm, of th valus of th odal rrors i, i,..., ; (c), th ma l -orm of th ratio of to th umbr of calculatig volums ; (d), th l -orm of us of l -orm o th odal valus i, i,..., ; (), th ma l -orm of th ratio of to th umbr of calculatig volums ; (f), th global rror, dtrmid by th us of l -orm, of th valus of th odal rrors i, i,..., ; (g), th ma l -orm of th ratio of to th umbr of calculatig volums ; (h) /, th l -orm of th ma squar odal rrors ( i, i,..., ), i.., / i i (6) (i), th applicatio of l -orm o th odal valus i, i,..., ; (j), th ma l -orm of th ratio of to th umbr of calculatig volums ; (j) /, th l -orm of th ma squar odal valus ( i, i,..., ), i.., / i i (7)

5 rocdigs of NIT 00 opyright 00 by ABM 3 th Brazilia ogrss of Thrmal Scics ad girig Dcmbr 05-0, 00, brladia, MG, Brazil (k), th global rror dtrmid by applicatio of th l -orm o th odal rrors, i,..., ; (l), th ma l -orm of th ratio of to th umbr of calculatig volums ; (m), th applicatio of th l -orm o th odal valus i, i,..., ; (), th ma l -orm of th ratio of to th umbr of calculatig volums. 3. Rsults Th rsults prstd blow cosist of th dtrmiatio of th practical ad thortical ordr of accuracy for th odal ad global rrors, cosidrig th us of th abov dscribd mtrics. Th practical ordrs of accuracy wr obtaid with ad (qs. (7) ad (8)). Th thortical ordrs of accuracy wr dtrmid cosidrig th dfiitios of accuracy ordr, local rror, global rror ad aalytical quivalc btw vctor orms (qs. (4), (5), (6), (9) (0), (), () ad (3)). Basd o th calculatio of ad, a ivstigatio was also mad of th practical ordrs of covrgc gratd by th applicatio of th sv mtrics o. Ths rsults wr cofirmd cosidrig th algbraic dvlopmt applicatio of th mtrics o q. (5). 3.. Accuracy ordr () of th odal rror i I th first, scod ad third-ordr umrical schms it was foud that, for th variabl [ ( )],,. Ths rsults cofirm th thortical ordr of accuracy () of th odal rror. 3.. Accuracy ordr of ad Aalytically, cosidrig qs. (6) ad (9), it is possibl to idtify th ffct causd by o th ordr of accuracy of th odal rror. I othr words, th applicatio of l -orm to obtai th global rror rsults i D c h (8) whr c ci, i,..., rprsts th umbr of volums ad D. h th siz of th calculatig domai. i i po ivstigatig for th first-ordr umrical schm, w foud that bhavior ca b vrifid i q. (8), whr lads to: wh h 0. This D c. I this cas, it was foud that: 0. 0 is du to th fact that th global rror dos ot approach zro with th rfimt of M ( h ) ( 0, h 0). Hc, th umrator of q. (7) bcoms ull. O th othr had, is justifid by th covrgt bhavior of. That is, for th first ordr schm, th umrator of q. (8) approachs th domiator, with h 0. or th scod ad third ordr schms, th valus obtaid for ad cofirmd th dgratio of o uit o th valu of, rsultig from th applicatio of (,, h 0). Aalytically, th us of th ma l -orm o th valus of th odal rrors lads to c h (9) This bhavior was also foud i th thr umrical schms (ig.). I othr words, for with h 0. o has:,, 3..3 Accuracy ordr of ad osidrig th us of th l -orm o q. (9), o has

6 rocdigs of NIT 00 opyright 00 by ABM 3 th Brazilia ogrss of Thrmal Scics ad girig Dcmbr 05-0, 00, brladia, MG, Brazil whr * c h (0) max{,..., }. or this mtric, i th thr umrical schms adoptd, w foud that th valu of * c c c was maitaid. I othr words,, for h 0. With rgard to th mtric was foud to icras by o uit. That is, for algbraic xprssio obtaid for i th thr umrical schms, th ordr of accuracy of th local discrtizatio rror, usig q. (0):, with h 0. This bhavior was also foud i th c D * h () 3 (Schm 3) (Schm 3) Ordr (Schm ) (Schm ) 0 (Schm ) (Schm ) log(h) igur. ad for 3..4 Accuracy ordr of, ad / W cosidrd hr th global rror dtrmid by, Aalytically, cosidrig qs. () ad (0), o fids that ad th mtric / dtrmid by q. (6). c h D c h () * * / I othr words, th ordr of accuracy rsultig from Th calculatio of th practical ordrs of blogs to th itrval [, ]. o th umrical schms with, ad 3, rspctivly, corroborats this rsult. I othr words, umrically, prstd, with h 0, for th thr umrical schms adoptd (ig. ). Aalogously to th obtaimt of q. (), for o has th followig rlatioship: c D c (3) D * * / h h

