REVIEW Lecture 16: Finite Volume Methods

Transcription

1 2.29 Numrcal Flud Mchancs prng 2015 Lctur 17 REVIEW Lctur 16: Fnt Volum Mthods Rvw: Basc lmnts of a FV schm and stps to stp-up a FV schm On Dmnsonal xampls d x j x j1/2 Gnrc quaton: Lnar Convcton (ommrfld qn): convctv fluxs 2 nd ordr n spac, thn 4 th ordr n spac, lnks to CD Unstady Dffuson quaton: dffusv fluxs Two approachs for 2 nd ordr n spac, lnks to CD Two approachs for th approxmaton of surfac ntgrals (and volum ntgrals) Intrpolatons and dffrntatons (xprss symbolc valus at surfacs as a functon of nodal varabls) Upwnd ntrpolaton (UD): Lnar Intrpolaton (CD): f j1/2 f j1/2 s (,) x t dx dt x j1/2 E Quadratc Upwnd ntrpolaton (QUICK), convctv flux Hghr ordr (ntrpolaton) schms f vn. 0 f vn. 0 x x E (1 ) whr xe x E x x x E (frst-ordr and dffusv) (2 nd ordr, can b oscllatory) g ( ) g ( ) U 1 D U 2 U UU x U D UU R x 2.29 Numrcal Flud Mchancs FJL Lctur 17, 1 3 D 3

2 TODAY (Lctur 17): Numrcal Mthods for th Navr-toks Equatons oluton of th Navr-toks Equatons Dscrtzaton of th convctv and vscous trms Dscrtzaton of th prssur trm Consrvaton prncpls Choc of Varabl Arrangmnt on th Grd Calculaton of th rssur rssur Corrcton Mthods A mpl Explct chm A mpl Implct chm Nonlnar solvrs, Lnarzd solvrs and ADI solvrs Implct rssur Corrcton chms for stady problms Outr and Innr tratons rojcton Mthods Non-Incrmntal and Incrmntal chms Fractonal tp Mthods: Exampl usng Crank-Ncholson 2.29 Numrcal Flud Mchancs FJL Lctur 17, 2

3 Rfrncs and Radng Assgnmnts Chaptr 7 on Incomprssbl Navr-toks quatons of J. H. Frzgr and M. rc, Computatonal Mthods for Flud Dynamcs. prngr, NY, 3 rd dton, 2002 Chaptr 11 on Incomprssbl Navr-toks Equatons of T. Cbc, J.. hao, F. Kafyk and E. Laurndau, Computatonal Flud Dynamcs for Engnrs. prngr, Chaptr 17 on Incomprssbl Vscous Flows of Fltchr, Computatonal Tchnqus for Flud Dynamcs. prngr, Numrcal Flud Mchancs FJL Lctur 17, 3

4 2.29 Intrpolatons and Dffrntatons (to obtan fluxs F = f ( ) as a functon of cll-avrag valus) Hghr Ordr chms (for convctv/dffusv fluxs) Intrpolatons of ordr of accuracy hghr than 3 mak sns f ntgrals ar also approxmatd wth hghr ordr formulas In 1D problms, f mpson s rul (4 th ordr rror) s usd for th ntgral, a polynomal ntrpolaton of ordr 3 can b usd: ( x) a a x a x a x = ymmtrc formula for : no nd for upwnd as wth 0 th or 2 nd ordr polynomals (donor-cll & QUICK) Wth (x), on can nsrt = (x ) n symbolc ntgral formula. For a unform Cartsan grd: Convctv Fluxs: => 4 unknowns, hnc 4 nodal valus (W,, E and EE) ndd For Dffusv Fluxs (1 st drvatv): Numrcal Flud Mchancs (smlar formulas usd for ϕ valus at cornrs) Ths FV approxmaton oftn calld a 4 th -ordr CD (lnar poly. ntrpol. was 2 nd -ordr CD) olynomals of hghr-dgr or of mult-dmnsons can b usd, as wll as cubc splns (to nsur contnuty of frst two drvatvs at th boundars). Ths ncrass th cost. y j+1 y j-1 y y j j WW x NW W W nw w sw N n n n s s x NE E y E x -1 x x a1 2a2x 3a3x for a unform Cartsan grd: x x 24 x E W EE ( Not: hghr-ordr, approach 1 approach 2! ) EE Notaton usd for a Cartsan 2D and 3D grd. Imag by MIT OpnCoursWar. E W EE FJL Lctur 17, 4

5 2 3 Ex. 1: obtan th coffcnts of by y W E ( x) a 0 a1x a2x a3x x fttng two valus and two 1 st drvatvs at th two nods on j x -1 x x +1 thr sd of th cll fac. Wth valuaton at x x : 4 th E x 4 ordr schm: O( x ) Notaton usd for a Cartsan 2D and 3D grd. 2 8 x x Imag by MIT OpnCoursWar. E If w us CD to approxmat drvatvs, rsult rtans 4 th ordr: E E W EE 4 O( x ) 2 16 Ex. 2: us a parabola, ft th valus on thr sd of th cll fac and th drvatv on th upstram sd (quvalnt to th QUICK schm, 3 rd ordr) 3 1 x U D x 2.29 Intrpolatons and Dffrntatons (to obtan fluxs F = f ( ) as a functon of cll-avrag valus) Compact Hghr Ordr chms olynomal of hghr ordr lad too larg computatonal molculs => us dfrrd-corrcton schms and/or compact (ad ) schms mlar schms ar obtand for drvatvs (dffusv fluxs), s Frzgr and rc (2002) Othr chms: mor complx and dffcult to program U Larg numbr of approxmatons usd for convctv fluxs: Lnar Upwnd chm, kwd Upwnd schms, Hybrd. Blndng schms to lmnat oscllatons at hghr ordr. Numrcal Flud Mchancs y j+1 y j y j-1 WW NW W nw w sw N n n n s s NE E y EE FJL Lctur 17, 5

6 Intgral Consrvaton Law for a scalar (from Lctur 8-N) d dt d dv dv s dv dt ( v. n )da fxd C q. n da C CV fxd CM CV C C CV Advctv fluxs Othr transports (dffuson, tc) (Adv.& dff. fluxs = "convctv" fluxs) um of sourcs and snks trms (ractons, tc) CV, fxd ρ,φ s Φ v q Applyng th Gauss Thorm, for any arbtrary CV gvs:.( v) ). q s t For a common dffusv flux modl (Fck s law, Fourr s law): q k Consrvatv form of th DE.( v ).( k ) s t 2.29 Numrcal Flud Mchancs FJL Lctur 17, 6

7 Cons. of Momntum: trong-consrvatv form of th Navr-toks Equatons ( v) d dt Applyng th Gauss Thorm gvs: vdv v( ( v. n ) da p nda C. nda C gdv CV F CV C C C CV p. g dv CV (from Lctur 8-N) ρ,φ v For any arbtrary CV gvs: v.( vv v ) p... g g t Wth Nwtonan flud + ncomprssbl + constant μ: Cauchy Mom. Eqn. CV, fxd s Φ q Momntum: Mass: v t. v 0 v.( vv v) ) p 2 v gg Equatons ar sad to b n strong consrvatv form f all trms hav th form of th dvrgnc of a vctor or a tnsor. For th th Cartsan componnt, n th gnral Nwtonan flud cas: Wth Nwtonan flud only: v u u u u 2 u j j gx t xj x j 3 x u.( v v ). p j 2.29 Numrcal Flud Mchancs FJL Lctur 17, 7

8 oluton of th Navr-toks Equatons In th FD and FV schms, w dalt wth th dscrtzaton of th gnrc consrvaton quaton Ths rsults apply to th momntum and contnuty quatons (th N quatons),.g. for ncomprssbl flows, constant vscosty Trms that ar dscrtzd smlarly Unstady and advcton trms: thy hav th sam form for scalar than for v Trms that ar dscrtzd dffrntly.( v) ). q s t v.( vv v) ) p 2 v gg t. v 0 Momntum (vctor) dffusv fluxs nd to b tratd n a bt mor dtals rssur trm has no analog n th gnrc consrvaton quaton => nds spcal attnton. It can b rgardd thr as a sourc trm (tratd non-consrvatvly as a body forc), or as, surfac forc (consrvatv tratmnt) Fnally, man varabl v s a vctor gvs mor frdom to th choc of grds 2.29 Numrcal Flud Mchancs FJL Lctur 17, 8

9 Dscrtzaton of th Convctv and Vscous Trms Convctv trm: Us any of th schms (FD or FV) that w hav sn (ncludng complx gomtrs). j and.nda x and j j.n n d j u u j j xj x If μ s constant, th vscous trm s as n th gnral consrvaton qn. for Vscous trm: For a Nwtonan Flud and ncomprssbl flows: If μ vars, ts drvatv nds to b valuatd (FD schm) or ts varatons accountd for n th ntgrals (for a FV schm) For a Nwtonan flud and comprssbl flow: Addtonal trms nd to b tratd,.g. Not that n non-cartsan coordnat systms, nw trms also ars that bhav as a body forc, and can thus b tratd xplctly or mplctly.g u r r 2 2 ( uu.( v v( ( v. n )d x u ( v. n )d u j ) v) ) and and j 2 u j 3 x 2.29 Numrcal Flud Mchancs FJL Lctur 17, 9 j

10 Dscrtzaton of th rssur trm For consrvatv N schms, gravty/body-forc trms oftn ncludd n th prssur trm, gvng: 2 2 u j.. ( p u p p ggr u gx ) 3 3 x j rssur thn part of th strss tnsor (shows up as dvrgnc n N qns.) Last trm s null for ncomprssbl flows In non-consrvatv N forms, th prssur gradnt s dscrtzd FD schms FD schms sn arlr ar drctly applcabl, but prssur can b dscrtzd on a dffrnt grd than th vlocty grd (staggrd grd) FV schms rssur usually tratd as a surfac forc (consrvatv form): For th uquaton: p. nd Agan, schms sn n prvous lcturs ar applcabl, but prssur nods can b on a dffrnt CV grd rssur can also b tratd non-consrvatvly: Dscrtzaton thn ntroducs a global non-consrvatv rror 2.29 Numrcal Flud Mchancs FJL Lctur 17, 10 V p dv.

11 2.29 Consrvaton rncpls for N Momntum and Mass Consrvaton Momntum s consrvd n any control volum n th sns that t can only chang bcaus of flow through th CV surfacs, forcs actng on ths surfacs or volumtrc body forcs Ths proprty s nhrtd n th CV formulaton (f surfac fluxs ar dntcal on both sds) mlar statmnts for Mass consrvaton Consrvaton of mportant scondary quantts,.g. nrgy Mor complx ssus In hat transfr, thrmal nrgy quaton can b solvd aftr momntum quaton has bn solvd f proprts don t vary much wth tmpratur T T s thn a passv scalar, wth on way couplng In ncomprssbl, sothrmal flows: kntc nrgy s th sgnfcant nrgy In comprssbl flows: nrgy ncluds comprssbl trms total nrgy s thn a sparat quaton (1 st law) but a scond drvd quaton can stll b wrttn, thr for kntc or ntrnal nrgy Numrcal Flud Mchancs FJL Lctur 17, 11

12 Consrvaton rncpls for N: Cont d Kntc Enrgy Consrvaton Drvaton of Kntc nrgy quaton Tak dot product of momntum quaton wth vlocty Intgrat ovr a control volum CV or full volum of doman of ntrst Ths gvs t whr 2 2 v v dv ( v g g.. v dv 2 2. n ) da p v. n da (.. v ).n n da : v p. v CV C C C CV p j j j s th vscous componnt of th strss tnsor In th volum ntgral of th RH, th thr trms ar zro f th flow s nvscd (trm 1 = dsspaton), ncomprssbl (trm 2) and thr ar no body forcs (trm 3) Othr trms ar surfac trms and kntc nrgy s consrvd n ths sns: dscrtzaton on CV should dally lad to no contrbuton ovr th volum om obsrvatons Guarantng global consrvaton of th dscrt kntc nrgy s not automatc snc th kntc nrgy quaton s a consqunc of th momntum quaton. Dscrt momntum and kntc nrgy consrvatons cannot b nforcd sparatly (th lattr can only b a consqunc of th formr) 2.29 Numrcal Flud Mchancs FJL Lctur 17, 12

13 2.29 Consrvaton rncpls for N, Cont d om obsrvatons, Cont d If a numrcal mthod s (kntc) nrgy consrvatv, t guarants that th total (kntc) nrgy n th doman dos not grow wth tm (f th nrgy fluxs at boundars ar null/boundd) Ths nsurs that th vlocty at vry pont n th doman s boundd: mportant stablty-rlatd proprty nc kntc nrgy consrvaton s a consqunc of momntum consrvaton, global dscrt kntc nrgy consrvaton must b a consqunc of th dscrtzd momntum quatons It s thus a proprty of th dscrtzaton mthod and t s not guarantd On way to nsur t s to mpos that th dscrtzaton of th prssur gradnt and dvrgnc of vlocty ar compatbl,.. lad to dscrt nrgy consrvaton drctly A osson quaton s oftn usd to comput prssur It s obtand from th dvrgnc of momntum quatons, whch contans th prssur gradnt (s nxt) Dvrgnc and gradnt oprators must b such that mass consrvaton s satsfd (spcally for ncomprssbl flows), and dally also kntc nrgy Numrcal Flud Mchancs FJL Lctur 17, 13

14 Consrvaton rncpls for N, Cont d om obsrvatons, Cont d Tm-dffrncng mthod can dstroy th nrgy consrvaton proprty (and mass consrvaton for ncomprssbl flud) Idally, nrgy consrvaton should b automatc from th numrcal schm Exampl: Crank-Nckolson V 1 Tm drvatvs ar approxmatd by: ( u n n (md-pont rul) u ) t n 1/ 2 If on taks th scalar product of ths quaton wth u (vl. for md-pont rul), whch n C-N s approxmatd by, n1/ 2 n1 n u ( u u ) / 2 th rsult s th dscrtzd rat of chang of th kntc nrgy quaton 2 n1 2 n V v v t 2 2 Ths th LH of C-N for kntc nrgy!!!! whr Wth propr chocs for th othr trms, th C-N schm s nrgy consrvatv v 2 u u (summaton mpld) 2.29 Numrcal Flud Mchancs FJL Lctur 17, 14

15 arabolc DE: Implct chms Lads to a systm of quatons to b solvd at ach tm-stp (rvw Lctur 14) B-C (Backward-Cntrd): 1 st ordr accurat n tm, 2 nd ordr n spac Uncondtonally stabl x -1 mpl mplct mthod t l+1 t l x x +1 Grd pont nvolvd n tm dffrnc Grd pont nvolvd n spac dffrnc B-C: Backward n tm Cntrd n spac Evaluats RH at tm t+1 nstad of tm t (for th xplct schm) Crank-Ncolson: 2 nd ordr accurat n tm, 2 nd ordr n spac Uncondtonally stabl x -1 Crank-Ncolson mthod t l+1 t l+1/2 t l x x +1 Grd pont nvolvd n tm dffrnc Grd pont nvolvd n spac dffrnc Tm: cntrd FD, but valuatd at md-pont 2 nd drvatv n spac dtrmnd at md-pont by avragng at t and t Imag by MIT OpnCoursWar. Aftr Chapra,., and R. Canal. Numrcal Mthods for Engnrs. McGraw-Hll, Numrcal Flud Mchancs FJL Lctur 17, 15

16 Consrvaton rncpls for N, Cont d om obsrvatons, Cont d nc momntum and kntc nrgy (and mass cons.) ar not ndpndnt, satsfyng all of thm s not drct: tral and rror n drvng schms that ar consrvatv Kntc nrgy consrvaton s partcularly mportant n unstady flows (.g. wathr, ocan, turbulnc, tc) Lss mportant for stady flows Kntc nrgy s not th only quantty whos dscrt consrvaton s dsrabl (and not automatc) Angular momntum s anothr on Important for flows n rotatng machnry, ntrnal combuston ngns and any othr dvcs that xhbt strong rotatons/swrl If numrcal schms do not consrv ths mportant quantts, numrcal smulaton s lkly to gt nto troubl, vn for stabl schms 2.29 Numrcal Flud Mchancs FJL Lctur 17, 16

17 Choc of Varabl Arrangmnt on th Grd Bcaus th Navr-toks quatons ar coupld quatons for vctor flds, svral varants of th arrangmnt of th computatonal ponts/nods ar possbl Collocatd arrangmnt Obvous choc: stor all th varabls at th sam grd ponts and us th sam grd ponts or CVs for all varabls: Collocatd grd N W w n s E W n w E s N Advantags: Collocatd arrangmnt of vlocty componnts and prssur on FD and FV grds. Imag by MIT OpnCoursWar. All (gomtrc) coffcnts valuatd at th sam ponts Easy to apply to multgrd procdurs (collocatd rfnmnts of th grd) 2.29 Numrcal Flud Mchancs FJL Lctur 17, 17

18 Choc of Varabl Arrangmnt on th Grd Collocatd arrangmnt: Dsadvantags Was out of favor and not usd much untl th 1980s bcaus of: Occurrnc of oscllatons n th prssur Dffcults wth prssur-vlocty couplng, and rqurs mor ntrpolatons Howvr, whn non-orthogonal grds startd to b usd ovr complx gomtrs, th stuaton changd Ths s bcaus th non-collocatd (staggrd) approach on nonorthogonal grds s basd on grd-orntd componnts of th (vlocty) vctors and tnsors Ths mpls usng curvatur trms, whch ar mor dffcult to trat numrcally and can crat non-consrvatv rrors Hnc, collocatd grds bcam mor popular wth complx gomtrs Numrcal Flud Mchancs (I) (II) (III) Vlocts rssur Imag by MIT OpnCoursWar. Varabl arrangmnts on a non-orthogonal grd. Illustratd ar a staggrd arrangmnt wth () contravarnt vlocty componnts and () Cartsan vlocty componnts, and () a colocatd arrangmnt wth Cartsan vlocty componnts. FJL Lctur 17, 18

19 2.29 Choc of Varabl Arrangmnt on th Grd taggrd arrangmnts No nd for all varabls to shar th sam grd taggrd arrangmnts can b advantagous (coupls p and v) For xampl, consdr th Cartsan coordnats Advantags of staggrd grds vral trms that rqur ntrpolaton n collocatd grds can b valuatd (to 2 nd ordr) wthout ntrpolaton Ths appls to th prssur trm (locatd at CV cntrs) and th dffuson trm (frst drvatv ndd at C cntrs), whn obtand by cntral dffrncs Can b shown to drctly consrv kntc nrgy Many varatons: partally staggrd, tc Numrcal Flud Mchancs Fully and partally staggrd arrangmnts of vlocty componnts and prssur. y j+1 y j y j-1 W nw w sw x -1 N n n s s E y W N nw n n w sw s s Imag by MIT OpnCoursWar. E W nw w X x x -1 x x +1 x -1 (a) (b) (c) Control volums for a staggrd grd for (a) mass consrvaton and scalar quantts, (b) x-momntum, and (c) y-momntum sw FJL Lctur 17, 19 N n s n Imag by MIT OpnCoursWar. s x E

20 Choc of Varabl Arrangmnt on th Grd taggrd arrangmnts: Exampl wth Cartsan coordnats, Cont d Trms can b valuatd (to 2 nd ordr) wthout ntrpolaton Ths appls to th prssur trm (normal at cntr of C). For xampl, along x drcton: Each p valu on th bnd of th vlocty grd s convnntly at th cntr th scalar grd: Dffuson trm (frst drvatv at C) obtand by cntral dffrncs. For xampl: E 2 2 xx u u u x x x E p p n. d p p y j+1 y j y j-1 W nw w sw w w N n s n s E y W nw w sw N n n s s E W nw w sw N n s n s E x -1 X x x x -1 x +1 x -1 x x-momntum solvd on CVs of u vl., y-momntum on CVs of v vl. and contnuty on CVs of Numrcal Flud Mchancs (a) (b) (c) Control volums for a staggrd grd for (a) mass consrvaton and scalar quantts, (b) x-momntum, and (c) y-momntum Imag by MIT OpnCoursWar. FJL Lctur 17, 20

21 MIT OpnCoursWar Numrcal Flud Mchancs prng 2015 For nformaton about ctng ths matrals or our Trms of Us, vst: