CHAPTER 12. Finite-Volume (control-volume) Method-Introduction

Transcription

1 CHAPR 12 Finit-Volum (control-volum) Mthod-Introduction

2 12-1 Introduction (1) In dvloing ht hs bcom knon s th finit-volum mthod, th consrvtion rincils r lid to fixd rgion in sc knon s control volum, r somht intrchngbly in th litrtur.

3 12-1 Introduction(2) In th finit-volum roch, oint of vi is tkn tht is distinctly diffrnt from finit-diffrnc mthod(or ylr-sris mthod ). In th ylr-sris mthod, cctd th PD s th corrct nd rorit from of th consrvtion rincil(hysicl i i l l) govrning our roblm nd mrly turnd to mthmticl tools to dvlo lgbric roximtions to drivtivs. W nvr gin considrd th hysicl l rrsntd by th PD. In th finit-volum mthod, th consrvtion sttmnt is lid in form licbl to rgion in sc (control volum).

4 12-1 Introduction(3) his intgrl form of th consrvtion sttmnt is usully ll knon from th first rincils, or it cn in most css, b dvlod dfrom th PDform of fth consrvtion from.

5 12-1 Introduction(4) h ftur of th FV mthod is shrd in common ith th finit-lmnt lmnt mthods. h FV rocdur cn, in fct, b considrd d s vrint of th finit-lmnt it t mthod, lthough it is, from nothr oint of vi, just rticulr ty of finitdiffrnc mthod.

6 12-1 Introduction(5) As n xml, considr unstdy 2-D ht conduction in rctngulr-shd solid. h roblm domin is dividd u into control volum ith ihssocitd oints. W cn stblish blihth control volums first nd lc grid oints in th cntrs of th volums (cll-cntrd mthod) or stblish th grid first nd thn fix th boundris of th control volums (cll-vrtx mthod) d)by, for xml lcing th boundris hlfy btn grid oints.

7 φ 12-1 Introduction(6) h Gnrl Diffrntil qution h diffrntil qution obying th gnrlizd consrvtion rincil cn b rittn by th gnrl diffrntil qution s ( ρφ ) t v ( ρvφ ) ( Γ φ ) s (1) :dndnt vribl, such s vlocity comonnts (u,v,), h or, k, ε concntrtion, tc.

8 12-1 Introduction(7) Γ : diffusion cofficints S: sourc trm h four trms of q.(1) r th unstdy trm, th convction trm, th diffusion trm nd th sourc trm. *Not: h consrvtion form of th PD is lso rfrrd to s consrvtion l form or divrgnc form, i.., ll stil drivtivs r urly s divrgncs.

9 12-1 Introduction(8) Consrvtion form of th govrning qutions of fluid flo ρ Mss : ( ρ v ) 0 t v ( ρ ) vv Momtum : ρ vv μ v t ( ρh) v nrgy : ( ρvh ) ( k ) S t ( ρc) v Scis : ( ρvc) ( D C) SC tt ( ) ( ) S M

10 12-1 Introduction(9) On-y nd to-y coordints : 1. Dfinitions: to-y coordint is such tht th conditions t givn loction in tht coordint r influncd by chngs in conditions on ithr sid of tht loction. A on-y coordint is such tht th conditions t givn loction in tht coordint r influncd by chngs in th conditions on only on sid of tht loction.

11 12-1 Introduction(10) 2. xmls: on-dimnsionl stdy ht conduction in rod rovids on xml of to-y coordint. h tmrtur of ny givn oint in th rod cn b influncd by chnging th tmrtur of ithr nd. Normlly, sc coordints r to-y coordints. im, on th othr hnd, is lys on-y coordint. During th unstdy cooling of solid, th tmrtur t givn instnt cn b influncd by chnging only ths conditions tht t rvild bfor tht instnt.

12 12-1 Introduction(11) 3. Sc s on-y coordint: If thr is strong unidirctionl i flo in th coordint dirction, thn significnt influncs trvl only from ustrm. h conditions t givn oint r thn ffctd lrgly by th ustrm conditions, nd vry littl by th donstrm ons. It is tru tht convction is on-y rocss, but diffusion (hich is lys rsnt) hs to-y influncs. Hovr., thn th flo rt is lrg, condition ovrors diffusion nd thus mk th sc coordint nrly on-y y.

13 12-1 Introduction(12) 4. Prbolic, llitic, hyrbolic: ) h trm rbolic indicts on y bhvior, hil llitic signifis th to-y conct. b) It ould b mor mningful if situtions r dscribd d s bing rbolic or llitic in givn coordint. hus, th unstdy ht condition roblm, hich is normlly clld rbolic, is ctully rbolic in tim nd llitic in ll coordint. A to-dimnsion boundry lyr is rbolic in th strm is coordint nd llitic in th cross-strm coordint

14 12-1 Introduction(13) c) A hyrbolic roblm hs kind of on-y bhvior, hich is, hovr, not long coordint dirctions but long scil-lins clld chrctristics. d) A sitution is rbolic if thr xists t lst on on-y coordint: othris, it is llitic. ) A flo ith on on-y sc coordint is somtims clld boundry-lyr-ty flo, hil flo ith ll to-yy coordint is rfrrd to s rcirculting flo.

15 12-1 Introduction(14) 5. Comuttionl imlictions: h motivtion for th forgoing discussion bout h motivtion for th forgoing discussion bout on-y nd to-y coordints is tht, it on-y coordint cn b idntifid in givn sitution, substntil conomy of comutr storg nd comutr tim is ossibl.

16 12-2 An Illustrtiv xml(1) h FV mthod usd th intgrl form of th consrvtion qution(q.1) s th strting oint: CV r ( Γ φ ) dv s dv n ( Γ φ ) da s dv (2) CV ϕ A Lt us considr stdy on-dimnsionl ht conduction govrnd by d d k s dx dx 0 (3) CV ϕ

17 12-2 An Illustrtiv xml(2) 1. Prrtion: o driv th discrriztion qution, shll mloy th grid-oint clustr shon in Fig.1. W focus ttntion on th grid oint P, hich hs th grid oints nd W s its nighbors.( dnots th st sid, hil W stnds for th st sid). h dshd lins sho th fcs of th control volum. h lttrs nd dnot ths fcs.

18 12-2 An Illustrtiv xml(3) (δ x) (δ x) W P x Fig. 1 For on-dimnsionl roblm undr considrtion, shll ssum unit thicknss in th y nd z dirctions. hus, th volum of th cv shon is x 1 1. If intgrt q(3) ovr th cv, gt k x k x sdx 0 (4)

19 12-2 An Illustrtiv xml(4) 2. Profil ssumtion: o mk furthr furthr rogrss, nd rofil ssumtion or n introltion formul. Hr, linr introltion functions r usd btn th grid oints, s shon in Fig 2. Fig. 2 W δx δx x

20 12-2 An Illustrtiv xml(5) 3. h discrriztion qution: If vlut th drivtivs d/dx in q.(4) from th icis-linr rofil, th rsulting qution ill b k ( ( ) ( ) k ( δx) ( δx) sx 0 (5)

21 12-2 An Illustrtiv xml(6) hr s is th vrg vlu of s ovr th cv. It is usful to cst th discrtiztion q(5) into th folloing form : b (6) Whr k W W, k W ( δx ) ( δx ), P W, b sx

22 12-2 An Illustrtiv xml(7) 4. Commnts: ) In gnrl, it is convnint to xtnd q.(6) into multidimnsionl form s P P b nb nb (7) hr nb dnots nighbor, nd th summtion is to b tkn ovr ll th nighbors. b) In driving q(6), hv usd th simlst rofil ssumtion tht nbld us to vlut d/dx. Of cours, mny othr introltion functions ould hv bn ossibl.

23 12-2 An Illustrtiv xml(8) c) Furthr, it is imortnt to undrstnd tht nd not us th sm rofil for ll quntitis. d) vn for givn vribl, th sm rofil ssumtion nd not b usd for ll trms in th qution.

24 12-2 An Illustrtiv xml(9) 5. rtmnt of sourc trm: h discrtiztion qutions ill b solvd by th tchniqus for linr lgbric qutions. h rocdur for linrizing givns~ rltionshi is ncssry. Hr, it is sufficint to xrss th ovrg vlu S s S S C S P P

25 12-2 An Illustrtiv xml(10) Whr S c stnds for th constnt rt of S, hil S is th cofficint of With th hil S is th cofficint of. With th linrizd sourc xrssion, th di ti ti ti ill b discrtiztion qution ill bcom b ( ) x k hr b W W P P δ ( ) ( ) x k W δ x S b x s C W P

26 12-3 h Four Bsic Ruls(1) Rul 1:Consistncy t control-volum fc -Whn fc is common to to djcnt control volums, th flux cross it must b rrsntd by th sm xrssion in th discrtiztion qutions for th to control volums Rul 2:Positiv cofficints -All cofficints ( nd nighbor cofficints nb ) must lys b ositiv.

27 12-3 h Four Bsic Ruls(2) Rul 3:Ngtiv-slo linriztion of th sourc trm -Whn th sourc trm is linrizd s SS C S P P, C P P th cofficint S P must lys b lss thn or qul to zro. Rul 4:Sum of th nighbor cofficints -W rquir P nb

28 CHAPR 13 h Finit Volum Mthod for Diffusion Problms

29 13-1 Stdy On-dimnsionl Condition(1) h Bsic qution d dx k d dx S 0 (1) h Discrtiztion qution P P hr b (2) W k δx b W ( ) k δ x S C W ( ) x W S P x

30 13-1 Stdy On-dimnsionl Condition(2) h Grid Scing 1. For th grid oints shon in 8.4, it it not ncssry tht th distncs (δx) nd (δx) b qul. Indd, th us of non-uniform grid scing is oftn dsirbl, for it nbls us to dloy comuting or ffctivly. In gnrl, shll obtin n ccurt solution o only hn th grid is sufficintly fin, but thr is no nd to mloy fin grid in rgions hr th dndnt vribl chngs rthr sloly ith x. On th othr hnd, fin grid is rquird hr th ~x vrition is st.

31 13-1 Stdy On-dimnsionl Condition(3) 2. A misconction sms rvil tht non- uniform grid ld to lss ccurcy thn do uniform grids. hr is no sound bsis for such n ssrtion. Also thr r no univrsl ruls bout ht mximum (or minimum) rtio th djcnt grid intrvls should mintin.

32 13-1 Stdy On-dimnsionl Condition(4) 3. Sinc th ~x distribution is not knon bfor th roblm is solvd, ho cn dsign n rorit non-uniform grid? First: On normlly hs som qulittiv xcttions bout th solution, from hich som guidnc cn b obtind. scond: rliminry cors-grid solutions cn b usd to find th ttrn of th ~x vrition; thn suitbl nonuniform grid cn b constructd.

33 13-1 Stdy On-dimnsionl Condition(5) h Intrfc Conductivity 1. h most strightforrd rocdur for obtining th intrfc conductivity k is to ssum linr vrition of k btn oints P nd (δx) P (δx) - (δx) x

34 13-1 Stdy On-dimnsionl Condition(6) hn, k f k (1 f ) k (3) hr f ( δx) ( δx) (4) ( ) If th intrfc r midy btn grid oints, f ould b 0.5, nd k ould b h rithmtic mn of k nd k.

35 13-1 Stdy On-dimnsionl Condition(7) 2.W shll shortly sho tht this siml-mindd roch lds to rthr incorrct imlictions in som css nd cnnot ccurtly hndl th brut chngs of conductivity i tht my occur in comosit mtrils. Fortuntly, much bttr ltrntiv is vilbl. 3.Our min objctiv is to obtin good rrsnttion for th ht flux q t th intrfc vi

36 13-1 Stdy On-dimnsionl Condition(8) 3.Our min objctiv is to obtin good rrsnttion for th ht flux q t th intrfc vi k ( ) q (5) ( δx) For th comosit slb btn oints P nd, stdy on-dimnsionl nlysis (ithout sourcs) ld to q ( δx ) ( δx ) k P P k (6)

37 13-1 Stdy On-dimnsionl Condition(9) Combintion of qs.(4) (6) yilds k 1 f f k k 1 (7) Whn th intrfc is lcd midy btn nd, hv f 0.5; thn ( 1 1 k k k ) or 2 k k k k q. (9) sho tht k is th hrmonic mn of k nd k, rthr thn th rithmtic mn. k (9) (8)

38 13-1 Stdy On-dimnsionl Condition(10) 4. ( δx) ( δx) k k A similr xrssion cn b rittn for W. 1

39 13-1 Stdy On-dimnsionl Condition(11) 5. h rcommndd intrfc conductivity formul (7) is bsd on th stdy, no-sourc, on-dimnsionl sitution in hich th conductivity vris in stis fshion from on control volum to th nxt. vn in situtions ith nonzro sourcs or ith continuous vrition of conductivity, it rforms much bttr thn th rithmtic- mn formul.

40 13-1 Stdy On-dimnsionl Condition(12) Itrtion 1. Strt ith guss or stimt for th vlus of f t ll grid oints. 2. From ths gussd s, clcult tnttiv vlus of th cofficints in th discrtiztion qution. 3. Solv th nominlly st of lgbric qutions to gt n vlus of. 4. With ths s s bttr gusss, rturn to st 2 nd rt th rocss until furthr rtitions cs to roduc significnt chngs in th vlus of.