7 rocdigs of NIT 00 opyright 00 by ABM 3 th Brazilia ogrss of Thrmal Scics ad girig Dcmbr 05-0, 00, brladia, MG, Brazil whr is ca b s that th ordr of accuracy rsultig from calculatio of th practical ordrs of h 0. Aalytically, th mtric blogs to th itrval [, ]. I th, for th thr umrical schms, it was foud that, with / ca b aalyzd cosidrig qs. (6), (9) ad (6), / c h (4) whr c ( c,..., c ). osidrig qs. (), (3), (8), (9) ad (0), o fids th followig rlatioship: * c h / c h (5) Th umrical rsults obtaid for / idicatd th prmac of th ordr of accuracy of th odal rror i all th umrical schms adoptd (,, h 0). It ca thrfor b statd that / maitais th ordr of accuracy of th umrical schm adoptd. I th works of alcão t al. (006) ad Roy (005), this mtric is usd for purposs of umrical vrificatio..5.0 (Schm 3) (Schm 3) Ordr.5.0 (Schm ) (Schm ) (Schm ) (Schm ) log(h) igur. ad for 3..5 ovrgc ordr of ad po ivstigatig th bhavior prstd by, i th thr umrical schms adoptd, was foud to b costat ( ). I this cas, could ot b dtrmid bcaus, h 0 (ulimitd aalytical solutio). Th aalytical xprssio for, cosidrig q. (5), is dtrmid by h.. h (6) sig liar approximatio basd o powr sris (Kryszig, 999) for th xpotial trm ( h ) with h 0, o has

8 rocdigs of NIT 00 opyright 00 by ABM 3 th Brazilia ogrss of Thrmal Scics ad girig Dcmbr 05-0, 00, brladia, MG, Brazil h (7) By aalogy to th accuracy ordr of th umrical rror, it ca b statd that Th aalytical xprssio obtaid for, o q. (5), rsults i has a ordr of covrgc -. ( ) ( ) (8) I th thr umrical schms, it was foud that for h 0, which cofirms th rsult of q. (8). As for th practical ordr of accuracy, it was foud that, with h 0 i th first ad scod-ordr schms. I th third-ordr umrical schm, w obsrvd:, with h 0. I othr words, for 3, thr was dgratio of o uit i th accuracy ordr of th umrical schm i rspos to th applicatio of ovrgc ordr of, ad / I th thr umrical schms, th mtric prstd costat ( / ). Similarly to th prvious cas ( ), could ot b calculatd. Th aalytical xprssio for, cosidrig q. (5), is dtrmid by h h 3 h h h 0 h (9) ad, by aalogy with th dfiitio of asymptotic ordr (q. 5), o has h (30) This xprssio corroborats th umrical rsults obtaid for ( ), i.., divrgc ( h 0 ). Distict bhaviors wr idtifid for, it was obsrvd that 0 rsult is cofirmd by dividig q. (30) by D / h. I th thr umrical schms, w foud that aalytical xprssio obtaid for ad for th mtric ad cofirms th gativ ordr of covrgc / dtrmid by q. (7). I th calculatio of ad, / with h 0, i th thr umrical schms. Aalytically, this / / for h 0. This rsult is cofirmd by th, cosidrig qs. (5) ad (7), i.., / 4 3 (3) Th practical ordr of covrgc ( ad ) of / prstd a bhavior similar to. That is, it prsrvd th accuracy ordr of th first ad scod-ordr schms, ad dgratd by o uit th valu of th accuracy ordr of th third-ordr schm.

