THE APPLICATION OF THE BOUNDARY ELEMENT METHOD TO SOLVE VISCOUS FLOW PROBLEMS BASED ON PRIMITIVE VARIABLES FORMULATION

Size: px

Start display at page:

Download "THE APPLICATION OF THE BOUNDARY ELEMENT METHOD TO SOLVE VISCOUS FLOW PROBLEMS BASED ON PRIMITIVE VARIABLES FORMULATION"

Gary Johnston
5 years ago
Views:

1 TE APPCATON OF TE BOUNARY EEMENT METO TO SOVE VSCOUS FO PROBEMS BASE ON PRMTVE VARABES FORMUATON M. F. C.. Santos UNESP - Facldad d Engnhara d aratngtá A. r. Arbrto Prra da Cnha, aratngtá - SP - Brazl flacaz@fg.nsp.br. R. Barbosa. F. F. M Carnro nsttto Tcnológco d Aronátca CTA-TA-EM/EME São osé dos Campos - SP - Brazl barbosa@mc.ta.br ABSTRACT Th bondary lmnt mthod s appld to th solton of ncomprssbl fld flow problms gornd by th contnty and Nar-Stoks qatons. Th dffrntal qatons ar transformd nto ntgral qatons. ndcaton of th transformaton s gn n dtal. Applcaton to smpl flow cass sch as th drn caty and forward facng stp s prsntd. Conrgnc dffclts ar ndcatd, whch ha lmtd th applcatons to flows of low Rynolds nmbrs.. Ky ords: Bondary lmnt, Nar-Stoks, nmrcal mthods NTROUCTON Th Bondary Elmnt Mthod (BEM) s a rlatly rcnt tchnq that has fond grat applcaton n sral flds of ngnrng. Ths tchnq has rcd ncrasng attnton of a grat nmbr of rsarchrs n mchancs of flds fld, as on can s by th ncrasng nmbr of artcls pblshd n th ltratr. Crtanly, that s th ara whr BEM srs ar facng on of th bggst obstacls, rlatd to th strong non-lnarty of th conct trms, that has bn makng th conrgnc of th soltons for hgh Rynolds nmbr flows a dffclt task. Bsds, many othr aspcts rlatd to th nmrcal mplmntaton ha also contrbtd to lmt th conrgnc of rslts and, n som cass, to rdc accracy, sch as, hypr and spr snglar ntgrals, qas-snglar ntgrals, cornr ffcts, dscrtzaton ffcts and so on. Ths work ams at th applcaton of ths mthod to th solton of stady stat, two-dmnsonal lamnar flow of ncomprssbl scos fld. Opton was mad to formlat th problm n prmt arabls (prssr and locty) that, althogh t sms to b mor stabl, t s rcognzd as on of th most dffclt approachs snc t s mor nolng from th mathmatcal pont of w, n ordr to obtan th ntgral qatons. Th ntgral qatons wr drd from th Nar-Stoks qatons, sng th ghtng Rsdal Mthod. All arabls nold n th solton wr obtand throgh th ntgral qatons, xcpt for th locty componnt drats n x- and y-drctons that wr obtand by th Fnt ffrnc Mthod for th sak of smplcty. Th bondary was dscrtzd by constant typ lmnts, and lnar tranglar clls wr adoptd n th doman dscrtzaton. Th nd for solton of th systm of partal dffrntal qatons whch modl th flow of a fld n channls lk pps, blad passags, nozzls and othrs, appard wth th ry day th fld flow was modld. Th dffclts nold n obtanng closd soltons, n for ry smpl flows, rqrd th dlopmnt of clr tchnqs bt only wth th applcaton of nmrcal soltons to that systm of qatons, som flows of practcal ntrst wr calclatd. Sral comptatonal tchnqs ha bn sd: fnt dffrnc, fnt lmnt, fnt olm and bondary lmnt to nam th most known. As nw algorthms wr dscord and fastr comptr was prodcd, ach of thos mthods old n all aras n th past yars. Fnt dffrnc mthods ha bn, and alrady ar, mplmntd to sol flow problms. Fnt lmnts gand attnton n th past dcads; n th snts t was stll crawlng. Both ar bass for commrcal cods for th solton of flows of almost ry knd. Comptr ffort has bn lmtng th applcaton of th nmrcal mthods n th sns that ry nw mthod of solton dscord clams rdcton n CPU tm and storag rqrmnts. Bt th rasons that ths mthods ar so CPU tm and storag hngr ar ntrnsc. Th bondary lmnt mthod, nrthlss, has progrssd dffrntly dpndng on th aras whr t has bn appld, fastst n aras rlatd to sold mchancs, slowst n th ons rlatd to fld mchancs. Th mthod s smlar to th fnt lmnt mthod. hl th lattr sols an algbrac systm of qatons obtand from ntgrals or lmnts n whch th olm s ddd (fnt lmnt), th formr sols ntgrals or th bondary of sch lmnts (bondary lmnt). Srfac ntgrals ar obtand by transformaton of olm ntgrals applyng rn-ass thorm. So, th bondary lmnt tchnq rdcs of on th nmbr of dmnsons of th problm. spt ntgral mthods wr aalabl many dcads ago for th applcaton to flow problms of practcal ntrst, a comprhns stdy of th formlaton and applcaton to flow problms ar stll bng consdrd 68 Engnhara Térmca (Thrmal Engnrng), No. 5, n 004 p

2 mor rcntly, as thy ar xpctd to allat th storag and hopflly CPU tm snsbly, bcas roghly th dgrs of frdom s rdcd n th bondary lmnt mthod. Althogh ths apparnt adantag, rqrng lss comptatonal ffort whn olm ntgrals ar transformd nto srfac ntgrals, som dsadantags ars, sch as hghr mathmatcal complxty n ordr to gt an sabl comptatonal formlaton; th nd for th calclaton of nfnt ntgrals; flly poplatd matrcs whos nrson s not so smpl as th band matrcs n th fnt dffrnc and fnt lmnt schms. n ordr to aldat th comptatonal modl th drn caty and stp channl flow problms wr adoptd as tst cass. Most mportantly, an attmpt to xtnd th applcaton of th mthod to sol th cascad flow of axal trbomachnry was mad, whch rprsntd th most challngng targt of ths work. Th man dffclty facd was d to th lack of conrgnc of rslts for hgh Rynolds nmbr flows. STATEMENT OF TE PROBEM t and b th doman and rspct bondary. Th doman can b a comprssor ntr-blad passag and th bondary th sold srfacs of two consct blads n a cascad and th hb and casng srfacs, for th xampl of on of th gomtry of ntrst. Ths s consdrd a ry complx flow whch n th most dlopd nmrcal mthods ha problms n prdctng accrat solton. Th fld, n ths cas, cold b ar, consdrd as a prfct gas, comprssbl and scos. t (x,y) b a pont of n R. An ncomprssbl stady flow can b modld by th consraton qatons of mass, x- and y-momntm, as gn by Eqs., and 3, rspctly, accordng to Santos and Barbosa and Santos : 0 () () (3) Th arabls ar, flow rlat locts n th x and y drctons;, P statc dnsty and prssr;, absolt and knmatcs scosts; k, c, c p thrmal condctty and spcfc hats at constant olm and prssr. n ordr to gnralz th problm, lt th followng chang of arabls (Eq. 4) tak ffct n th consraton qatons: x x V P P P V V y y V V R whr th proprts wth th nfnty symbol ndx rfr to th far stram condton. R and Pr ar th Rynolds and Prandtl nmbrs, rspctly. Th qatons of consraton of mass, x- and y- momntm bcom, aftr th chang of arabls, approprat collcton of trms and droppng th orbar: 0 (4) (5) R R (6) R R (7) Th arabls of th problm, consdrng a prfct gas fld, ar, and P. Thrfor, th systm of partal dffrntal qatons can b rwrttn n matrx form as Eq. 8: U B (8) whr s a lnar partal dffrntal oprator, U s th nknown ctor P t and B th ctor of nonlnar trms. pndng on th assmptons mad, and B can tak dffrnt forms. For nstanc, ctor B can b lnarsd and ts lnar trms ncldd n. and dnots ln and colmns, rspctly, of th matrcs. t, for th momnt, all non-lnar trms b ncldd nto B. Thn t U P (9) For th sak of smplfcaton, lt th dffrntal oprators, dfnd by Eq. 0, b ncorporatd n th qatons, Engnhara Térmca (Thrmal Engnrng), No. 5, n 004 p

3 (0) and gnor th trms contanng th araton of and k. Thn, R R, 0 U and P B R R 0 () On looks for th solton of Eq. 8 sbjct to th bondary condtons approprat for th problm. Thos condtons wll b statd latr, whn th applcatons ar shown. TE METO Thr s no known analytcal solton for th Eq. 8. t U ~ b an approxmat solton n th sns U ~ B R 0 () that s, U ~ dffrs ry lttl from U bt t s not qal to U. On sks to fnd U ~ sch that U Bd Rd 0 (3) whr s a wght fncton. Th nam bondary lmnt mthod coms aftr th way s dfnd and th way Eq. 3 s sold. TE ET FUNCTON Th wght fncton, sally calld fndamntal solton, s dtrmnd from Eq. 4. t AX E (4) b a lnar systm of qatons whr A and E ar known matrcs, s a mltplyng scalar and X th nknown solton ctor. Vctor X s obtand mltplyng both mmbrs of Eq. 4, to th lft, by A, whr A dt A cofa t. t b calclatd aftr th axlary problm gn by Eq. 7, ddcd from Eqs. 5 to 7 blow. Thn or Fnally, X Y (5) t X Y cof dt X Y (6) t cof, whr dt X Y dt X Y (7) (8) Ths, f on fnds th fncton dfnd by th charactrstc Eq. 7, on blds th wght fncton. Ths procsss ar nthr straghtforward nor asy. Matrcs and M t K cof, th adjont of and ts transposd cofactor ar: M R K so that dt[ R R R R R R R 4 R 0 (9) ] R (0) Th Eq. 8 has th solton gn by Eq. : X, Y r lnr R 8 () Thrfor, th wght fncton s dtrmnd. Nxt stp s th ntgraton of Eq Engnhara Térmca (Thrmal Engnrng), No. 5, n 004 p

4 SCRETZATON Th rn-ass Thorm appld to Eq. 3 allows th doman ntgral b transformd nto an ntgral or th bondary. Ths contor s sally ddd nto n bondary lmnts. Thrfor n U Bd FU, B, d 0 () whch ndcats that th systm of dffrntal qatons (Eq. ) has bn transformd nto a systm of algbrac qatons (Eq. ) that nols th als of th arabls at ach bondary lmnt. Th solton on th bondary may b fond prodd th als of th arabls at th lmnts on th bondary ar dtrmnd. Aftr th applcaton of th rn-ass thorm and ntgraton by parts or, Eq. 3, that holds also for ry bondary lmnt, s dtrmnd: U d XS X,Y X X,Yd X K K X X, Yd X B k k (3) n a lmt procss at whch Y approachs X, consdrng th snglarts of K, on arrs at Eq. 4, sally calld th bondary ntgral qaton: C K YUY XS X, Y X X, Y B k X X, Y d X k k (4) whr smmaton s mpld by rpatng ndx. t s worth mntonng that th scond mmbr of Eq. 4 comprss ntgrals or th bondary and or th doman, ths d to th non-lnar conct trms. C K (y) s th tnsor coffcnt dpndnt on th bondary gomtry. f y ls or th bondary and th bondary s thr rglar, ts al s ½; f th pont y ls n th doman, ts al s ; f y s n th bondary and thr s dscontnty of th drat of, thn ts al dpnds on th angl formd by th lft and rght tangnts. tals of th trms of Eq. 3 ar: S K d X, Y j R j K K,j jk, ~ 3K n j X Rp and j j j j whr coma (,) ndcats drat wth rspct to th followng ndx and smmaton s mpld by rpatng ndcs. Slctng th constant lmnt, on has n C n K (5) Thrfor, qaton (4) bcoms, aftr ntgraton YU Y S X,Y X,Y d k X X, Yd X B k k (6) t s worth notng that th trms corrspondng to th arabls als n ach lmnt can b factord so that th systm of algbrac qatons bcoms thn dnt. MPEMENTATON Bondary t th bondary b ddd nto m constant lmnts, wth th collocaton ponts (nods) placd at th md pont of ach lmnt. Applcaton of Eq. 6 to thos m nods gs a st of 3m qatons wth 3m nknowns. For th solton of ths systm of qatons two axlary matrcs ar assmbld, for ach lmnt: (X Y) Sk X, Y X Y X, Y wth, =,. from whch s drd Eqs. 8. g h (Y) (Y) k X Y d X Y d wth, =,, 3. (7) (8) ntgrals of Eq. 8 ar carrd ot nmrcally, sng th ass mthod straght, whn X Y. hn X = Y or f X Y th ntgrands bcom nfnt, rqrng th calclaton n th sns of Cachy prncpal al. Sral tchnqs ar aalabl to prform ths calclatons, all rqrng carfl calclaton of sch ntgrals. oman To calclat th ntgrals or, th doman s ddd nto M lmnts by an approprat nt. Usng th lnarty of ntgrals, on obtans Eq. 8: B X X, YdX k = M j B k X, Yj S (9) j whr j s th ass wght fncton at pont j, S s Engnhara Térmca (Thrmal Engnrng), No. 5, n 004 p