41 13-1 Stdy On-dimnsionl Condition(13) Sourc-rm Linriztion *: th guss vlu or th rvious-itrtionitrtion vlu of xml 1: Givn S5-4 -S c 5, S -4 rcommndd -S c 5-4 *S *, 0 not imrcticl -S c 57 *,S -11 str S~ rltionshi, ill slo don th convrgnc

42 13-1 Stdy On-dimnsionl Condition(14) xml 2: Givn S37 1. S c 3,S 7 this is not cctbl, s it mks S ositiv. h rsnc of ositiv S mny cus divrgnc. 2. S c 37 *, S 0 this is th rctic on should follo. 3. S c 39 *, S -2 this is n rtificil crtiv S. It ill, in gnrl, slo don th convrgnc.

43 13-1 Stdy On-dimnsionl Condition(15) xml 3: Givn S S c 4-5 * 3, S 0 this 0 is th lzy-rson roch. 2. S c 4, S -5 * 2 this givn S~ curv is str thn this imlis.

44 13-1 Stdy On-dimnsionl Condition(16) 3. Rcommndd mthod: S S * * ds d ( ) * ( ) * 3 * 2 * hus, S c * 3 15 * * 3, S 15 his linriztion rrsnts th tngnt to th S~ curv t * * 2

45 13-1 Stdy On-dimnsionl Condition(17) 4. S c 420 *, S -25 * 2 his givns str S~ curv, hich ould slo don convrgnc S (1) (2) (4) (3)

46 13-1 Stdy On-dimnsionl Condition(18) Boundry Conditions: 1. yiclly, thr kinds of boundry conditions r ncountrd in ht condition. hs r -Givn boundry tmrtur. t -Givn boundry ht flux -Boundry ht flux scifid vi ht trnsfr cofficint nd th tmrtur of th surrounding fluid.

47 13-1 Stdy On-dimnsionl Condition(19) 2. If th boundry tmrtur is givn, no rticulr difficulty riss, nd no dditionl qutions r rquird. Whn th boundry tmrtur is not givn, nd to construct n dditionl qution for B. his is don by intgrting th diffrntil qution ovr th hlf control volum shon djcnt to th boundry in th folloing Figur.

48 13-1 Stdy On-dimnsionl Condition(20) Hlf C.V. B I W P Fig 1 yicl C.V. (δx) i q B B x i Fig 2 I

49 13-1 Stdy On-dimnsionl Condition(21) 3. Aly th rincils of nrgy consrvtion x s dx d k dx d k dx s dx d k dx d B i i B B i Alying th rincils of nrgy consrvtion ovr C.V. of Fig.2 nd noting g g tht th ht flux q stnds for -k(d/dx), gt x S S q q B c i B 0 ) ( g ( ) ( ) ( ) b x S S dx k q B c i I B i B B c i B 0 ( ) x S q x S b x k hr b P I B B c i i I I I B B,, δ

50 13-1 Stdy On-dimnsionl Condition(22) 4. If q B is scifid in trms of ht trnsfr cofficint h nd surrounding-fluid tmrtur f such tht q B h( f - B ) hn, th qution for B bcoms B B II b ki hr I, b SCx hf, δx ( ) B I SP x i h

51 13-1 Stdy On-dimnsionl Condition(23) Solution of th Linr Algbr qution (DMA) 1. h discrtiztion qutions cn b rittn s i i b c d i i 1 i i1 i (1)

52 13-2 Unstdy On-Dimnsionl Condition(1) h gnrl Discrtiztion qution 1. Unstdy on-dimnsionl ht-conduction qution if ( ρc ) k t x x ρ,c r constnt, ρc t k x x (1) q.(1) bcoms (2)

53 13-2 Unstdy On-Dimnsionl Condition(2) 2. h discrtiztion qution W P ρc ρc tt t dtdx t tt t t dtdx ρc t tt t x x k x ( 0 ) P dxdt (3)

54 13-2 Unstdy On-Dimnsionl Condition(3) ( ) ( ) ( ) t t dt k k x c (4) 0 ρ ( ) ( ) ( ) [ ] t t t P t f f dt dt x x x c (5) ) (1 (4) 0 δ δ ρ hr f is ighting fctor btn 0 nd 1 from qs (4) nd (5) cn gt [ ] t t f f dt ) (5 ) (1 from qs.(4) nd (5), cn gt ( ) ( ) ( ) k k f x ( ) ( ) ( ) ( ) ( ) k k x x f t c δ δ ρ ( ) ( ) ( ) ( ) ( ) 1 (6) x k x k f δ δ

55 13-2 Unstdy On-Dimnsionl Condition(4) hr 0 0 [ f ( ) ] f W f W ( 1 f ) ( f ) ( 1 f ) k [ 0 ] 1 W ( ) δx ( x ) cx 0 f f δ ρ t [ ] 0 0 P 1 W P (6) k

56 13-2 Unstdy On-Dimnsionl Condition(5) 1. xml: Crnk-Nicolson, nd Fully Imlicit Schms f f f ( x) ρc 0 : x licit schm stbility critri : t < 2k 0.5 : Crnk Nicolson schm unconditionlly stbl 1: Fll Fully imlicit i schm 2

57 13-2 Unstdy On-Dimnsionl Condition(6) 2. Vrition of tmrtur ith tim for thr diffrnt schms 0 xlicit Fully imlicit it Crnk-Nicolson t tt t

58 13-2 Unstdy On-Dimnsionl Condition(7) 3. Why ould rfr th fully imlicit schm? ) For f0 (xlicit) schm ) 0 ( W P his mns tht P is not rltd to othr unknos such s or W W,, but is xlicitly obtinbl in trms of knon tmrtur P0, 0, W0.

59 13-2 Unstdy On-Dimnsionl Condition(8) b) For f0.5, th cofficint of P0 in q(6) bcoms P0 -( W )/2. For uniform conductivity nd uniform grid scing, this cofficint cn b sn to b ρc( x/ t)-k/ x. Whnvr th tim st is not sufficintly i smll, this cofficint i could bcom ngtiv, ith its otntil for hysiclly unrlistic rsults. c) For f1, th cofficint of P0 in q(6) must nvr bcom ngtiv. It is for this rson tht shll dot th fully imlicit schm in this book.

60 13-2 Unstdy On-Dimnsionl Condition(9) h Fully Imlicit Discrtittion qution 1 b 1. P P W W b Whr ( ) ( ) t x c x k x k P W W W 0,,, ρ δ δ x S b C x S P P W P 0 2. It cn b sn tht, s t, this qution rducs to our stdy-stt discrtiztion qution.

61 13-3 o-and hr- Dimnsionl Situtions(1) Discrtiztion qution for o Dimnsions ( ) c ρ ( ) s y k y x k x t c ρ ( ) ρ thn constnt, c, if ( ) s y k y x k x t c ρ

62 13-3 o-and hr- Dimnsionl Situtions(2) N x n W y s S y x

63 13-3 o-and hr- Dimnsionl Situtions(3) P P W W N N b h hr x k x k y k y k s W n N W W,,,, ( ) ( ) ( ) ( ) y x c y y x x P S W n N W W,,,, 0 ρ δ δ δ y x S b t C P, 0 0 y x S P P N S W P Discrtiztion qution for hr Dimnsions 0 b B B S S N N W W P P

64 13-4 Ovrrlxtion nd Undrrlxtion(1) Ovrrlxtion is oftn usd in conjunction ith th Guss-Sidl mthod, th rsulting schm bing knon s succssiv Ovr-Rlxtion (SOR). With th lin-by-lin mthod, th us of ovrrlxtion is lss common. Undrrlxtion is vry usful dvic for nonlinr roblm. It is oftn mloyd to void divrgnc in th itrtiv solution of strongly nonlinr qutions.

65 13-4 Ovrrltion nd Undrrlxtion(2) h gnrl discrtiztion qution of th form is P P * P or P P * P nb nb nb P nb b (1) b (2) is th vlu of P P α * P P α nb nb P P nb b * P (3) from th rvious itrtion. nbnb b * P (4) P P * nb b 1α P (4b α ( ) )

66 13-4 Ovrrltion nd Undrrlxtion(3) At first, it should b notd tht, hn th itrtions, tht is, P bcoms qul to P*. q. (4) imlis tht th convrgd vlus of do stisfy th originl q.(1). Any rlxtion schm, of cours must ossss this rorty; th finl convrgd solution, lthough obtind through th us of rbitrry rlxtion fctors or similr dvics, must still stisfy th originl discrtiztion qution.

67 13-4 Ovrrltion nd Undrrlxtion(4) hr r no gnrl ruls for choosing th bst vlu of α. h otimum vlu dnds uon numbr of fctors, such s th ntur of fth roblm, th numbr of grid oints, th grid scing, nd th itrtiv rocdur usd. Usully, suitbl vlu of α cn b found by xrinc.

68 CHAPR 14 Convction nd Diffusion

69 14-1 Convction-Diffusion rm h convction trm hs n insrbl connction ith th diffusion trm, nd thrfor, th to trms nd to b hndld s on unit. Govrning gqutions ρ t x j ( ρu ) ρu j 0 j ( ρφ ) ( ) φ ρ φ Γ S t x j u j x j x j

70 14-2 Stdy On-Dimnsionl Convction nd Diffusion(1) Govrning qutions d dx d dx ( ρu) 0 or ρu cons tn t d φ dx xx ( ρuφ ) Γ (1)

71 14-2 Stdy On-Dimnsionl Convction nd Diffusion(2) A Prliminry Drivtion 1. Intgrtion of q (1) ovr th C.V. shon in Fig.1 givs dφ dx dφ dx ( ρuφ ) ( ρuφ ) Γ Γ (2) C.V. W P (δx) (δx)

72 14-2 Stdy On-Dimnsionl Convction nd Diffusion(3) 2. Diffusion trm: h sm y of chtr Convction trm: 1 ( 1 φ φ ) nd φ ( φ φ ) φ P nd P φ 2 2 h fctor ½ riss from th ssumtion of th intrfcs bing midy. No, q (1) cn b rittn s 1 2 Γ 1 P 2 φ φp Γ φp φ δx δx ( ρ u) ( φ φ ) ( ρu) ( φ φ ) ( ) ( ) ( W ) ( ) P W W

73 14-2 Stdy On-Dimnsionl Convction nd Diffusion(4) 4. ) Γ F ρu, D δx b) Both hv th sm dimnsions; F indicts th strngth of th convction (or th flo), hil D is th diffusion conductnc. c) D lys rmins ositiv, F cn tk ithr ositiv or ngtiv vlus dnding on th dirction of th fluid flo.

74 14-2 Stdy On-Dimnsionl Convction nd Diffusion(5) 5. Discrtiztion qution: (3) F W W P P φ φ φ ) (3 2 F F D Whr ) (3 2 F F b F D W ( ) ) (3 2 2 c F F F D F D P ( ) ) 3 ( c F F W

75 14-2 Stdy On-Dimnsionl Convction nd Diffusion(6) 6. Discussion: ) Sinc by continuity FF, do gt th rorty P W b) q q( (3) is lso knon s th cntrl-diffrnc schm nd is th nturl outcom of ylorsris formultion.

76 14-2 Stdy On-Dimnsionl Convction nd Diffusion(7) c) xml: D D 1 nd F F 4 Considr to sts of vlus: i. If Φ 200 nd Φ W 100, thn Φ P 50! ii. If Φ 100 nd Φ W 200, thn Φ P250! Sinc, in rlity, cnnot fll outsid th rng of 100~200 stblishd by its nighbors, ths rsults r clrly unrlistic.

77 14-2 Stdy On-Dimnsionl Convction nd Diffusion(8) d) qs (3)-(3c) indict tht th cofficints could, t tims, bcom ngtiv. Whn F xcds 2D, thn, dnding on hthr F is ositiv or ngtiv, hr is ossibility of or W bcoming ngtiv. his ill b violtion of on of th bsic ruls.

78 14-2 Stdy On-Dimnsionl Convction nd Diffusion(9) h Uind Schm 1. h uind schm rcognizs tht th k oint in th rliminry formultion is th ssumtion tht th convctd rorty Φ t th intrfc is th vrg of Φ nd Φ P, nd it rooss bttr rscrition. h formultion of th diffusion trm is lft unchngd, but th convction trm is clcultd from th folloing ssumtion:

79 14-2 Stdy On-Dimnsionl Convction nd Diffusion(10) h vlu of Φ t n intrfc is qul to th vlu of Φ t th grid oint on th uind sid of th fc. thus, Φ Φ if F>0 (4) nd, Φ Φ if F<0 (4b) h vlu of Φ cn b dfind similrly. W P ρuφ ρuφ ( ) ( ) ( Fφ ) ( Fφ )

80 14-2 Stdy On-Dimnsionl Convction nd Diffusion(11) 2. W shll dfin A,B to dnot th grtr of A nd B. hn, th uind schm imlis F φ [ F, 0] φ [ F,0] (5) φ

81 14-2 Stdy On-Dimnsionl Convction nd Diffusion(12) 3. h discrtiztion qution bcoms φ hr φ φ W P D D D W W (6) [ F ],0 (6 ) [ F,0] (6b) [ F,0] D [ 1 F,0] W ( F F ) (6 c )

82 14-2 Stdy On-Dimnsionl Convction nd Diffusion(13) 4. Discussion: It is vidnt from qs.(6) tht no ngtiv cofficints ould ris, thus, th solutions ill lys b hysiclly rlistic.

83 14-2 Stdy On-Dimnsionl Convction nd Diffusion(14) h xct Solution x x φ x ( ρuφ ) Γ (1) cn b solvd xctly ifγγ is tkn to b constnt.(ρu is constnt)

84 14-2 Stdy On-Dimnsionl Convction nd Diffusion(15) If domin 0 x L is usd, ith th boundry conditions At At x 0 φ φ0 x L φ φ h solution of q(1) is x( ( ) φ φ P x 1 0 L φ φ x P 1 L 0 L ( ) (7) hr P is Pclt numbr dfind by ul ρ Γ It cn b sn tht P is th rtio of convction nd diffusion. th strngths of

85 14-2 Stdy On-Dimnsionl Convction nd Diffusion(16) 2. h ntur of th xct solution: ф L -P>>1 ф ф 0 P -1 P0 P1 P>>1 Fig.2 xct solution for th on-dimnsionl convctiondiffusion roblm 0 L x

86 14-2 Stdy On-Dimnsionl Convction nd Diffusion(17) ) In th limit of zro Pclt numbr, gt th ur-diffusion (or conduction) roblm, nd th ф~x vrition is linr. b) Whn th flo is in th ositiv x dirction (i.., for ositiv vlus of P), th vlus of ф in th domin sm to b mor influncd by th ustrm vlu ф 0. c) For lrg ositiv vlu of P, th vlu of ф rmins vry clos to th ustrm vlu ф 0 ovr much of th domin.

87 14-2 Stdy On-Dimnsionl Convction nd Diffusion(18) d) Whn th fluid flos in th ngtiv x dirction, Φ L bcoms th ustrm vlu, hich domints th vlus of Φ in th domin. ) Whn lrg ngtiv P, th vlu Φ of ovr most of th rgion is vry nrly qul to Φ L.

88 14-2 Stdy On-Dimnsionl Convction nd Diffusion(19) 3. Imlictions: ) It is sy to s hy our rliminry drivtion fild to giv stisfctory formultion. h Φ ~x rofil is fr from bing linr xct for smll vlus of P. b) Whr P is lrg, th vlu of Φ t xl/2 (th intrfc is nrly qul to th vlu of Φ t th uind boundry. his is rcisly th ssumtion md in th uind schm; but thr it is usd for ll vlus of P, not just for lrg vlu.

89 14-2 Stdy On-Dimnsionl Convction nd Diffusion(20) c) Whr P is lrg, dφ/dx is nrly zro t xl/2. hus th diffusion is lmost bsnt. h uind schm lys clcults th diffusion trm from linr Φ~x rofil nd thus ovrstimts diffusion i t lrg vlu of P.

90 14-2 Stdy On-Dimnsionl Convction nd Diffusion(21) h xonntil schm 1. It is usful to considr totl flux J tht is md of th convction flux ρuφ nd th diffusion flux -Γdφ /dx. hus, J dφ ρ u φ Γ (8) ) dx ith this dfinition q (1) bcoms dj dx 0 (9) hich, hn intgrtd ovr th C.V. shon in fig.1 givs J J 0 (10)

91 14-2 Stdy On-Dimnsionl Convction nd Diffusion(22) 2. h substitution of q. (9) into q. (8) ould giv th xrssion for J : φ φ J F φ ( ) (10) x 1 ( ρu) ( ) δx F hr Γ D

92 14-2 Stdy On-Dimnsionl Convction nd Diffusion(23) 3. Finlly, substitution of q. (11) nd similr xrssion for J into q. (10) lds to xrssion for J into q. (10) lds to (12) 0 F F W φ φ φ φ φ φ ( ) ( ) (13) (12) 0 1 x 1 x F F W W P P W φ φ φ φ φ ( ) ) (14 1 / x D F F hr ( ) ( ) ) (14 1 / x / x b D F D F F ( ) ) 14 ( c F F W

93 14-2 Stdy On-Dimnsionl Convction nd Diffusion(24) 4. Discussions: ) hs cofficints xrssions dfin th xonntil schm. Whn usd for th stdy on-dimnsionl roblm, this schm is gurntd to roduc th xct solution for ny vlu of th Pclt numbr nd for ny numbr of grid oints.

94 14-2 Stdy On-Dimnsionl Convction nd Diffusion(25) b) Dsit its highly dsirbl bhvior, it is not idly usd bcus i. xonntils r xnsiv to comut. ii. Sinc th schm is not for to- or thr-dimnsionl situtions, nonzro sourcs, tc., th xtr xns of comuting th xonntils dos not sm to b justifid.

95 14-2 Stdy On-Dimnsionl Convction nd Diffusion(26) h Hybrid schm 1. o rcit th connction btn th xonntil schm nd th hybrid schm, shll lot /D vs. s follos:

96 14-2 Stdy On-Dimnsionl Convction nd Diffusion(27) D 1 D 2 D From q(14), dduc tht D x( ) 1 xct Fig ( ) 1 D 0

97 14-2 Stdy On-Dimnsionl Convction nd Diffusion(28) 2. From Fig 3, cn gt ) For D b) For D c) At 0, th tngnt is 0 D 1 2

98 14-2 Stdy On-Dimnsionl Convction nd Diffusion(29) 3. h hybrid schm is md u of ths thr stright lins of Fig.3, so tht stright lins of Fig.3, so tht 2 ) < For ) 2 ) < For b D For 0 2 ) ) > For c D For b 0 2 ) > D For c

99 14-2 Stdy On-Dimnsionl Convction nd Diffusion(30) 4. hs xrssions cn b combind into comct form s follos: comct form s follos: 0 1 D 0 0, 2 1, F D F or D 0, 2, D F or

100 14-2 Stdy On-Dimnsionl Convction nd Diffusion(31) 5. h significnc of th hybrid schm cn b undrstood by obsrving tht ) It is idnticl ith th cntrl-diffrnc schm for 2 < < 2 b) Outsid this rng it rducs to th uind schm in hich th diffusion hs bn st qul to zro. c) h nm hybrid is indictiv of combintion of th cntrl-diffrnc nd uind schm, but it is bst to considr th hybrid schm s th thrlin roximtion to th xct curv, shon in Fig.3

101 14-2 Stdy On-Dimnsionl Convction nd Diffusion(32) 6. h convction-diffusion discrtiztion for th hybrid schm cn no b rittn s th hybrid schm cn no b rittn s W W P P φ φ φ F hr F F D F 0, 2, F D F 0, 2, ( ) W P F F

102 14-2 Stdy On-Dimnsionl Convction nd Diffusion(33) h Por-L schm 1. It cn b sn from Fig.3, tht th drtur of th hybrid schm from th xct curv is rthr lrg. A bttr roximtion to th xct curv is givn by th Por-l schm.

103 14-2 Stdy On-Dimnsionl Convction nd Diffusion(34) 2. h Por-l xrssions for cn b rittn s ) For < 10 D b ) For -10 (15 ) 5 < 0 ( ) (15 b ) c ) For 0 10 D D 5 ( ) (15 c )

104 14-2 Stdy On-Dimnsionl Convction nd Diffusion(35) A Gnrlizd Formultion 1. h gnrl convction-diffusion formultion cn b rittn s φ φ hr D D A φ ( ) [ F ],0 ( ) [ F,0 ] A ( F F )

105 14-2 Stdy On-Dimnsionl Convction nd Diffusion(36) 2. h function A( ) for diffrnt schms Cntrl diffrnc Uind Hybrid Por l xonntil( xct) [ 0,1 0.5 ] ( ) 5 [ ] 0, [ x( ) 1]

106 14-3 Discrtiztion qution for o Dimnsions(1) Discrtiztion qution of 2-D t J J x y ( ρφ ) s (1) x y hr J x nd J y r th totl (convction lus diffusion) fluxs dfind by J x J y ρuφ Γ ρvφ Γ φ x φ y ( 1) ( 1b)

107 14-3 Discrtiztion qution for o Dimnsions(2) h intgrtion of q(1) ovr th c.v. shon in Fig.1 ould giv N x n W J J s J n J s S Fig.2 J y y x

108 14-3 Discrtiztion qution for o Dimnsions(3) ( ) 0 0 y x φ ρ φ ρ ( ) J J J J t y x s n φ ρ φ ρ ( ) ) (2 y x S S c φ

109 14-3 Discrtiztion qution for o Dimnsions(4) In similr mnnr, cn intgrt th continuity q continuity q 0 F F y x ρ 0 y x t y x ρ ovr th c.v. nd obtin ovr th c.v. nd obtin ( ) F F F F y x ρ ρ (3) 0 0 u hr F F F F F t s n ρ (3) 0 ρ

110 14-3 Discrtiztion qution for o Dimnsions(5) (3) (2) 0 y x ρ φ ( ) ( ) ( ) ( ) ( ) ( ) ) (4 0 y x S S F J F J F J F J t y x φ φ φ φ φ ρ φ φ ( ) ( ) ( ) ) (4 y x S S F J F J c s s n n φ φ φ

111 14-3 Discrtiztion qution for o Dimnsions(6) h finl discrtiztion qution J J F φ F φ ( φ ) φ ( φ φ ) ( ) [ F ],0 ( ) [ F,0] hr D A W DW A h 2 - D discrtiz tion qution cn no b rittn s φ φ Wφ Nφ N SφS b

112 14-3 Discrtiztion qution for o Dimnsions(7) hr ( ) [ ] F A D 0, ( ) [ ] ( ) [ ] F A D F A D W 0,0 ( ) [ ] ( ) [ ] F A D F A D s s s S n n n N,0,0 t y x s s s S 0 0 ρ y x S b t C 0 0 y x S S N 0

113 14-4 On-Wy Sc Coordint(1) im is on-y coordint. h convction- diffusion formultion rvls tht sc coordint cn lso bcom on-y. Wht mks sc coordint On-Wy? Whn th Pclt numbr xcds10, th Por- l schm ill st th donstrm-nighbor ihb cofficint qul to zro. (th hybrid schm dos this for Pclt numbr grtr thn 2.)

114 14-4 On-Wy Sc Coordint(2) Suos tht, in th 2-D sitution shon in Fig.2, thr is high h flo rt in th ositiv x dirction. hn, for ll th grid oints long y-dirction lin, th cofficint ill b zro. In othr ords, φ ill b dndnt on φ W, φ N, nd φ S, but not on. hus th x coordint ill bcom on-y coordint sinc th φ vlu t ny oint ill b uninfluncd by ny of th donstrm vlus.

115 14-4 On-Wy Sc Coordint(3) N S Fig.2

116 14-4 On-Wy Sc Coordint(4) h outflo Boundry Conditions At th outflo boundry shon in Fig.3, for xml, on my not kno th tmrtur or th ht flux. Ho cn thn solv th roblm? h nsr is surrisingly siml: No boundrycondition informtion is ndd t n outflo boundry. For ll grid oints nxt to th outflo boundry, th cofficint ill b zro, nd hnc no boundry vlus ill b ndd. In othr ords, th rgion nr th outflo boundry xhibits, for lrg Pclt numbr, locl on-y bhvior.

117 14-4 On-Wy Sc Coordint(5) outflo boundry Inflo boundry Fig 3 W N P S

118 14-4 On-Wy Sc Coordint(6) A rticulrly bd choic of n outflo- boundry loction is th on in hich thr is n inflo ovr rt of it. An xml of fthis shon in Fig.4. For such bd choic of th boundry, no mningful solution cn b obtind.

119 14-4 On-Wy Sc Coordint(7) Bd Fig. 4 Good

120 CHAPR 15 Clcultion of h Flo Fild

121 15-1 Nd for Scil Procdur h rl difficulty in th clcultion of th vlocity fild lis in th unknon rssur fild. h rssur grdint forms rt of sourc trm for momntum qution.

122 15-2 Som Rltd Difficultis(1) Rrsnttion of th Prssur-Grdint rm 1. o intgrt d/dx ovr th control volum shon in Fig.1, cn gt -. W P x c.v. Fig. 1

123 15-2 Som Rltd Difficultis(2) 2. o xrss - in trms of th grid-oint rssur, my ssum icis-linr rofil for rssur. hrfor, cn gt W 2 2 W 2 his mns tht th momntum qution ill contin th rssur diffrnc btn to ltrnt grid oints, nd not btn djcnt ons.

124 15-2 Som Rltd Difficultis(3) 3. hr is nothr imliction tht is fr mor srious. It cn b sn from Fig.2, hr rssur fild is roosd in trms of th grid- oint vlus of rssur Fig. 2 such zig-zg fild cnnot b rgrdd s rlistic; but for ny grid oint, th corrsonding W - cn b sn to zro, sinc th ltrnt rssur vlus r vryhr qul.

125 15-2 Som Rltd Difficultis(4) Rrsnttion of th continuity qution If intgrt t th continuity it qution ovr th c.v. shon in Fig1, hv u -u 0

126 15-2 Som Rltd Difficultis(5) Onc gin, th us of icis-linr rofil for u nd of th midy loctions of th control-volum fcs lds to u u 2 u W u 2 0 or u u hus, th discrtizd continuity qution dmnds th qulity of vlocitis t ltrnt grid oints nd not t djcnt ons. W 0

127 15-3 A Rmdy : h stggrd Grid(1) h difficultis dscribd so fr cn b rsolvd by rcognizing tht do not hv to clcult ll th vribls for th sm grid oints. W cn, if ish, mloy diffrnt grid for ch dndnt vribl.

128 15-3 A Rmdy : h stggrd Grid(2) Stggrd grid: 1. h vlocity comonnts r clcultd for th oints tht li on th fc of th control volum, s shon in Fig Othr vribls r clcultd for th grid oints(smll circls). Fig.3 u i,, v i, othr vribls

129 15-3 A Rmdy : h stggrd Grid(3) 3. h imortnt dvntgs r tofold: ) For tyicl c.v. shon in Fig.3, it is sy to s tht th discrtizd continuity qution ould contin th diffrnc of djcnt vlocity comonnts, nd this ould rvnt vy vlocity fild. b) h scond imortnt dvntg of th stggrd grid is tht th rssur diffrnc btn to djcnt grid oints no bcoms th nturl driving forc for th vlocity comonnt loctd btn ths grid oints.

130 15-4 h Momntum qutions(1) h discrtiztion qution (2-D) cn b rittn s n u v n similrly nb nb t u v nb nb t b b ( P ) A ( 1) ( P N ) An (1b ) b ( ) A (1 c ) nb nb P t t

131 15-4 h Momntum qutions(2) n Fig.4 c.v. for u Fig.4b c.v. for v h rssur grdint is not includd in th sourc-trm S C nd S P.

132 15-4 h Momntum qutions(3) h momntum qution cn b solvd hn th rssur fild is givn or is somht stimtd. Unlss th corrct rssur fild is mloyd, th rsulting vlocity fild ill not stisfy th continuity i qution. Such n imrfct fild ldbsd on gussd rssur fild * ill b dnotd by u *,v *, *. his strrd vlocity fild ill rsult from th solution of th folloing qutions: n t u v * * n * t nb nb nb u v * nb * nb * nb b b b ( * * ) P A (2) ( * * ) P N A n (2 b ) ( * * ) A (2c) P t t

133 15-5 h Prssur nd Vlocity Corrctions(1) On im is to find y of imroving th gussd rssur * such tht th rsulting strrd vlocity fild ill rogrssivly gt closr to stisfying th continuity qution. Lt us roos tht th corrct rssur is obtind from * (3) hr ill b clld th rssur corrction.

134 15-5 h Prssur nd Vlocity Corrctions(2) Similrly, cn gt u u * u (4), v v * v (4 b ), u, v, * vlocity (4 c ), corrctions

135 15-5 h Prssur nd Vlocity Corrctions(3) If (1)-(2), hv ( ) A (5) u nbunb At this oint, shll boldly dcid to dro th trm nb u' nb from q.(5) nd th rsult is u hr ( ) or u d ( ) d A (6)

136 15-5 h Prssur nd Vlocity Corrctions(4) q.(b) ill b clld th vlocity-corrction formul, hich cn lso b rittn s * u d ( ) (7 ) u Similrly, v n t v * n * t d n d t ( ) n (7b) ( ) (7c) t

137 15-6 h Prssur-Corrction qution(1) h discrtiztion q. of continuity q: ( ) ( ) ( ) 0 z y v x u t ρ ρ ρ ρ ( ) ( ) ( ) [ ] 0 z y u u z y x z y x t ρ ρ ρ ρ ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) [ ] ) 0 (8 y x x z v v y t b t s n ρ ρ ρ ρ ρ ρ

138 15-6 h Prssur-Corrction qution(2) Substituting qs(7) into q(8), cn obtin obtin,, (9) z y d z y d hr b W B B S S N N W W ρ ρ,,,, z y d z y d z y d z y d b b B t t s s S n n N ρ ρ ρ ρ ( ) ( ) ( ) [ ] * * 0 z y u u z y x b B S N W ρ ρ ρ ρ ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) [ ] ) (8 0 * * * * y x x z v v y t t b n s ρ ρ ρ ρ ρ ρ

139 15-6 h Prssur-Corrction qution(3) If b is zro, it mns tht th stind vlocity, in conjunction ith th vilbl vlu of (ρ 0 - ρ ), do stisfy th continuity qution nd no rssur corrction is ndd. h trm b this rrsnts mss sourc, hich th rssur corrctions must b nnihilt.

140 15-7 h SIMPL Algorithm(1) SIMPL stnds for Smi-Imlicit Mthod for Prssur-Linkd qutions. Squnc of ortions: 1. Guss th rssur *. 2. Solv th momntum qs, such s (2)~(2c), to obtin u *,v *, *. 3. Solv ' q. 4. Clcult * ' 5. Clcult u,v, from (7)~(7c) 6. Solv othr Φ s. 7. rt th corrctd rssur s n gussd rssur *, rturn to st (2) nd rt th hol rocdur until convrgd solution is obtin.

141 15-7 h SIMPL Algorithm(2) Discussion of th Prssur-Corrction qution 1. If xrssions such s nb u' nb r rtind, thy ould ldhv to b xrssd din trms of th rssur corrctions nd th vlocity corrctions t th nighbors of u nb. h omission of th nb u' nb trm nbls us to cst th qution in th sm form s th gnrl Φ qution, nd to dot squntil, on-vribl-t--tim, solution rocdur.

142 15-7 h SIMPL Algorithm(3) 2. h ords smi-imlicit in th nm SMPL hv bn usd dto cknoldg th omission i of th trm nb u' nb. his trm rrsnts on indirct or imlicit influnc of th rssur corrction on vlocity. Prssur corrctions t nrby loctions cn ltr th nighboring vlocitis nd thus vlocity corrction t th oint undr considrtion. W do not includ this influnc nd thus ork ith schm tht is only rtilly, nd not totlly, imlicit.

143 15-7 h SIMPL Algorithm(4) 3. h omission of ny trm, ould of cours, b uncctbl if it mnt tht t th ultimt t solution ould not b tru solution of th discrtizd forms of th momntum nd continuity qution. It lso hns tht th convrgd solution givn by SIMPL dos not contin ny rror rsulting from th omission of nb u' nb

144 15-7 h SIMPL Algorithm(5) 4. h mss sourc b srvs s usful indictor of th convrgnc of th fluid-flo solution. h itrtions should b continud until th vlu of b vryhr bcoms sufficintly smll. 5. h rssur-corrction cn b sn to b mrly n intrmdit lgorithm tht lds us to th corrct rssur fild, but hs no dirct ffct on th finl solution.

145 15-7 h SIMPL Algorithm(6) 6. h rssur-corrction qution is ron to divrgnc unlss som undr-rlxtion is usd. A gnrlly succssful rctic cn b dscribd s follos: undr-rlx rlx u *,v *, * hil solving th momntum qutions (ith rlxtion fctor α0 0.5). Also, mloy * α, α 0.8 It is not imlid tht ths vlus r th otimum ons or ill vn roduc divrgnc for ll roblms. h otimum rlxtion fctor vlus r usully roblm-dndnt.

146 15-7 h SIMPL Algorithm(7) Boundry Conditions for th Prssur- Corrction qution 1. hr r to kinds of conditions t boundry. ithr th rssur t th boundry is givn (nd th vlocity is unknon) or th vlocity comonnt norml to th boundry is scifid. 2. Givn rssur t th boundry: If th gussd rssur fild * is rrngd such tht t boundry * givn, thn th vlu of ' t th boundry ill b zro.

147 15-7 h SIMPL Algorithm(8) 3. Givn norml vlocity t th boundry: As shon in Fig.1, th vlocity u is givn. It th drivtion of th ' qution for th c.v. shon, th flo rt cross th boundry fc should not xrssd in trms of u * nd corrction, but in *Not: trms of u itslf N * u u u, W W S S N N hn, ' ill not r or ill b zro in th ' qution. hus no informtion bout ' ill b ndd. b S Fig.1 u ( (givn)

148 15-7 h SIMPL Algorithm(9) h Rltiv Ntur of Prssur 1. Sinc no boundry rssur is scifid nd ll th boundry cofficints such s ill b zro, th ' qution is lft ithout ny mns of stblishing th bsolut vlu of '. h cofficints of th ' qution r such tht P nb ; this mns tht ' nd ' c (c is n rbitrry constnt) ould both stisfy th ' qution.

149 15-7 h SIMPL Algorithm(10) 2. Only diffrnc in th rssur r mningful (*'), nd ths r not ltrd by n rbitrry constnt to th ' fild. Prssur is thn rltiv vribl, not bsolut on.

150 15-7 h SIMPL Algorithm(11) 3. In mny roblms, th vlu of th bsolut rssur is much lrgr thn th locl ldiffrncs in rssur tht r ncountrd. If th bsolut vlus of rssur r for, round-off off rrors ould ris in clculting diffrncs lik -. It is, thrfor, bst to st 0 s rfrnc vlu t suitbl oint nd to clcult ll othr vlus of s rssurs rltion to strt from '0 s guss for ll oint, so tht th solution for ' dos not cquir lrg bsolut vlu.

151 15-8 A Rvisd Algorithm: SIMPLR(1) Motivtion 1. SIMPLR stnds for SIMPL Rvisd. 2. In most css, it is rsonbl to suos tht th rssur-corrction qution dos firly good job of corrcting th vlocitis, but rthr oor job of corrcting th rssur.

152 15-8 A Rvisd Algorithm: SIMPLR(2) h Prssur qution 1. h momntum qution is first rittn s u u b No, dfin nb nb d ( ) (1) sudovlocity û ˆ nbunb b u (2) u uˆ d ( ) (3 ) similrly, cn rit v n t vˆ n ˆ t d d first rittn s n t ( N ) (3b) ( ) (3c) substituting (3) ~ (3c) into continuity q., cn gt b W W N N S S B B (4)

153 15-8 A Rvisd Algorithm: SIMPLR(3) 2. Although th rssur q nd ' q r lmost idnticl, thr is on mjor diffrnc: No roximtions hv bn introducd in th drivtion of th rssur qution. hus, if corrct vlocity fild r usd to clcult th sudo-vlocitis, i th rssur qution ould t onc giv th corrct rssur.

154 15-8 A Rvisd Algorithm: SIMPLR(4) h SIMPLR Algorithm h rvisd lgorithm consists of solving th rssur qution to obtin th rssur fild nd solving th rssur-corrction qution only to corrct th vlocitis.

155 15-8 A Rvisd Algorithm: SIMPLR(5) h squnc of ortions cn b sttd s: 1. Strt ith gussd vlocity fild. 2. Clcult sudo-vlocity û,v,ŵ from qs (1),(2). 3. Solv rssur qution by q.(4) to obtin th rssur fild. 4. rt this rssur fild s *, solv momntum q to obtin u *,v *, * solv ' q. 5. Solv ' q 6. Corrct th vlocity fild (u u * d (' -' ),tc), but do not corrct th rssur. 7. Solv othr Φ s if ncssry. 8. Rturn to st 2 nd rt until convrgnc.

156 15-8 A Rvisd Algorithm: SIMPLR(6) Discussion 1. In gnrl, sinc th rssur-corrction qution roducs rsonbl vlocity filds, nd th rssur qution orks out th dirct consqunc of givn vlocity fild, convrgnc to th finl solution should b much fstr.

157 15-8 A Rvisd Algorithm: SIMPLR(7) 2. In SIMPL, gussd rssur fild lys n imortnt rol. On th othr hnd, SIMPLR dos not us gussd rssurs, but xtrcts rssur fild from givn vlocity fild. 3. Bcus of th clos similrity btn th rssur qution nd th rssur-corrction qution, th discussion in rvious sction bout boundry conditions for q. is lso rlvnt to th rssur qution.

158 15-8 A Rvisd Algorithm: SIMPLR(8) 4. Although SIMPLR hs bn found to giv fstr convrgnc thn SIMPL, it should b rcognizd tht on itrtions of SIMPLR involvs mor comuttionl ti ffort. Sinc SIMPLR rquirs fr itrtions for convrgnc, th dditionl ffort r itrtion is mor thn comnstd by th ovrll sving of ffort

159 15-9 h SIMPLC Algorithm SIMPLC stnds for SIMPL-Consistnt. It follos th sm sts s th SIMPL lgorithm. h u-vlocity corrction qution of SIMPLC is givn by d ( ) A d u, nb ( not : ( ) ) u nbunb A

160 15-10 Convrgnc Critrion Φ nb Φ nb b Rsidul R nb Φ nb b- Φ Obviously, hn th discrtiztion q is stisfid, R ill b zro. A suitbl convrgnc critrion is to rquir tht th lrgst vlu of R b lss thn crtin smll numbr.