9 rocdigs of NIT 00 opyright 00 by ABM 3 th Brazilia ogrss of Thrmal Scics ad girig Dcmbr 05-0, 00, brladia, MG, Brazil 3..7 ovrgc ordr of ad I th thr umrical schms mployd hr, th calculatios of ad ld to: ad 0 for h 0. Ths valus ar cofirmd by th coditio imposd o th right boudary of th problm modl (Sctio 3.). I this cas, th rfimt of M ( h) ld to,. Aalytically, this rsult ca b valuatd cosidrig th us of th l -orm i q. (5) ad liar approximatio basd o powr sris (Kryszig, 999). 4. ONLSIONS A aalysis was mad of th ffct causd by th us of vctor p-orms to obtai th global discrtizatio rror, whr th discrtizatio rror was dfid as th diffrc btw th xact aalytical solutio ad th umrical solutio for a giv variabl of itrst. With th dfiitios of th ffctiv ordr, appart ordr ad asymptotic ordr of th rror, ad with th covrgc of ths ordrs i th umrical xprimts, th paramtr ordr of accuracy () was adoptd. Th ivstigatio th focusd o th bhavior of this ordr i th local ad global rrors. Accordig to th dfiitio of th rror, th variatio of is qual to th variatio of its rror (), with th rfimt procss of M ( h ), i..,. Howvr, sic th vctor orms addrssd i this work (rprstd by ) do ot charactriz liar oprators, o has:. As a rsult, th rspctiv ordrs of accuracy (for k k k ) ad of covrgc (for ) did ot prst a dirct rlatioship. k k By mas of algbraic dvlopmt ad umrical xprimtatio, w foud that ca b dgratd, lvatd or maitaid usig vctor p-orms. Basd o th rsults obtaid i th calculatio of th global rror, w foud that th mtrics:, / ad maitai th ordr. 5. AKNOWLDGMNTS Th authors thak Th NISAÇO rogram of th AB (Brazilia Spac Agcy), Nq (oslho Nacioal d Dsvolvimto itífico Tcológico, Brazil) ad udação Araucária (araá) for thir fiacial support. Th first author is scholarship of Nq. 6. RRNS arptr, M.H., Kdy,.A., Hstr, B., Vik, S.A. ad Vatsa, V.N., 005, ourth-ordr Rug-Kutta Schms for luid Mchaics Applicatios, Joural of Scitific omputig, Vol. 5, No., pp alcão, J.A.., rrira d Souza,.J.A., ad Bosschrs, J., 006, A vrificatio study o low-ordr thr dimsioal pottial-basd pal cods, omputrs ad luids, Vol. 35, pp rzigr, J.H. ad ric, M., 00, omputatioal Mthods for luid Dyamics, 3ª d., Sprigr, Brli. Garby, M. ad icard,., 008, A cod-idpdt tchiqu for computatioal vrificatio of fluid mchaics ad hat trasfr problms, Acta Mch Si, Vol. 4, pp Golub, G. H. ad Va Loa,., 996, Matrix omputatios, 3ª d., Johs Hopkis rss. Ju, L., Tia, L. ad Wag, D., 009, A postriori rror stimats for fiit volum approximatios of lliptic quatios o gral surfacs, omput. Mthods Appl. Mch. grg, Vol. 98, pp Kryszig,., 999, Advacd girig Mathmatics, Wily, Nw York. Marchi,.H. ad Silva, A..., 00, idimsioal umrical solutio rror stimatio for covrgt appart ordr, Numrical Hat Trasfr, part B, Vol. 4, pp Mathou, G., atao,. ad Dimotakis,.., 008, Vrificatio of a fluid-dyamics solvr usig corrlatios with liar stability rsults, Joural of omputatioal hysics, No. 7, pp Myrs, J., Gurts, B.J. ad Sagaut,., 007, A computatioal rror-assssmt of ctral fiit-volum discrtizatios i larg-ddy simulatio usig a Smagorisky modl, Joural of omputatioal hysics, Vol. 7, pp Roach,.J., 998, Vrificatio ad validatio i computatioal scic ad girig, Hrmosa, Albuqurqu, SA. Roy, J.., 005, Rviw of cod ad solutio vrificatio procdurs for computatioal simulatio, Joural of omputatioal hysics, No. 05, pp Simos,.D. ad Str,., 003, Vrificatio ad validatio of RANS mauvrig simulatio of sso Osaka: ffcts of drift ad ruddr agl o forcs ad momts, omputrs & luids, Vol. 3, pp Vrstg, H.K. ad Malalaskra, W., 007, A Itroductio to omputatioal luid Dyamic, Th iit Volum Mthod, ª d., Harlow, glad.

10 rocdigs of NIT 00 opyright 00 by ABM 3 th Brazilia ogrss of Thrmal Scics ad girig Dcmbr 05-0, 00, brladia, MG, Brazil 7. RSONSIBILITY NOTI Th authors ar th oly rsposibl for th pritd matrial icludd i this papr.