2.29 Numerical Fluid Mechanics Fall 2011 Lecture 23

Transcription

1 2.29 Numrcal Flud Mchacs Fall 2011 Lctur 23 REVIEW Lcturs 22: Compl Gomtrs Grd Grato Basc cocpts ad structurd grds trtchd grds Algbrac mthods Gral coordat trasformato Dffrtal quato mthods Coformal mappg mthods Ustructurd grd grato Dlauay Tragulato Advacg Frot mthod Ft Volum o Compl gomtrs Computato of covctv flus Computato of dffusv flus Commts o 3D 2.29 Numrcal Flud Mchacs FJL Lctur 23, 1

2 TODAY (Lctur 23): Grd Grato (d), FV o Compl Gomtrs ad oluto to th Navr-toks Equatos Grd Grato Basc cocpts ad structurd grds Algbrac mthods (strtchd grds), Gral coordat trasformato, Dffrtal quato mthods, Coformal mappg mthods Ustructurd grd grato Dlauay Tragulato Advacg Frot mthod Ft Volum o Compl gomtrs Computato of covctv flus Computato of dffusv flus Commts o 3D oluto of th Navr-toks Equatos Dscrtzato of th covctv ad vscous trms Dscrtzato of th prssur trm Cosrvato prcpls 2.29 Numrcal Flud Mchacs FJL Lctur 23, 2

3 Rfrcs ad Radg Assgmts Chaptr 8 o Compl Gomtrs ad Chaptr 7 o Icomprssbl Navr-toks quatos of J. H. Frzgr ad M. rc, Computatoal Mthods for Flud Dyamcs. prgr, NY, 3 rd dto, 2002 Chaptr 9 o Grd Grato ad Chaptr 11 o Icomprssbl Navr-toks Equatos of T. Cbc, J.. hao, F. Kafyk ad E. Laurdau, Computatoal Flud Dyamcs for Egrs. prgr, Chaptr 13 o Grd Grato ad Chaptr 17 o Icomprssbl Vscous Flows of Fltchr, Computatoal Tchqus for Flud Dyamcs. prgr, Rf o Grd Grato oly: Thompso, J.F., Wars Z.U.A. ad C.W. Mast, Numrcal Grd Grato, Foudatos ad Applcatos, North Hollad, Numrcal Flud Mchacs FJL Lctur 23, 3

4 2.29 Grd Grato: Ustructurd Grds Gratg ustructurd grd s complcatd but ow rlatvly automatd classc cass Ivolvs succsso of smoothg tchqus that attmpt to alg lmts wth boudars of physcal doma Dcompos doma to blocks to dcoupl th problms Nd to df pot postos ad coctos Most popular algorthms: Dlauay Tragulato Mthod Advacg Frot Mthod Two schools of thought: structurd vs. ustructurd, what s bst for CFD? prgr. All rghts rsrvd. Ths cott s cludd from our Cratv Commos lcs. For mor formato, s tructurd grds: smplr grd ad straghtforward tratmt of algbrac systm, but msh grato costrats o compl gomtrs Ustructurd grds: gratd fastr o compl domas, asr msh rfmts, but data storag ad soluto of algbrac systm mor compl Numrcal Flud Mchacs FJL Lctur 23, 4

5 Dlauay Tragulato (DT) Grd Grato: Ustructurd Grds Us a smpl crtro to coct pots to form coformg, o-trsctg lmts Mamzs mmum agl ach tragl Not uqu Task of pot grato s do dpdtly of cocto grato Basd o Drchlt s doma dcomposto to a st of packd cov rgos: For a gv st of pots, th spac s subdvdd to rgos such a way that ach rgo s th spac closr to tha to ay othr pot = Drchlt tssllato Ths gomtrcal costructo s kow as th Drchlt tssllato Th tssllato of a closd doma rsults a st of o-ovrlappg cov rgos calld Voroo rgos/polygos Th sds of th polygo aroud s mad of sgmts bsctors of ls og to ts ghbors: f all par of pots wth a commo sgmt ar od by straght ls, th rsult s a Dlauay Tragulato Each vort of a Voroo dagram s th crcumctr of th tragl formd by th thr pots of a Dlauay tragl Crtro: th crcumcrcl ca ot cota ay othr pot tha ths thr pots Not: at th d, pots ar at summts of tragls (a) atsfs th crtro (b) Dos ot Imag by MIT OpCoursWar Imag by MIT OpCoursWar. Numrcal Flud Mchacs FJL Lctur 23, 5

6 Grd Grato: Ustructurd Grds Advacg Frot Mthod I ths mthod, th ttrahdras ar bult progrssvly, ward from th boudary A actv frot s matad whr w ttrahdra ar formd For ach tragl o th dg of th frot, a dal locato for a w thrd od s computd Rqurs trscto chcks to sur tragls do t ovrlap 2.29 prgr. All rghts rsrvd. Ths cott s cludd from our Cratv Commos lcs. For mor formato, s I 3D, th Dlauay Tragulato s prfrrd (fastr) Numrcal Flud Mchacs FJL Lctur 23, 6

7 Ft Volums o Compl gomtrs FD mthod (classc): Us structurd-grd trasformato (thr algbrac-trasft, gral, dffrtal or coformal mappg) olv trasformd quatos o smpl orthogoal computatoal doma FV mthod: tarts from cosrvato qs. tgral form o CV d dt CV C C CV dv ( v. )da q. da s dv Advctv (covctv) flus Othr trasports (dffuso, tc) W hav s prcpls of FV dscrtzato Covctv/dffusv flus, from 1 st - 2 d ordr to hghr ordr dscrtzatos Ths prcpls ar dpdt of grd spcfcs, but, um of sourcs ad sks trms (ractos, tc) vral w faturs ars wth o-orthogoal or arbtrary ustructurd grds 2.29 Numrcal Flud Mchacs FJL Lctur 23, 7

8 Eprssg flus at th surfac basd o cll-avragd (odal) valus: ummary of Two Approachs ad Boudary Codtos t-up of surfac/volum tgrals: 2 approachs (do thgs oppost ordr) 1. () Evaluat tgrals usg classc ruls (symbolc valuato); () Th, to obta th ukow symbolc valus, trpolat basd o cll-avragd (odal) valus ( ) F f da F ( ) G F F ( ) H ( ' s) H ( ' s) mlar for othr tgrals: 1 ( s dv, dv, tc) V V V 2. () lct shap of soluto wth CV (pcws appromato); () mpos volum costrats to prss coffcts trms of odal valus; ad () th tgrat. (ths approach was usd th ampls). ( ) a ( ) J ( ) a ( ) ( ) a ( ) ( ) a F V F ( ) F f da Boudary codtos: Drctly mposd for covctv flus O-sd dffrcs for dffusv flus ( ' s) ( ' s) mlar for hghr dmsos: (, y) J (, y); tc (, y ) ; tc a a (From lctur 19) 2.29 Numrcal Flud Mchacs FJL Lctur 23, 8

9 2.29 Appro. of urfac/volum Itgrals: Classc symbolc formulas urfac Itgrals 2D problms (1D surfac tgrals) Mdpot rul (2 d ordr): Trapzod rul (2 d ordr): mpso s rul (4 th ordr): 3D problms (2D surfac tgrals) Mdpot rul (2 d ordr): Hghr ordr mor complcatd to mplmt 3D Volum Itgrals: 2D/3D problms, Mdpot rul (2 d ordr): 2D, b-quadratc (4 th ordr, Cartsa): F f da 1 s dv, dv V V V Notato usd for a Cartsa 2D ad 3D grd. Imag by MIT OpCoursWar. 2 ( ) F f da f f O y f ( f fs) 2 F f da O( y ) 2 ( f 4 f fs) 4 F f da O( y ) 6 F f da f O y z 2 2 (, ) s dv s V s V V (summary from Lctur 18) y 16s 4ss 4s 4sw 4s ss ssw s sw 36 Numrcal Flud Mchacs FJL Lctur 23, 9 y y +1 y y -1 WW NW W W w w sw N s s NE E y E EE

10 FV: Appromato of covctv flus ( v. )d ) C Advctv (covctv) flus For compl gomtrs, o oft uss th mdpot rul for th appromato of surfac ad volum tgrals Cosdr frst th mass flu: =1: f 1 v. Aga, w cosdr o fac oly: ast sd of a 2D CV (sam approach appls to othr facs ad to ay CV shaps). Md-pot rul for mass flu: m f m f1 d f f O( 2 ) ( v v.. ) y NW w w W sw N η η ξ E s E ξ Th ut ormal vctor to fac ad ts surfac ar dfd as: y s ) (y y s ) ( whr ( ) ( y ) 2 2 Hc, mass flu s: y y ( m v.( ( v v.. ) ) ( u v ) Imag by MIT OpCoursWar Numrcal Flud Mchacs FJL Lctur 23, 10

11 FV: Appromato of covctv flus, Cot d Mass Flu Th mass flu for th md-pot rul: m y ( u v) What s th dffrc btw th Cartsa ad oorthogoal grd cass? I o-orthogoal cas, ormal to surfac has compots all drctos All vlocty compots thus cotrbut to th flu (ach compot s multpld by th procto of oto th corrspodg as) η NW N y η ξ w ξ E w W s sw E Imag by MIT OpCoursWar Numrcal Flud Mchacs FJL Lctur 23, 11

12 FV: Appromato of covctv flus, Cot d Mass flu for md-pot rul: Covctv flu for ay trasportd Is usually computd aftr th mass flu. Usg th md-pot rul: F ( v. ) d f ( ( v. ) m whr How to obta?, us thr: = valu at ctr of cll fac A lar trpolato btw two ods o thr sd of fac (also 2 d ordr) bcoms trapzodal rul Ft to a polyomal th vcty of th fac (pcws shap fucto) Cosdratos for ustructurd grd: y m ( u v ) m ( Bst comproms amog accuracy, gralty ad smplcty s usually: Lar trpolato ad md-pot rul Idd: facltats us of local grd rfmt, whch ca b usd to achv hghr accuracy at lowr cost tha hghr-ordr schms. Howvr, hghrordr FE or compact FD ar ow bg usd/dvlopd 2.29 Numrcal Flud Mchacs FJL Lctur 23, 12

13 FV: Appromato of dffusv flus C k. d Dffusv Flus For compl gomtrs, w ca stll us th mdpot rul Md-pot rul gvs: F d ( 2 k.d f f O( ) (k.. ) Hr, gradt ca b prssd trms of global Cartsa coordats (, y) or local orthogoal coordats (, t) y W NW w w sw η N η ξ s E Imag by MIT OpCoursWar. E ξ I 2D: t y t Thr ar may ways to appromat th drvatv ormal to th cll fac or th gradt vctor at th cll ctr As always, two ma approachs: Appromat surfac tgral, th trpolat pcfy shap fucto, costrats, th tgrat 2.29 Numrcal Flud Mchacs FJL Lctur 23, 13

14 FV: Appromato of dffusv flus, Cot d 1) If shap fucto (, y) s usd, wth md-pot rul, ths gvs: (k. ) ) k k k y F d y Ca b valuatd ad rlatvly asy to mplmt plctly Implctly ca b hardr for hgh-ordr shap fct (, y) (mor cll volvd) 2) Aothr way s to comput drvatvs at CV ctrs frst, th trpolat to cll facs (2 stps as for computg from ) y W NW w w sw η N η ξ s E Imag by MIT OpCoursWar. E ξ ) O ca us avrags + Gauss Thorm locally Drvatv at ctr avrag drvatv ovr cll CV dv dv Gauss thorm for / (smlar for y drvatv): dv.d c c CV C 4facs c 2.29 Numrcal Flud Mchacs FJL Lctur 23, 14

15 FV: Appromato of dffusv flus, Cot d y 2.29 Hc, th gradt at th CV ctr wth rspct to s obtad by summg th products of ach c wth th procto of ts cll surfac oto a pla ormal to, ad th dvdg th sum by th CV volum For w ca us th appromato for th covctv flus c W ca th trpolat to obta th gradt at th ctrs of cll facs For Cartsa grds ad lar trpolato, o rtrvs ctrd FD W NW w w sw η N η ξ s E Imag by MIT OpCoursWar. E ξ CV dv dv dv 4 c facs ) Cll-ctr gradts ca also b appromatd to 2 d ordr assumg a lar varato of locally: E. (r r ) E Thr ar as may such quatos as thr ar ghbors to th cll ctrd at d for last-squar vrsos (oly drvatvs D) Issus wth ths appromato ar oscllatory solutos ad larg computatoal stcls for mplct schms us dfrrd-corrcto approach Numrcal Flud Mchacs c c FJL Lctur 23, 15

16 FV: Appromato of dffusv flus, Cot d ) Dfrrd-Corrcto Approach: Ida bhd dfrrd-corrcto s to dtfy possbl optos ad comb thm to rduc costs ad lmat u-dsrd bhavor. om optos: If w work local coordats (, t): (k. ) ) k F d re r (1) 1 2 r E rw 2 r EE r (2) E If grd was clos to orthogoal Cartsa, usg CD: E W 1 tr: EE If trpolat th gradt at th cll c y 2.29 W NW w w sw η N η ξ s E Imag by MIT OpCoursWar. E ξ Oscllatory solutos do ot cotrbut to ths thrd hghrordr choc, but gradts at cll facs would stll b larg => oscllatos do occur: Th obvous soluto: Numrcal Flud Mchacs mplct r1 plct mplct (1) (2) (1) wll oscllat Nd to fd othr optos/soluto WW φ W W W φ E φ φ EE E EE Imag by MIT OpCoursWar. FJL Lctur 23, 16 r

17 FV: Appromato of dffusv flus, Cot d ) Dfrrd-Corrcto Approach, Cot d (Muzafra, 1994): y If l coctg ods ad E s arly orthogoal to cll fac, drvatv w.r.t ca b appromatd wth drvatv w.r.t ξ (as bfor): appromato clos to 2 d ordr F k k k d E r E r If grd s ot orthogoal, th dfrrd corrcto trm should cota th dffrc btw th gradts th ad ξ drctos W NW w w sw η N η ξ s E Imag by MIT OpCoursWar. E ξ d F k k E whr th frst trm s computd as: k k re r Brackt trm trpolatd from cll ctr gradts (thmslvs obtad from Gauss thorm) Hc:. ad. F d old k k old E r E r 2.29 Numrcal Flud Mchacs FJL Lctur 23, 17

18 FV: Appromato of dffusv flus, Cot d ) Dfrrd-Corrcto Approach, Cot d (Muzafra, 1994): I th formula: old F k k d E r E r Th dfrrd corrcto trm s (clos to) zro wh grd (clos to) orthogoal,.. ad ξ drctos ar th sam (clos to ach othr). y It maks th computato of drvatvs smpl (amouts to sums of ghbor valus), rcall that: old old trpolatd from, η N NW η w ξ ξ E th lattr gv by.g. c 4 c facs c dv w W s sw E Imag by MIT OpCoursWar. rvts oscllatos sc basd o sums of c, wth postv coffcts W rmad Cartsa coordats (o d to trasform coordats, w ust d to kow th ormals & surfacs), whch s hady for compl turbult modls 2.29 Numrcal Flud Mchacs FJL Lctur 23, 18

19 om commts o FV o compl gomtrs L -E dos ot always pass through th cll ctr d som updats that cas othrws, schm s of lowr ordr (.g. appromato s ot scod ordr aymor) chms ca b tdd to 3D grds but som updats ca also b dd For ampl, cll facs ar ot always plaar 3D Block B w N V 4 R R s R ' 1 ' V 3 Block-Itrfac l+1 l Cll-fac ctr L Block A Cll vrt Cll ctr l-1 Th trfac btw two blocks wth o-matchg grds. Collocatd arragmt of vlocty compots ad prssur o FD ad FV grds. NE E' E V 5 V Block-structurd grds ad std grds also d spcal tratmt For ampl, matchg at boudars (usually trpolato ad avragg) Numrcal Flud Mchacs 4 C 4 C 1 V 1 3 C 3 C 2 2 Cll volum ad surfac vctors for arbtrary cotrol volums. Imag by MIT OpCoursWar. FJL Lctur 23, 19

20 Itgral Cosrvato Law for a scalar (from Lctur 6) d dt d dv dv. s dv dt ( v. )da fd C q da C CVfC fd CM CV C C CV Advctv flus Othr trasports (dffuso, tc) ("covctv" flus) um of sourcs ad sks trms (ractos, tc) CV, fd ρ,φ s Φ v Applyg th Gauss Thorm, for ay arbtrary CV gvs:.( v) ). q s t q For a commo dffusv flu modl (Fck s law, Fourr s law): q k Cosrvatv form of th DE.( v ).( k ) s t 2.29 Numrcal Flud Mchacs FJL Lctur 23, 20

21 Cos. of Momtum: trog-cosrvatv form of th Navr-toks Equatos ( v) d dt Applyg th Gauss Thorm gvs: vdv v( ( v. ) da p da. gdv C da C CV F CV C C C CV p. g dv CV (from Lctur 6) ρ,φ v For ay arbtrary CV gvs: v.( vv v ) p.. g t Wth Nwtoa flud + comprssbl + costat μ: CV, fd s Φ q Momtum: Mass: t. v 0 v.( vv v) ) p 2 v gg Equatos ar sad to b strog cosrvatv form f all trms hav th form of th dvrgc of a vctor or a tsor. For th th Cartsa compot, th gral Nwtoa flud cas: Wth Nwtoa flud oly: v u u u 2 u u g t u.( v v ). p Numrcal Flud Mchacs FJL Lctur 23, 21

22 oluto of th Navr-toks Equatos I th FD ad FV schms, w dalt wth th dscrtzato of th grc cosrvato quato.( v) ). q s t Ths rsults apply to th momtum ad cotuty quatos (th N quatos),.g. for comprssbl flows, costat vscosty t. v 0 v.( vv v) ) p 2 v gg Trms that ar dscrtzd smlarly Ustady ad advcto trms: thy hav th sam form for scalar tha for = v Trms that ar dscrtzd dffrtly Momtum (vctor) dffusv flus d to b tratd a bt mor dtals rssur trm has o aalog th grc cosrvato quato => ds spcal attto. It ca b rgardd thr as a sourc trm (tratd o-cosrvatvly as a body forc), or as, surfac forc (cosrvatv tratmt) Fally, ma varabl v s a vctor gvs mor frdom to th choc of grds 2.29 Numrcal Flud Mchacs FJL Lctur 23, 22

23 Dscrtzato of th Covctv ad Vscous Trms Covctv trm: Us ay of th schms (FD or FV) that w hav s (cludg compl gomtrs). ad.da C ad. d u u If μ s costat, th vscous trm s as th gral cosrvato q. for Vscous trm: For a Nwtoa Flud ad comprssbl flows: If μ vars, ts drvatv ds to b valuatd For a Nwtoa flud ad comprssbl flow: Addtoal trms d to b tratd,.g. Not that o-cartsa coordat systms, w trms also ars that bhav as a body forc, ad ca thus b tratd plctly or mplctly.g u r r 2 2 ( uu.( v v( ( v. )d u ) v ) ad ad u ( v. )d 2 u Numrcal Flud Mchacs FJL Lctur 23, 23

24 Dscrtzato of th rssur trm For cosrvatv N schms, gravty/body-forc trms oft cludd th prssur trm, gvg: 2 2 u.. ( p u p p ggr u g ) 3 3 rssur th part of th strss tsor (shows up as dvrgc N qs.) Last trm s ull for comprssbl flows I o-cosrvatv N forms, th prssur gradt s dscrtzd FD schms FD schms s arlr ar drctly applcabl, but prssur ca b dscrtzd o a dffrt grd tha th vlocty grd (staggrd grd) FV schms rssur usually tratd a surfac forc (cosrvatv form): For th u quato: p. d Aga, schms s prvous lcturs ar applcabl, but prssur ods ca b o a dffrt CV grd rssur ca also b tratd o-cosrvatvly: Dscrtzato th troducs a global o-cosrvatv rror 2.29 Numrcal Flud Mchacs FJL Lctur 23, 24 V. p dv

25 2.29 Cosrvato rcpls for N Momtum ad Mass Cosrvato Momtum s cosrvd ay cotrol volum th ss that t ca oly chag bcaus of flow through th CV surfacs, forcs actg o ths surfacs or volumtrc body forcs Ths proprty s hrtd th CV formulato (f surfac flus ar dtcal o both sds) mlar statmts for Mass cosrvato Cosrvato of mportat scodary quatts,.g. rgy Mor compl ssus I hat trasfr, thrmal rgy quato ca b solvd aftr momtum quato has b solvd f proprts do t vary much wth tmpratur T T s th a passv scalar, wth o way couplg I comprssbl, sothrmal flows: ktc rgy s th sgfcat rgy I comprssbl flows: rgy cluds comprssbl trms two quatos ca b wrtt, o for ktc or tral rgy ad o for th total rgy Numrcal Flud Mchacs FJL Lctur 23, 25

26 Cosrvato rcpls for N: Cot d Ktc Ergy Cosrvato Drvato of Ktc rgy quato Tak dot product of momtum quato wth vlocty Itgrat ovr a cotrol volum CV or full volum of doma of trst Ths gvs t whr 2 2 v v dv ( v g g.. v dv 2 2. ) da p v. da (. v ). da : v p. v CV C C C CV p s th vscous compot of th strss tsor Hr, th thr RH trms th volum tgral ar zro f th flow s vscd (trm 1 = dsspato), comprssbl (trm 2) ad thr ar o body forcs (trm 3) Othr trms ar surfac trms ad ktc rgy s cosrvd ths ss: dscrtzato o CV should dally lad to o cotrbuto ovr th volum om obsrvatos Guaratg global cosrvato of th dscrt ktc rgy s ot automatc sc th ktc rgy quato s a cosquc of th momtum quato. Dscrt momtum ad ktc rgy cosrvatos caot b forcd sparatly Numrcal Flud Mchacs FJL Lctur 23, 26

27 MIT OpCoursWar Numrcal Flud Mchacs Fall 2011 For formato about ctg ths matrals or our Trms of Us, vst: