SOCRATES Tachng Staff Moblty Program 999-000 DMA-URLS Lctur not on A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry Alssandro Corsn Dpartmnto d Mccanca Aronautca Unvrsty of Rom La Sapnza - Octobr 999 -
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry Indx Acknowldgmnts 3. Introducton. An ntroducton to th ntgral mthods 6 3. Wak rsdual form of th Navr-Stoks problm 8. Fnt lmnt mthod and dscrtzaton 0 5. Fnt lmnt ntrpolaton functons 5 5. Intrpolaton spacs n XENIOS 8 6. Stablzd fnt lmnt formulaton for advctv-dffusv flows 6. Th stablzaton of convcton domnatd flows 6. Th stablzaton of dffuson domnatd flows 3 7. Rfrncs 3
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry Acknowldgmnts Th prsnt manuscrpt contans th nots of th lctur gvn by th author th of Octobr 999 at th Dpartmnt of Flud Mchancs - Tchncal Unvrsty of Budapst. Th work has bn carrd out wthn th framwork of th SOCRATES tachng staff moblty program launchd btwn th Unvrstà d Roma La Sapnza and th Tchncal Unvrsty of Budapst. Th frst author wshs to acknowldg Prof. Tamas Lajos that hosts hm wthn th Ph.D. cours "Flud Mchancs" (TUB cod BMEGEATA08) and Dr. Janos Vad n hs qualty of SOCRATES ducatonal rsponsbl for ts hgh qualty organzaton. Th author s also ndbtd to Prof. V. Naso that n hs qualty of CIRPS (Italan Intrunvrsty Rsarch Cntr for Intrnatonal Co-opraton) drctor has bn th crator of such coopraton. 3
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry. Introducton Th fnt lmnt mthod (FEM) s an approxmat mthod wll known to th ngnrng communty for th soluton of partal dffrntal quatons (PDE) govrnng boundary and ntal valu problm (Hughs, 987a) (Chung, 978). Such a mthod transforms contnuum problms nto systms of algbrac quatons procdng va N N th us of varatonal prncpls or wghtd rsdual mthods to transform PDE nto ntgral quatons; th sub-dvson of computatonal doman nto many small lmnts of convnnt shaps so that a bass for th varabl ntrpolaton could b dfnd. Th rsdual ntgral FE formulaton s abl to asly handl complx gomtrs and drvatv typ (Numann) boundary condtons du to ts ntrnsc proprts of workng of unstructurd doman dscrtzaton. Th FEM orgnally dvlopd n th 950's for arcraft structural analyss has bn succssfully appld to th fld of non-structural problm such as flud flows and lctromagntsm startng from th work of Znkwcz (965). Th clos rlatonshp btwn fnt lmnt analyss and classcal mathmatcal approachs, such th calculus of varatons or th wghtd rsdual mthods, has stablshd th FEM as an mportant branch of approxmaton thory. Varatonal prncpls, usd n th Raylgh-Rtz mthod, unfortunatly cannot always b found n flud problms, partcularly whn th dffrntal quatons ar not slf-adjont. Thus th wghtd rsdual mthod s oftn appld snc t rqurs no varatonal prncpls. Such a mthod utlzs a concpt of orthogonal projcton of a rsdual of a dffrntal quaton onto a subspac spannd by crtan wghtng functons. In th fnt lmnt mthod, w may us thr varatonal prncpls whn thy xst, or wghtd rsduals through approxmatons. Th wghtd rsduals formulaton s basd on th charactrzaton of two sts of functonal spacs. Th frst s to b composd of th approxmaton or tral functons, whl th scond collcton has to contan th varatons or wghtng functons. Although thr ar svral ways of choosng such classs of functons, n fnt lmnt applcatons to flud dynamcs th Galrkn or Bubnov-Galrkn mthod, whr th ntroducd functonal spacs ar composd by dntcal bass functons, s consdrd th most convnnt tool for formulatng FE modls. Two man nstablts orgns could b dtctd n cas of a straghtforward applcaton of th Galrkn FEM n th fld of flud dynamcs of ncomprssbl flows. Such nstablts, drctly rlatd to th mathmatcal charactr of modlng quatons, could b n prncpl controlld by choosng propr fnt lmnt dscrtzatons (n trms of both th msh rfnmnt and th par of prmtv varabls ntrpolatons). Oftn th dynamc rspons of ncomprssbl turbulnt flows (.g. typcal of rotatng and statonary flow phnomna n turbomachns) s controlld by strong advctv and/or dffusv mchansms, that n practc nd prohbtv lvl of dscrtzaton f an accurat soluton has to b achvd. That s som knd of stablzaton s mandatory, and th prsnt work wll b focusd at th dscusson of consstnt wghtd rsdual mthods of achvng accurat and stabl solutons.
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry Th frst nstablty orgn s lnkd to th lack of dffusv trm n th contnuty quaton so that th advctv-dffusv Navr-Stoks systm of quatons s ncomplt parabolc. As a mattr of fact th ncomprssblty condton lads to an ndtrmnaton of th systm of govrnng quatons bcaus th unknown prssurs hav to b computd out of th contnuty quaton, whch plays th rol of an addtonal constrant to th vlocty fld. From a numrcal pont of vw, th soluton to such an ndtrmnaton gvs ras to th 'mxd mthod' whr dffrnt functonal spacs should b chosn for th vlocty and prssur ntrpolaton so that th satsfacton of a stablty condton (known as th Babuska-Brzz condton) s nforcd. Although numrous convrgnt combnatons of vlocty and prssur 'lmnts' hav bn dfnd (.g quadratc-vlocty and lnar-prssur), t s far to not that n gnral thy ar not attractv from an mplmntaton standpont partcularly for thr-dmnsonal computatons. Bsds th prssur nstablty n cas of dffuson-domnatd flows, thr s a scond on orgn typcal of th advcton-domnatd flows drctly rlatd to th numrcal approxmaton of flow phnomna. In othr words, th modlng of non-symmtrc advctv trms mployng symmtrc oprators (such as cntrd fnt dffrncs or Galrkn bass functon) lads to vlocty soluton corruptd by spurous oscllatons 'wggls'. Th magntud of oscllatons s rlatd to th convcton ntnsty (hgh Rynolds or Pclt numbrs) or to th prsnc of downstram boundary condtons forcng rapd chang of flow bhavor. Th only way to lmnat th wggls wthout changng th rsdual formulaton rqurs a svr rfnmnt of computatonal msh n rgons whr strong gradnts occur (.g. boundary or shar layrs), such that locally th flow s govrnd only by dffuson. As a consqunc th computatonal load (CPU tm and storag rqurmnt) rss dramatcally. Th lmts mplct n th classcal rmdal stratgs for th lmnaton of ncomprssbl flow numrcal nstablts, hav provdd th motvatons for dvlopmnt of an altrnatv to th Galrkn formulaton. In rcnt yars th Ptrov-Galrkn wghtd rsdual formulatons hav bn dvlopd as dvcs for th nhancmnt of stablty wthout upsttng of consstncy. Th ladng da s to stablzd an orgnal Galrkn formulatons addng balancng trms that manat from a prturbaton of wghtng functons, gvng rs to Ptrov-Galrkn formulatons abl to crcumvnt th Babuska-Brzz condton (Tzduyar t al.,99) (Hansbo, 995) (Hughs t al., 986) or to ntroduc stramws artfcal dffusvty (Hughs t al., 979) (Hughs, 987b). Th prsnt work s amd at prsntng th stablzaton stratgy mplmntd n th framwork of th nhous mad fnt lmnt basd Navr-Stoks solvr XENIOS (Rspol and Sclan, 99), (Corsn, 996), (Borllo t al., 997b) and dvlopd on th bass of alrady mntond consstnt Ptrov-Galrkn approachs. Th not that follows s frst brfly ntroducd th radrs to th ntgral mthods wth partcular mphass to th wghtd rsduals mthods (Chaptr and 3). Thn, n Chaptr and 5, ar rspctvly dscussd th fnt lmnt dscrtzaton tchnqu and som basc nformaton concrnng th fnt lmnt spacs adoptd wthn th cod XENIOS. Fnally n Chaptr 6, th stablzd formulatons for both advctv and dffusv flow lmts ar prsntd and dtals concrnng th Ptrov-Galrkn stablzd of Navr-Stoks problm for turbulnt and ncomprssbl flow ar also dscussd. 5
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry. An ntroducton to th ntgral mthods Boundary valu problms dfnd by systm of PDE could b xprssd n an quvalnt ntgral form manly on th bass of two approachs. Th frst s th varatonal mthods whch transform th orgnal dffrntal problm ntroducng an ntgral functonal GI (calld varatonal prncpl) constructd as th nnr product of th PDE onto a varaton of th unknown varabl. GI thn could b consdrd as a vrtual work, and th ntgral form of th problm smply clams a mnmum or statonary nrgy condton. Such mthod, on of th most powrful mthods of soluton for ngnrng problms, ncssarly bgns wth fndng th varatonal prncpl. varatonal prncpl dscrtzaton quatons of Raylgh -Rtz PDE nnr product rsdual prncpl quatons of Galrkn Fg.. - Varatonal mthods vrsus wghtd rsdual mthods Gnrally th physcal law that modls th dynamc of vscous flow could not b transformd n a varatonal prncpl, contrary to phnomna such as Stoks or potntal flow. For such a rason th fnt lmnt formulaton of Navr-Stoks problm rls on th us of th scond ntgral approach, that s th wghtd rsduals mthod n ts Galrkn dfnton. Th wghtd rsduals mthod dfns a mor unvrsal approach wth rspct to th varatonal on, bcaus th dfnton of th ntgral problm s obtand rqurng that th nnr product of th rsduals of th PDE to a subspac of wghts s qual to zro. Such orthogonalty condton s quvalnt to forcng th rror of th approxmat dffrntal quaton to b zro n an avrag sns (just a fnt condtons of orthogonalty could b mposd) (Chung, 978). Lt rcall that th nnr product btwn functons s quvalnt to th scalar product of vctors n a Cartsan fram. Th nnr product of orthogonal sts of functons s thrfor dfnd as whr G 3k s th dlta of ronckr. I, I ³ I I G G Q N Q N QN (.) Lt consdr now a gnrc dffrntal quaton of th form subjct to th followng boundary condtons Lu f 0 n R nsd, Drchlt condtons u = g on * g von Numann condtons u, n = h on * h whr nsd dfnng th numbr of spac dmnsons and (.) (.3.a) (.3.b) 6
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry L, s th lnar dffrntal oprator, u, s th unknown varabl, f, s th sourc trm. Th boundary * of computatonal doman s consdrd as th composton of two substs * g and * h, dfnd as follows * * * g g h * ˆ * h (.) whr th symbol dfns th mpty st. Th dfnton of a rsdual formulaton nvolvs th ntroducton of two classs of functons. Th frst class S s ntroducd to dfn an approxmaton of th soluton and t s composd by th canddat tral or tral functons u, that must satsfy th ssntal st of boundary condtons (.3.a). Th scond class W contans th varatons or wghtng functons w, usd for constructng th rsdual orthogonal projctons. Ths collcton s vry smlar to th tral solutons wth th xcpton that thy hav to satsfy th homognous countrpart of th Drchlt boundary condtons (.3.a). Th mpossblty of dfnng ovr contnuous tral functons u and varatons w ( * ), gvs rs to th constructon of fnt-dmnsonal approxmaton of S and W. Ths collcton of functons ar dnotd by S h and W h, rspctvly. Th suprscrpt h rfrs to th assocaton of th approxmat functon spacs to th dscrtzaton of th doman, whch s paramtrzd by a charactrstc lngth scal. Th ntroducton of approxmat tral soluton ~ u nto th (.) wll not satsfy xactly th govrnng dffrntal quaton, and a rsdual or rror appars that s proportonal to th fnt-dmnsonal dscrtzaton of th orgnal contnuous problm Lu~ f (.5) Th applcaton of wghtd rsduals mthod rqurs, as alrady dscussd, to construct th nnr product of (.5) to a st of wghtng functons ~ w w, ~ ³ w ~ d 0 (.6) Th abov condton of orthogonalty prmts th dfnton of a rsdual formulaton consstnt wth th orgnal dffrntal formulaton that s abl to guarant th bst approxmaton proprty (Hughs and Brooks, 98). Th numrcal analyss tool of flow phnomna takng plac n turbomachnry dvlopd wth th fnt lmnt cod XENIOS s focusd on th soluton of th systm of quatons that modls th stady-stat dynamc rspons of ncomprssbl turbulnt fluds. For such a rason t s manngful to brfly rcall th * 7
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry boundary valu problm xprssd for th followng st of flud proprts (x -, x - and x 3 - momntum componnts, prssur, turbulnt kntc nrgy k and vscous dsspaton rat H). Hr th usd frst ordr turbulnc closur follows a two quatons approach mplmntd both n ts sotropc (Jons and Laundr, 97) and cubc non-lnar (Craft t al., 993) vrsons (Corsn, 996) (Borllo t al., 997a). Lt consdr th followng vctor form of th Navr-Stoks boundary valu problm ovr a doman R nsd uju, j V j, j f, momntum quatons (.7) u kk, 0, contnuty quaton and,j,k =,nsd wth followng st of boundary condton ovr * * g * h u g o* g V j n h o* h (.8) In dtal, th advctv-dffusv quatons modlng th consrvaton of turbulnt varabls mployd n a standard ddy-vscosty approach ar U ª P º k, P k, j P UH V k ¹ ¼ u j j t ª P U H P V H ¹ u j, j t, j k, j º H C P U H H ¼ k k, j (.9) wth followng st of boundary condton ovr * * g * h u n 0 o* h wk wn H 0 o* C / P k ky n h 3/ o* g (.0) 3. Wak rsdual formulaton of Navr-Stoks problm To dfn and buld an ntgral fnt lmnt basd Navr-Stoks problm formulaton a wak-global approach (Hughs, 987a) s hr proposd. Th pcularty of such a global approach s that th fnt lmnt formulaton s obtand as a consqunc of th dscrtzaton th wghtd rsdual form of th problm ovr th * In ordr for u to b th xact soluton of (.) t s ncssary to mpos th orthogonalty of th rsduals to an nfnt st of projctng drcton. 8
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry computatonal doman * dcomposd nto small lmntary doman (). In ordr to achv a manngful fnt lmnt rsdual formulaton n vw of th charactr of govrnng quatons (.9), that contan both frst and scond ordr trms, th classs of tral and wghtng functons should satsfy gnralzd proprts of contnuty and ntgrablty ovr th closd doman * R nsd. Such a global approach s quvalnt to th local on suggstd by Chung (978), whr th fnal form of th ntgral problm s a consqunc of th composton of local wghtd rsdual formulaton appld n ach lmntary doman on th bass of functons ~ u and ~ w dfnd on as w~ w~ w~ C 0 ^ ^ ` ` (3.) u~ uu ~ ~ C 0 that s a pcws contnuty condton on th dscrtzd doman. Th dfnton of a wak ntgral formulaton for th problm undr xam rqurs th ntroducton of two functon spacs S and W and thr fnt dmnsonal approxmatons. S h s th approxmat collcton of tral functons dfnd as S h = ^u ~ u ~ h,u ~ g o ` H * g (3..a) whl W h s th class of wghtng functons W h = ^w ~ w ~ h,w ~ 0 o ` H * g (3..b) It s worth to not that th componnt functons satsfy complmntary sts of boundary valus. Furthrmor to prsrv th sns of th applcaton of Grn-Gauss thorm to th momntum quaton, such spacs must b composd by functons that must b drvabl and wth th drvatv squar-ntgrabl ovr th global doman. Synthtcally t s now possbl to stat th followng sutabl wak formulaton of Navr-Stoks problm obtand from th gnral xprsson (.6) ns d ( ) d f d ³ w u u ³ w V ³ w j,j ns j, ns (3.3a) Intgratng by parts th dffusv flux ntgral (wth scond ordr drvatvs) and by vrtu of th Grn-Gauss thorm, th wak form rads as d d f d h d* ³ w u u ³ w V ³ w ³ w ³ w ns c j, j u d 0, u g o* g ns, j ns * h (3.3b) whr w ns ndcats th class of wghts appld to th momntum quatons (togthr wth th two advctvdffusv consrvaton quatons of th turbulnt varabls), and w c th wghts appld to th scalar contnuty quaton. Fnally, w h ndcats th rstrcton of wghtng functons on th boundary of computatonal doman. 9
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry Th frst thng to gv attnton s that th boundary condton V j n h o* h s not xplctly mntond n th wak statmnt (3.3), but t s mpld n th varatonal formulaton that s only fulflld n an approxmat sns du to th wghtd rsdual approach. Boundary condtons of ths typ ar rfrrd as natural or Numann condtons. Othrws th Drchlt or ssntal boundary condtons ar xactly satsfd by th approxmat soluton functon (3..a). It s also mportant to hghlght that th ntgral proprts mposd to th classs S h and W h gv sns to th ntgral trm wns, j d ³ V, whr xplctly appars th dvatorc strss tnsor. By vrtu of th common ntgraton doman and of th ncomprssbly condton, th systm of ntgral quatons (3.3) could b also transformd as ³ w ns d d f d d* d 0 u j u, j ³ w ns, V j ³ w ns ³ w * h h ³ w u whr th contnuty condton s addd to th momntum quaton as an addtonal constrant mposng to th vlocty fld a zro dvrgnc, and ts wghtng functon w c plays th rol of a Lagrangan multplr n th boundd problm (3.). c, (3.) As alrady mntond n th ntroductory Chaptr, th par of functons usd to ntrpolat and wght vlocty and prssur must n prncpl satsfy th Babuska-Brzz stablty condton. Th mxd fnt lmnt formulaton mplmntd n XENIOS adopts a quadratc varaton for w ns and lnar varaton for w c.. Fnt lmnt mthod and dscrtzaton Frst stp toward th fnt lmnt form of th wak formulaton (3.) s th dfnton of th functons that compos th spac of solutons S and wghts W, usng an approxmat rprsntaton of th contnuum doman on th bass of th nformaton locatd n a sutabl numbr of ponts (calld nods) wthn ach lmnts. In th fnt lmnt mthodology mplmntd n XENIOS, th gnrc tral functon s S h s buld on * wth th followng structur s v g (.) whr g dfns th whol st of ssntal boundary condtons and v s a functon dfnd on *. Imposng th Drchlt condtons to (.) follows s v / * g g o v 0 * g (5.) thus t s possbl to conclud that th approxmat bass functon v, as th fundamntal componnt of S h dfnton, satsfy th condtons statd for th wght spac of xstnc W h (3..b). As a consqunc of th proprty (5.) follows th possblty of choosng dntcal bass functon n ordr to approxmat th soluton and to wght ts rsdual, and th ntgral mthod basd on such a choc of functon 0
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry collcton s namd th Galrkn mthod. Such fundamntal componnt functon s calld bass or shap functon. Lt consdr agan th closd computatonal doman * R nsd, and a st of nodal ponts l. In th Galrkn rsdual mthod th structur adoptd for th bass functon rsts on a dscrt rprsntaton of doman that uss a fnt numbr l of nodal ponts (that s a fnt numbr of nformaton) to buld th nsung polynomal structurs whr c l, nods c l I l l w o wghtng functons nods d l I l l ar th nodal valus of wghts, u o tral functons (.3) (.) d l, I l, ar th nodal valus of unknown varabls, th bass functons dfnd n ach nods to ntrpolat th bhavor of soluton and of varaton on th computatonal doman. Each functon blongng to th collctons S h and W h could b thus approxmatly dfnd as a lnar polynomal wth constant nodal coffcnts and ntrpolatng shap functons I l, that fulfll th followng proprts, for ach nod l I l l o l m I 0 m ] z l (.5) Fg.s. and. show, n th cas of a lnar approxmaton (lnar shap functon) ovr a -D closd doman =,0], th form of ach nodal ntrpolatng functons and th ffct of thr lnar combnaton to dfn on th whol doman th rsultng ntrpolatd soluton or varaton. A n+ Fg.. Nodal lnar shap functons
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry Fg.. Rsultng ntrpolatd functon It s worth to not that ntroducng th dfnton of tral (.) and wghtng (.3) functons wthn th wak rsdual formulaton (.6) or (3.), th st of nodal valus of th varatons c l could b smplfd as common constant factors on th nnr product. As a consqunc of such poston n th dscrtzd rsdual Galrkn formulaton, th constructon of nnr product btwn th rsduals of systm of quatons and th st of wghts s smply obtand wth rfrnc to th nodal shap functons. That s wth rfrnc to th lmntary porton of th contnuum doman locatd by th st of nods. Lt now consdr th ffct of th applcaton of th doman subdvson nto lmnts, shown n Fg..3 and just dscussd, to th wak formulaton of Navr - Stoks problm (3.). nods Fg..3 Computatonal doman fnt lmnt dscrtzaton Th doman s shard nto a fnt numbr NUMEL of lmntary sub-domans, wth =,..., NUMEL. * s th boundary of th lmnt and th dscrtzaton fulflls th nsung proprts ˆ (.6) (.7) Fnally an ntror boundary could b dfnd as * * * nt (.8)
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry Furthrmor, lt assum th contnuty of wghts and soluton functons n ach lmnt, w~ w~ w~ C 0 ^ u~ uu ~ ~ C 0 whras thr frst ordr drvatvs could b dscontnuous across th ntr-lmnt boundary. ^ ` ` (.9) Th dscrtzd formulaton, wth a smplfd notaton, rads as u 6 ³ w u u d 6 ³ w d 6 ³ w n ] d 6 ³ w u d 6 ns U j, j ns, j ns j * c, * nt ns d h d* * h o * g ³ w f ³ w h g 8 8 (.0) whr s th lmntary ntgraton doman, * nt s th composton of ntr-lmnt boundars that fall n th ntror of, > j j@ V n V n V n j + j j - j, s th balanc of dffusv fluxs across ntr-lmnt boundars, thr ntroducton n th dscrtzd form of th dffusv ntgral s a consqunc of th lmnt-ws contnuty charactrstc of th usd shap functons (.9). Lt consdr, now, th lmntary doman * gnrcally locatd wthn th doman. Th us of approxmat tral and wghtng functons, wth an lmntary doman of dfnton (.9), allows th formulaton of local lmntws rsdual formulaton of Navr-Stoks problm as ³ wns U uj u, j d ³ wnsvj, j d ³ wns f d ³ wcu, d 0 (.) Th prsnc of Cauchy tnsor dvrgnc 8 j, ncludng prssur and dffusv trms, lads to th applcaton of an ntgraton by parts and of th Grn-Gauss thorm n ordr to lowr th scond ordr dffrntal trm and accordng to th local contnuty proprts of usd bass functons. Th rsdual du to th strsss dstrbuton wthn ach lmnt s thn computd as th sum of volum contrbuton drctly lnkd to th strss tnsor tslf and a contrbuton dpndng from th fluxs of normal and tangntal strsss on th lmnt boundary * ³ w V d ³ w V d ³ w V n d* ns j, ns, j ns j j * (.) Substtutng (.) n (.0) th lmntary rsdual formulaton rads as ³ w u u d ³ w d ³ w n d* ³ w f d ns U j, j ns, V j nsv j ns * ³ w u c, d 0 (.3) 3
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry Th composton of (.3) wrttn for ach lmnt lads to th followng global formulaton 8 8 $ w u u d $ w d $ w n ] d $ w u d ³ ns U j, j ³ ns, j ³ ns j * ³ c, * nt 8 $ ³ wns f d ³ wh j n d* * h (.) whr th collcton of lmntary fluxs across th ntr-lmnt boundars gvs rs to th followng ntgrals > @ 8 8 8 ³ w n d ³ w n d ³ w n d 6 ns j * 6 ns j * h j * * * nt * h (.5) In (.5) s ntrstng to dstngush a frst ntr-lmnt balanc trm (consqunc of th lmnt-ws contnuty of bass functon) and a scond flux ntgral across th global boundary porton * h wr th natural condtons ar appld. It s worth to not that approachng th ntr-lmntary boundars th frst ordr drvatvs of th adoptd tral and wghtng functons gv rs to qual absolut valus wth oppost sgn. Such condtons show th prfct balanc of dffusv ntr-lmntary ntgrals Vn V n o > V j n j@ 0 (.6) As a consqunc th dscrtzd rsdual formulaton of Navr-Stoks problm (.) could b wrttn as 8 w u u d w d w u d w f d w h d 6 ³ ns U j, j 6 ³ ns, j 6 ³ c, 6 ³ ns ³ h * * h (.7) As a mattr of fact th substtuton of natural boundary condtons (V j = h on * h ) n (.), has prmttd to dmonstrat th quvalnc btwn th Navr-Stoks rsdual formulatons undr xam N N wak formulaton, dfnd on th global computatonal doman as a sort of nrgtc condton to fnd th soluton; rsdual formulaton, dfnd on an lmnt bass as a local orthogonalty condton btwn th rsduals and th varatons * It s thus possbl to conclud that th modlng of flow bhavor usng an approxmat fnt lmnt mthod usually rqurs th nsung stps N N N th approxmat rprsntaton of contnuous computatonal doman, as th composton of small subdomans calld lmnts ; th dfnton of approxmat soluton, obtand by th ntrpolaton of th unknown valus n a fnt numbr of nodal ponts dfnd n ach lmnt usng a collcton of bass functons; th dfnton of an ntgral quaton for ach unknown varabl, by us of a rsdual prncpl.
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry 5. Fnt lmnt ntrpolaton functons On of th crucal aspct n th applcaton of fnt lmnt mthods to CFD concrns wth th choc of shap or bass functons, that s th ntrpolatng functons usd to approxmat th solutons and thr varatons on th dscrtzd computatonal doman. Th structur of such functons and thr ordr dtrmn th accuracy of FEM. Th shap functons could b gvn by both polynomals (wth svral ordrs) and product of polynomals to trgonomtrc or xponntal functons. For nstanc th adopton of polynomal xprssons allows a lnar approxmaton of a varabl n a quadrangular two-dmnsonal lmnt by us of ts valus at th four cornr nods. Agan n a quadrangular two-dmnsonal lmnt a polynomal could smulat a quadratc law just addng fv nods, (four nods n th mddl of ach lmnt dg and th last n th lmnt cntr). Smlarly work polynomal xprssons mor complx as th Lagrang ons, allowng dffrnt ordr of approxmaton n ach Cartsan drctons. As a consqunc of th ncrasng complxty, shap functons wth growng ordr prmts a bttr approxmaton of unknowns bhavor and thus ladng to th us of a rducd numbr of lmnts and nodal ponts (that s th numbr of problm DOFs), prsrvng th accuracy of th FEM. On th othr hand th us of hghr-ordr polynomals for th collcton of fnt lmnt shap functons lads to ncrasd codng dffcults. Such consdratons stat clarly that th choc of fnt lmnt and of th assocatd functon spacs must b th consqunc of a comproms btwn th accuracy of th FEM and ts computatonal cost (n trms of codng complxty, CPU tm and storag rqurmnts). As far as th gomtrc shap of th lmnts s concrnd, oftn th dscrtzd modlng of complx gomtrs ntroducs dstorton of th sub-domans (partcularly wth quadrangular lk shap), n ordr to smulat clos th doman boundars. From a computatonal vwpont, t s thus mandatory to ntroduc a mappng opraton abl to smply corrlat th spatal dscrpton of ach lmnt (Cartsan, sphrcal, tc. nods coordnats) wth a normal or logc systm of coordnats. Wth th grat advantag that n a normalzd gomtry rprsntaton, ndcatd as (,, ]), th cornr nods coordnats hav always unt valu (ngatv or postv). Fg. 5. shows a two-dmnsonal logc lmnt, whl th followng Fg. 5. dscrb th corrspondnc Cartsan and logc coordnat systms through a mappng oprator for two-dmnsonal gomtrs. Fg. 5. - Two-dmnsonal lmnt n logc rfrnc 5
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry Fg. 5. Lnk btwn Cartsan (x-y) and logc ( ) fram of rfrncs Th mappng oprator s gnrally stablshd through an analytcal corrspondnc btwn th global and th local normalzd systms of coordnats (Fg. 5.) wth th followng gnral functonal rlatonshp x½ ½ y¾ f ¾ z 9 Whr th functon f dfns analytcally th coordnats transformaton oprator and t ntroducs a bunvocal corrspondnc btwn th lmnts of th dscrtzd doman and th rctangular normalzd lmnts. Th smplcty of th logc rprsntaton of lmntary doman togthr wth thr unvrsal applcablty to lmnt of gnrc shap, suggst n a FE basd CFD cod th us of th logcal fram of rfrnc as th bass for th dfnton of th approxmat ntgral formulaton (.g. local shap functon to buld both tral and wghtng functons collctons). Elmnt by lmnt, th ral Cartsan gomtry would b thn constructd usng approprat transformaton oprator. In dtal th lnk btwn th ntroducd systm of coordnats could b xprssd as follows nl x(,, ) N (,, ) x (5.) 9 9 nl y(,, ) N (,, ) y (5.3) 9 9 nl z(,, ) N (,, ) z (5.) 9 9 whr nl s th numbr of lmnt nods, N ar th approprat ntrpolatng functons, (x,y,z ) ar th nodal coordnats l coordnat n th global fram of rfrnc. (5.) For nstanc, wth rfrnc to a two-dmnsonal lmnt wth four nods (Fg. 5.) th rlatd lnar shap functons N could b dfnd as 6
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry x =D 0 + D + D + D 3 (5.5) y = E 0 + E +E +E 3 (5.6) wth th polynomal coffcnts D and E that could b obtand applyng th (5.5) and (5.6) at ach cornr nods whr th global-to-logn coordnat transformaton s known. Such a poston lads to th followng rlatons N x(, ) N (, ) x x,,,,, N,,,,, y(, ) N (, ) y y (5.7) (5.8) Wrtng th abov rlatons for ach lmnt nod a systm of quatons for th unknown polynomal coffcnts D and E (n thr-dmnsons D, E and J) s fnally obtand. Th shap functons could b dfnd usng an altrnatv procdur calld drct formulaton, that s basd on th constructon of th collcton of ntrpolatng bass accordng to th st of gnral proprts thy hav to fulfl x th functon rlatd to th gnrc lmnt nod mst b such that N ()= and N (j)= 0 for jz ; x th bhavor of th functon along th lmnt boundars must guarant th contnuty wth nghbourng lmnts. Just to gv an xampl agan wth rfrnc to th lmnt shown n Fg. 5., th shap functon for th uppr-rght nod (normalzd coordnats (,)) could b dfnd as th product ( + )( + ) /. As a mattr of fact t assums th unt valu at th poston (,) and zro valus at th othr nodal postons whr at last on of th normalzd coordnats s st qual to -. Furthrmor th dfnd functon shows a lnar varaton on th lmnt dgs and by that way guarants th contnuty wth th nghbourng domans. Th soparamtrc lmnts ar commonly adoptd n th dvlopd FE cods. Th nam 'soparamtrc' drvs from th fact that th sam paramtrc functon whch dscrbs th gomtry may b usd for ntrpolatng spatal varatons of a varabl (u, v, T, p,...) wthn an lmnt. In gnral nl x( 9,, ) N (9,, ) x u( 9,, ) (9,, ) u and for an soparamtrc lmnt ) (,, ]) = N (,, ]). (5.9) nl ) (5.0) Th last problm to b solvd n th constructon of FE basd CFD cod, s th dfnton of shap functon drvatvs n th global systm of coordnats (x, y, z) ncssary to th dfnton of lmnt coffcnt matrcs. Such matrcs contan th global unknowns and varatons drvatvs, that ar of dffcult drct computaton whl thy offr a smpl dfnton n th logc fram of rfrnc. It s thus mandatory to fnd a logc-to-global transformaton tool for th drvatvs. Lt comput th shap functon frst drvatvs wth rfrnc to th normalzd coordnats,, and ] t s possbl to wrt 7
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry w) w w) wx wx w w ) wy w ) w y w w z w z w (5.) w) w w) wx wx w w w ) wy w ) y w w z w z w (5.) w) w] w) wx wx w] w ) wy w ) w y w] w z w z w] (5.3) or n a matrx form w ) ½ w w) ¾ w w ) w9 ª wx wy wzº w ) ½ w w w wx wx wx wx w) w w w ¾ w y wx wy wz w ) w] w] w] ¼ w z =J w ) ½ wx w) ¾ w y w ) w z (5.) whr J s calld th Jacoban and dfnd th sarchd transformaton oprator. As a mattr of fact th global drvatvs could b dfnd aftr th computaton of th nvrs Jacoban J -, as w ) ½ wx w) ¾ w y w ) w z =J - w ) ½ w w) ¾ w w ) w9 (5.5) From an algorthmc pont of vw, th dmonstratd smplcty of tratng th ntrpolatng functons and thr drvatvs n th logc rfrnc lads to th computaton of th ntgral trms that dfn th coffcnt matrx (on an lmnt lvl) n a normalzd control doman. Such a tchnqu rqurs of cours a coordnat transformaton abl to transfr th dffrntal ara or volum from th global to th logc rfrnc. Of cours th sam transformaton must nvolv th xtrms of ntgraton. Thus dfnd as a gnrc trm of th lmnt matrx th transformaton of ntgral could b wrttn as V > @ ³ dv ³ ³ ³ dt J d d d 9 (5.6) 5. Intrpolaton spacs n XENIOS A sampl of th shap functon mplmntd n th CFD cod XENIOS s hr gvn. Th shap functon ar dfnd wth thr frst ordr drvatvs for two-dmnsonal lmnt wth both b-lnar or b-quadratc approxmaton. Such functons hav bn dfnd by us of th drct formulaton alrady dscussd. Th four nods quadrangular lmnt s shown n Fg. 5.3, togthr wth th nod numbrng. 8
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry 9 Fg. 5.3 - Quadrangular lnar lmnt, local nod numbrng Th rlatd b-lnar shap functons ar ¼ º ª (5.7) Th frst ordr drvatvs of th b-lnar shap functons (5.7) ar ¼ º ª w w I (5.8.a) ¼ º ª w w I (5.8.b) Th b-quadratc nn nods lmnt has th local numbrng shown n Fg. 5.. 3
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry 6 7 5 8 9 3 Fg. 5. - Quadrangular lnar lmnt, local nod numbrng Th rlatd b-quadratc shap functons ar wth th followng frst ordr drvatvs ª º ¼ (5.9) 0
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry ¼ º ª w w I (5.0.a) ¼ º ª w w I (5.0.b)
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry 6. Stablzd fnt lmnt formulaton for advctv-dffusv flows Th stablzaton mthods, that wll b dscussd n th followng Chaptrs, dfn numrcal tools abl to corrct th nstablty orgns that affct orgnally th fnt lmnt Galrkn formulaton of ncomprssbl Navr-Stoks problm. Thr prsntaton s hr carrd out wth rfrnc to two dffrnt xtrm flow condtons N N th purly advctv flow lmt, charactrzd by th localzaton of advctv transport of varabl along th stramlns; th purly dffusv flow lmt (Stoks flow), charactrzd by prssur nstablty rlatd to th ncomprssblty constrant. 6. Th stablzaton of convcton domnatd flow Th soluton of flud dynamc problms usng symmtrc Galrkn rsdual formulaton loss th bst approxmaton proprts showd n structural and thrmal problms (Hughs t al., 98). As a consqunc of th convctv transport of flud varabls through th computaonal doman th coffcnt matrx assocatd wth th govrnng quatons contans frst ordr non symmtrc trms. Th dffcults that ars n smulatng vlocty flds, strongly asymmtrc, usng symmtrc oprators such that th lmntary Galrkn shap functons (or quvalntly cntrd fnt dffrncs stncl) gv rs to th prsnc of spurous oscllatons of vlocty on contguous nod (calld 'wggls'). A smpl way to comprhnd such a numrcal bhavor could b sarchd on th bass of th dmonstratd quvalnc btwn lnar Galrkn fnt lmnt approxmaton and th cntrd fnt dffrnc (Hughs t al, 98) (Lonard, 979). Th modlng wth cntrd fnt dffrnc (that s symmtrc oprators) of frst ordr drvatvs dos not lad to an ntrnsc stablty. Lt consdr, for nstanc, a convctv on-dmnsonal problm whr th vlocty u transport a scalar M through th doman R u wm wx Th stablty condton gnrally rqurs that ach wrong varatons of th valu of th transportd scalar M should rsults n a varatons of th convctv govrnng trm abl to corrc and compnsat such rror. That s th stablty condton could b xprssd as w convctv.trm (6.) 0 wm (6.) Lt now consdr th form that (6.) assums f a cntrd fnt dffrnc approxmaton s usd. Indcatng wth th sub-scrpts (-) and (+) th grd ponts locatd rspctvly upwnd and downwnd from th pont () whr th trm has to b computd, t s possbl to wrt
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry wm M u u wx M ' x that clarly shows that th snstvty of convctv trm to th varatons of scalar M L s zro. Thus, t could b concludd that th modlng of convctv trm, apparng n th momntum quatons as wll as n th consrvaton quatons of turbulnt varabls, carrd out wth oprators spatally symmtrc ntroduc n th algorthm a nutral stablty. That s th modld convctv trm s unabl to fl th drcton of propagaton along th grd of th convctv sgnals. (6.3) Th frst da dvlopd for rcovrng th algorthm stablty procds modlng th convctv trm by us of upwnd fnt dffrncs schm, so that th spatal drvatvs of th varabl n ach nodal poston () xplctly dpnds from th valu assumd at th nod tslf () and at th nod wth an upwnd locaton. In such a way that a drctonal stablty s rcovrd at xpns of accuracy of th formulaton. Lt now dmonstrat that an quvalnt stablzaton could b carrd out by th composton of a convctv trm modld wth cntrd fnt dffrnc stncl and an artfcal dffusv lk trm. Wth rfrnc to a ondmnsonal problm t could b wrttn wm M M M M u' x M M M u u u wx ' x ' x ' x (6.) whr ~ k u'x dfns a numrcal dffusvty that drctly dpnds on th magntud of convctv phnomna rlatv to th charactrstc grd dmnson ('x). Th ntrprtaton of th upwnd dffrncng tchnqu usng th artfcal dffusvty approach, rprsnts th lnk that orgnally gvs rs to th possblty of mplmntaton of such stablzaton mthods n th framwork of a rsdual fnt lmnt formulatons. It s thrfor wth th goal of corrctng th typcal undrdffusvty of Galrkn schm that th stablzaton tchnqu dscrbd n th followng Chaptr hav bn dvlopd. Non-consstnt stablzaton mthods, Upwnd and Stramln Upwnd Th stablzaton of Galrkn fnt lmnt formulaton s basd on th ntroducton of an artfcal balancng ntgral abl of corrctng th ngatv dffusvty of th rsdual mthod. An approprat choc of th artfcal contrbuton ntnsty could lad, n a on-dmnsonal cas, to th smulaton of xact numrcal soluton. Such upwnd schms ar calld optmal (Brooks and Hughs, 98). Th artfcal dffusvty s n th optmal upwnd scalar schm dfnd usng th followng xprssons 3
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry u h k ~ ] ¹ ] coth( D ) D (6.5) D u h k whr, for a manngful xtnson to mult-dmnsonal cas u, s th absolut valu of th local vlocty; h, s th charactrstc lmnt dmnson; D, s th lmntary or grd Pclt numbr; k, s th physcal flud dffusvty; ], s a stablzaton coffcnt abl to modfy th ntnsty of artfcal dffusvty wth rfrnc to th magntud of convctv transport phnomna. Th applcaton of upwnd scalar schms to mult-dmnsonal flow condtons lads oftn to non controlld ovr-dffusd soluton n partcular showng non physcal cross wnd dffuson, bcaus of th sotropc charactr of th balancng oprator. In ordr to lmnat th drawbacks shown by th scalar schms has bn thn dvlopd th stramln upwnd tchnqu whr th upwnd ffct s concntratd n prncpl along th stramln drcton. In such a mthod th balancng oprator, agan wth th form of a dffusv trm, acts xclusvly n th stramln drcton as an ansotropc artfcal dffusvty. Th artfcal dffusvty assums thrfor a tnsoral charactr and could b xprssd as follows whr ~ ~ k k u u j j u u u j j, dfns th vlocty componnts unt vctor, u u u, s th vlocty norm, ~ k, s th artfcal dffusvty alrady dfnd wth rfrnc to th scalar upwndng tchnqus. (6.6) It s now ntrstng th analyss of th form of th tnsoral balancng trm. Lt concntrat th analyss to stady and ncomprssbl Navr-Stoks quatons. Consdr th dvrgnc of strss tnsor apparng n th molcular dffusv trm, ts symmtrc part could b wrttn as ( k ( u u )), ( k ( u )), ( k ( u )), j,j j, j j,j j j j, j (6.7) By smply rvrs th drvaton ordr of th scond trm and mposng th ncomprssblty condton, dvrgnc fr vlocty fld, th (6.7) could b modfd as follow
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry wth ( k ( u )), ( k ( u )), ( k ( u )), ( k ( u )), j, j j j j, j j, j j j j, j ( k ( u )), ( k ( u )), ( k ( u )), j, j j j j, j j, j j u j,j = 0 (6.8) On th bass of th dvlopmnt, th dffusv ntgral trm n th Galrkn rsdual formulaton of Navr Stoks problm (.), (.7) could now b wrttn as ~ ³ w V d ³ w k k u d ns, j ns, j j j, whr xplctly appars a dffusvty obtand as th sum of physcal and artfcal contrbuton ns, j j ns, j j, ~ ³ w k V d ³ w k u d (6.9) Introducng n (6.9) th artfcal tnsoral dffusvty xprsson th stablzaton ntgral bcoms and substtutng th unt vctor ³ w k ~ u u u ns, j, j u j wth ts dfnton u u u j j d (6.0) t s possbl to wrt ³ w u u ~ ns, k u ju, j d (6.) What clarly appars from th (6.) s that th tnsoral stablzaton trm has th form of th convctv ntgral that must b controlld. Th analyss carrd out has dmonstratd th quvalnc btwn th two dffrnt approach dvlopd to obtan a stramln upwnd stablzaton n th ambt of a fnt lmnt rsdual formulaton from on hand th classcal approach that procds addng a dffusv balancng ntgral, from th othr hand th ntrvnton on th convctv ntgral n such a way that th orgnal Galrkn wght s modfd by a prturbaton dpndng from th stablzaton paramtrs. 5
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry Rsdual stablzd formulaton Stramln Upwnd - Ptrov Galrkn (SU/PG) Th consstncy of th stablzaton mthods could b rcovrd xtndng th wghts prturbaton, lmtd to th convctv ntgral n th stramln upwnd schm, to ach trm that s contand n th rsdual Navr- Stoks problm formulaton (.7). In such a way th buld rsdual structur assums th charactr of a Ptrov- Galrkn formulaton, du to th ntroducton of dffrnt functon spacs usd for th approxmaton of th soluton and of th varatons. Such a rsdual stablzd fnt lmnt formulaton for convcton domnatd flows s calld stramln upwnd - Ptrov Galrkn (SU/PG) (Brooks and Hughs, 98). Th stablshd consstncy of ntgral govrnng quatons mov dfntvly th stramln upwnd - Ptrov Galrkn from th classcal upwnd mthods closly lnkd to th fnt dffrnc scalar upwnd concpt. In such a way that th SUPG formulaton s not subjct to th artfcal dffuson crtcsm assocatd to th arly stablzaton tchnqu. Lt consdr a flow rgon R nsd (nsd s th numbr of spac dmnson), whch has a boundary * dfnd by pcws contnuous functons. Consdr also a pont x ( =,..., nsd) blongng, and dfn n as th componnt along drcton of th unt vctor normal to * (postv orntaton toward th nnr of th doman). Th doman boundary * s shard n two subst * g and * h, that satsfy th followng rlaton * * * g (6.) h * ˆ * (6.3) g h As alrady mntond, th fnt lmnt mthod procds by subdvdng th doman nto a fnt numbr of lmnts numl, dov =,..., numl. Lt now dfn * as th boundary fo th lmnt, th dscrtzaton follows th nsung proprts wth an ntror boundary such as ˆ * * * (6.) (6.5) nt (6.6) Rcall also th PDE that govrns a stady and ncomprssbl Navr-Stoks boundary problm Uu u wth th followng st of constrants j j, V j, j f (6.7) u, 0, th ncomprssblty constrant; (6.8) u g o* g, ssntal Drchlt boundary condtons; V j n ho* h, natural Numann boundary condtons. Th classcal Galrkn rsdual mthod adopts dntcal collctons of tral and wghtng functons, so that th wghts ar thn contnuous across th ntr-lmnt boundars. As a mattr of fact such proprty s lost 6
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry whn a Ptrov-Galrkn (SUPG) formulaton s usd du to th prturbaton of th st of wghts that modfy th orgnal Galrkn functons on an lmnt bass as follows wns' wns p (6.9) ns whr w ns s th Galrkn wght appld to th momntum quatons and p ns s th stablzng stramln upwnd lk contrbuton. Th applcaton of such a prturbaton on contguous lmnts ntroducs th dscontnuty of wghtng functons alrady mntond. Howvr th prturbaton functon stll fulflls th ntgrablty proprty on an lmnt scal. Lt consdr a pont x blongng to th ntror boundary * nt and, arbtrarly, stablsh a postv orntaton for th normal drcton across th boundary. Dfn also n + and n - as th unt vctor normal to * nt n th consdrd nodal poston x, t could b wrttn that n - = - n + Introducng for smplcty a trm that accounts for th sum of convctv and dffusv fluxs Uu u,j V a d j j (6.0) Is possbl to show that th jump of F at th consdrd nodal pont boundary x across th nghborng lmnts, dfnd as > @ n (6.) n n n s an nvarant wth rspct to th adoptd sgn convnton for * nt. On th bass of such an ntroducton, th applcaton of prturbd wghtng functon on th bass of th SUPG mthod to th Navr-Stoks problm lads to th followng rsdual stablzd formulaton U d U 8 8 w u u w w f ] p u u p p f ] d 6 ³ ns j, j ns, j ns 6 ³ ns j, j ns j, j ns 8 6 ³ wns j n] d* ³ whhd* 0 * nt * h or n an quvalnt way 8 8 8 w u u f ] d w n ] d w ( n h ) d 6 ³ cns U j, j j, j 6 ³ ns j * ³ h j * nt * h * (6.) 0 (6.3) It s worth to not that from th obtand rsdual form of th ntgral problm s agan possbl to xtract th orgnal dffrntal xprsson of th Navr-Stoks boundary problm. Imposng th annhlaton of th whol ntgrand functons, th Eulr-Lagrang condtons of th rsdual formulaton could b xtractd gvng th followng systm of quatons u u U j, j j, j 8 j * h j f ( n h ) 0 o 8 n ] 8 0 o * nt 0 (6.) Th dffrntal formulaton obtand (6.) s smply modfd wth rspct to th orgnal PDE form by a contnuty condton of dffusv fluxs across th ntr-lmnt boundary. 7
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry Th consstncy of th SU/PG stablzaton mthod rmans thus dmonstratd. Th fundamntal aspct that charactrzd th obtand stablzd fnt lmnt formulaton n comparson to th classcal upwnd schms s that th stramln upwnd prturbaton functon p ns plays s rol lmtd to th lmnt ntror whr th prturbaton tslf s contnuous. Th functon p ns dos not altr th contnuty condton of dffusv flux across th lmnt boundars as wll as th natural boundary condton h, whl t acts mplctly on th ffct of th Drchlt ssntal condtons through th convctv trm (Brooks and Hughs, 98). It s furthrmor ntrstng to not that, n cas of adopton of lnar shap functons and th doman dscrtzaton lads to rctangular lmnts th dvrgnc of th dffusv fluxs s qual to zro V j,j 0 that s th functon p ns dos not play any stablzaton ffct on th dffusv ntgral. Th xprsson of stramln upwnd prturbaton p ns of wghts s thn dfnd startng from th modfcaton of convctv wght obtand dscussng th non consstnt stramln upwnd mthod (6.). Th shown quvalnc btwn th modfcaton of th convctv wght and th ntroducton of a tnsoral dffusv balancng ntgral prmts to wrt th followng p ns ~ k u j wns, j / u (6.5) Fg. 6. shows th ffct of th prturbaton of a lnar Galrkn functon. Fg. 6.. Comparson btwn lnar Galrkn and Ptrov-Galrkn wghts Th coffcnt ~ k agan s an artfcal dffusvty that could b dfnd n svral ways. Stll rmans vald, as vrfd by numrcal computatons, that ts absolut valu as lttl mportanc f compard to th structur of th prturbaton functon p ns dfnd n (6.5). Howvr, s hr rportd th xprsson for ~ k mplmntd n XENIOS for mult-dmnsonal computatons wth ~ k u h u h 9 u9 h9 ¹ (6.6) 8
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry and coth( D ) D coth( D ) D 9 coth( D ) D u h D k 9 9 and u h D k D 9 u9 h 9 k u = u, u = u, u ] = ] u. In dtal, and ] ar th unt vctors of logc lmntary drctons, and ] whl h h h] ar approprat charactrstc lmntary lngth scals. Clarly f th wght w ns s contnuous across th lmnt boundars, th prturbaton p ns and as a consqunc th modfd wght w ns ' wll b dscontnuous. Th Stramln Upwnd - Ptrov Galrkn formulaton mplmntd n XENIOS Th mplmntaton of a SUPG stablzd formulaton n th fnt lmnt cod XENIOS has rqurd two fundamntal changs wth rspct to th formulaton orgnally proposd by Brooks and Hughs (98). Th us n XENIOS of mxd stabl lmnts, wth a quadratc ntrpolaton for th vlocty and turbulnt quantts and a lnar on for th prssur, gvs rs to an orgnal ntrprtaton of th consstnt stablzaton schm. As a mattr of fact th us of scond ordr bass functon lads to th prsnc n th advctv-dffusv quatons of th flow modl of stablzng ntgral wth non-zro scond ordr drvatvs. Such as ~ k ³ pnsv j, jd ³ wns j, j V j, jd u u (6.7) Wth a drct rfrnc to th Navr Stoks formulaton, two trms of th stablzd ntgral formulaton hav bn changd ³ p d ³ p p d ³ p ( k ( u u )) d (6.8) nsv j, j ns ns, j j, j j,, j 9
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry Th frst chang nvolv th ntgral trm of th momntum quaton proportonal to th prssur gradnt. Th structur of th SUPG stablzng ntgral n an lmntary doman s ~ u j ³ k p, jd u w ns,j (6.9) Th stablzng ffct s thrfor a consqunc of th product n ach lmnt of th spatal gradnt of quadratc and lnar functons. Lt consdr, for nstanc, th bhavor of spatal frst ordr drvatvs of th functons w ns (parabolc) and p (lnar) valuatd for th cntral nod of a on-dmnsonal quadratc lmnt Fg. 6.. p, j A w ns, j Fg. 6.. Frst ordr drvatvs of quadratc and lnar shap functons From th comparson, hr lmtd to on dmnson, bcoms vdnt that th stablzng contrbuton du to th mxd spacs of ntrpolaton assums an lmntary valu that s always postv both downwnd and upwnd of th nod A. Such a crcumstanc s that th trm loss ts ablty of ntroducng a stramln upwnd prturbaton. Its prsnc sms to ntroduc xclusvly an artfcal dsturb that n th dvlopd formulaton s lmnatd. Th scond chang has nvolvd th dffusv trm whch mplctly nfluncs th st of natural boundary condtons and thr fulfllmnt. Th local lmntary contnuty of th ntroducd prturbaton functons prmts th applcaton of th Grn- Gauss rul on an lmnt scal, such that a boundary ntgral orgns that corrsponds to an artfcal dffusv flux of th form ³ p V d ³ p V d ³ p V d ns j, j ns, j j * ns j n j (6.30) Such a trm s prfctly balancd along th ntr-lmnt boundars n th ntror of th computatonal doman, as a mattr of fact ach lmntary artfcal flux s annhlatd by fluxs qual and oppost from th nghborng lmnts. On th contrary t should b st qual to zro on th physcal boundary of th doman whr t dfns unphyscal flux condtons. 30
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry Th mplmntd stablzd formulaton thrfor adopts a tchnqu abl to nforc th st of problm natural condtons (Hansbo, 995). Such tchnqu mposs th annhlaton of th prturbaton ntgral n th vcnty of th doman boundary and could b formulatd as G b p ns, (6.3) ³ V j j d whr G b s an swtch factor buld as an array wth a dmnson qual to th numbr of nods that stor unt valus f th nods blong to th ntror of th doman or zro valus f th nods blong to th doman boundary G b () = o * G b () = 0 o * 6. Th stablzaton of dffuson domnatd flow Th mthods that attmpt to control th nstablty orgn rlatd to th flow ncomprssblty crcumvntng th Babuska-Brzz condton, hav as a common stratgy th rlaxng of fr-dvrgnc constrant on th vlocty fld. In such a way that th dvrgnc of th vlocty fld s st qual to a small postv trm, mad proportonal for nstanc to th prssur Laplacan (Brzz and Ptkaranta, 98) u, v ( p, ), (6.3) Such a tchnqu lads of cours to a los of consstncy of th stablzd rsdual formulaton wth th orgnal PDE boundary problm. Wth rfrnc to th Stoks flow problm, that dfns th dffusv lmt of th mor gnral Navr-Stoks formulaton, t s possbl to analyz rsdual formulatons abl to ntroduc a rlaxaton of th ncomprssblty constrant prsrvng th consstncy. Lt rcall th dffrntal form of th Stoks problm for ncomprssbl flows n a doman R nsd, whr * s th doman boundary dfnd by pcws contnuous functons shard n two substs * g and * h, such that * * * g (6.33) h * ˆ * (6.3) g h Th ncomprssbl Stoks problm could b formulatd as V j, j u, 0 f 0 wth th followng boundary condtons u g o* g V j n ho* h (6.35) 3
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry Th rlatv rsdual Galrkn formulaton on th dscrtzd doman s 8 8 w w f ] d w n ] d w h d w u d 0 6 ³ s, j s 6 ³ s j * ³ h * 6 ³ c, * nt * h (6.36) whr w s dfns th Galrkn wght appld to th Stoks quatons whl th w c dfns th wght appld to th contnuty quaton. Th ntroducton of a prturbaton of th wght w s, whch s contnuous nto ach lmntary doman, lads to th dfnton of a consstnt stablzd rsdual that gvs rs to a rlaxaton of th ncomprssblty constrant. Lt ntroduc a wght functon wth th followng structur wc w D h w, (6.37) s s c whr D dfns a non-dmnsonal postv stablty paramtr and h s th lmnt lngth scal (Hughs t al., 986). Th prsnc of such prturbd wghtng functons transforms th orgnal Galrkn rsdual formulaton nto a mor gnral Ptrov - Galrkn structur 8 8 6 ³ ws, j ws f ] d 6 ³ D h wc, j, f ] d 8 w n ] d w h d w u d 6 ³ s j * ³ h * 6 ³ c, * * nt h 0 (6.38) Lt now consdr th structur of Cauchy tnsor 8 j V pg k ( u u ) (6.39) j j j, j j, Th stablzng trm s lnkd to th spatal gradnt of th prssur and assums th followng xprsson ³ D h w, p, d c (6.0) Transformng th (6.0) by us of an ntgraton by parts th ntgral bcoms D c,, D c(, ), D * c, n ³ h w p d ³ h w p d ³ h w p d* (6.) such xprsson d facto gvs rs to th prsnc of an ntgral trm proportonal to th prssur Laplacan that du to th appld wght could b ntrprtd as a stablzaton ntgral actng on th rsdual contnuty quaton. Th abov dvlopmnts show clarly th rol of th stablzaton n altrng th ncomprssblty constrant ntroducng a non-zro vlocty fld dvrgnc. Rsdual stablzd formulaton Prssur Stablzd - Ptrov Galrkn (PS/PG) Th Ptrov-Galrkn mthod for th consstnt stablzaton of ncomprssbl Stoks flow, could b xportd n th mor gnral framwork of th Navr-Stoks flows as a tchnqu to prturb th wght functon appld 3
A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry to th contnuty quaton. Such a stablzaton mthod s calld Prssur Stablzd - Ptrov Galrkn PS/PG (Tzduyar, 99). Th PS/PG trm appld to th Galrkn rsdual contnuty quaton s dfnd as (Hughs t al., 986) (Tzduyar, 99) 8 6 ³ t w u u f ] d U pspg c, U j, j j, j (6.) whr t pspg s a stablty factor, n whch all th rlaton wth problm varabl ar cumulatd. Such a factor n th cod XENIOS assums th followng xprsson t h U pspg J whr h, s a global lngth scal dpndng from th doman dscrtzaton; U, s th global scalng vlocty; R U, s a Rynolds numbr rfrrd to h and U; J(R U ) = coth(r U )-/R U, s th law of dpndnc btwn th stablzaton ntnsty and th flow rgm. (R U ) It s worth to not that n th dscussd PSPG formulaton th rsduals of th momntum quatons s usd to buld th stablzaton ntgral. That s th rlaxaton of th ncomprssblty constrant s mad proportonal to th rror affctng th vlocty fld soluton. * Th stablzaton mthod hr dscussd has bn orgnally dvlopd to crcumvnt th Babuska - Brzz stablty condton and th nd to us mxd fnt lmnt vlocty-prssur spacs, thus allowng th us of qual-ordr ntrpolaton spacs. Its mplmntaton s rcommndabl also n prsnc stabl par of ntrpolaton spacs (such th quadratc-vlocty/lnar-prssur) for th nsung rasons N N N n prsnc of stabl mxd spacs th PSPG stablzd formulaton prsrvs th convrgnc proprts of th orgnal Galrkn on (Hughs t al., 986); th PSPG stablzaton allows th lmnaton of zro dagonal ntrs n th global coffcnt matrx orgnally causd by th ncomprssblty constrant; th PSPG stablzaton prmts th achvmnt of fastr convrgnc hstory n cas of adopton of tratv solvr (Hughs t al., 986). 33