5 th lmnt ara, and s th nmbr of ass ntgraton ponts. Comptr Program n ordr to mplmnt a comptr program, th followng stps ar sggstd: dscrtzaton of th bondary n n lmnts calclaton of th gomtrc proprts for th lmnts n th bondary assmbly of matrcs g and h for ach lmnt and assmbly of matrcs and for all lmnts n th contor. To do ths, dfn th ctors of nknown arabls at th bondary: t m m m m and wrt Eq. 9 n th form t (30) C (3) or, makng C, on arrs at Eq. 3. (3) applcaton of th bondary condtons to Eq. 3. To do ths, not that lmnts of and ha som als prscrbd. Rarrang and sch that th nknown arabls com frst and thn th prscrb als, as ndcatd by Eq. 33. p t t p. (33) ntrnal nods Eqaton 30 ar sd for th calclaton of th ntrnal nods, for whch th constants n C ar qal to. Th solton at th ntrnal nods starts wth a gss of th flow fld and mposton of th bondary condtons. rd nraton For th problms ndcatd blow, th grd was gnratd sng orthogonal straght lns lnkng th oppost srfacs, whch wr ddd nto a nmbr of qal lmnts. For complx bondars t s ndd a spcal grd gnraton rotn. Ths sbjct s not dscssd n ths work, althogh t may b a ry complx task. APPCATON To dmonstrat th applcablty of th mthod, th flow facng a forward stp and th drn caty flow wr slctd. Flow n a box wth mong sld Th drn caty, that s, th flow n a box wth th ppr srfac mong to th rght, was tstd sng th softwar dlopd. A pctr of th flow n th box s shown n Fg. blow. Th bondary condtons ar th no-slp condtons n th box bondars, that s, zro at th nonmong srfacs and th locty of th mong sld at th ppr srfac. Tsts wr carrd ot wth dffrnt grds to chck th snstty of th algorthm to th grd sz. Comparson of th rslts wr carrd ot wth th pblshd data of Tosaka t al 3, as ndcatd n Fg., for th poston x=0.5, Rynolds nmbr of 00. rarrangmnt of matrcs, and accordngly, sch that Eq. 33 arrs: p p pp p p p pp p p (34) Fgr. rn caty flow - Vlocty ctor fld. rarrangmnt of Eq. 34 sch that only th nknowns ar n th lft sd, that s, AX=BP+ (35) Matrx A s almost flly poplatd (dns) so that nrson s tm consmng. ts nrson may b carrd ot by th ass lmnaton algorthm. Solton of Eq. 35 gs th als of th nknowns at th bondary. 7 Engnhara Térmca (Thrmal Engnrng), No. 5, n 004 p

6 Tosaka t al. 3 rslts ths work rslts Fgr. Vlocty profl at th rtcal cntr ln of th caty. (R = 00) REFERENCES Barbosa,. R., Santos, M. F. C.., 998, Th Bondary Elmnt Mthod n Fld Mchancs, n: ATCYM 98, Vol., pp Santos, M. F. C.., 998, Bondary Elmnt Mthod Appld to Fld Flow Problms, Ph.. Thss (n Portgs), nsttto Tcnológco d Aronátca, São Palo, Brazl. Tosaka, N., Kakda, K., Onsh, K., 985, Bondary Elmnt Analyss of Stady Vscos Flows Basd on P-V- U Formlaton, n: Proc. 7nth Confrnc of BEM n Engng., ak Como, pp:7-80(9), taly. Tosaka, N., 989, ntgral Eqatons Formlaton wth th Prmt Varabls for ncomprssbl Vscos Fld Flow Problms, Comptatonal Mchancs, Vol. 4, pp: Flow n a stppd channl Th flow n a forward facng stp s shown n Fgr 3 blow, for th bondary condtons of no-slp on sold srfacs and parabolc profl for th axal locty at channl nlt. Th rslts shown rfr to a grd of 7x5 ponts, Rynolds nmbr of 0. Fgr 3. Channl flow - Vlocty ctor fld CONCUSON Th bondary lmnt mthod can b appld to th calclaton of comprssbl scos flows as dmonstratd throgh th abo xampls. Althogh a ry coars grd was sd, th rslts ar qt satsfactory, captrng rgons of rrs flows as n th cas of th stppd channl. CPU tm was not rcordd for ths work bt, comparng wth othr fnt dffrnc schms sd for th sam prpos, t s consdrably lowr. t s thrfor mportant to nstgat othr applcatons sch as th flow n blad passags, th ltmat goal for th rsarch ndr way. Engnhara Térmca (Thrmal Engnrng), No. 5, n 004 p

Introduction. Nomenclature. Statement of the Problem

Introduction. Nomenclature. Statement of the Problem R. G. R. Camacho and. R. Barbosa R. G. R. Camacho and. R. Barbosa Insttto cnológco d Aronátca 8-9 São osé dos Campos, S. Brazl ramrz@ta.br rgramrz65@hotmal.com barbosa@ta.br h Bondar Elmnt Mthod Appld