Chapter 11 Calculation of

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1 Chtr 11 Clcultion of th Flow Fild OUTLINE 11-1 Nd for Scil Procdur 11-2 Som Rltd Difficultis 11-3 A Rmdy : Th stggrd Grid 11-4 Th Momntum Equtions 11-5 Th Prssur nd Vlocity Corrctions 11-6 Th Prssur-Corrction Eqution 11-7 Th SIMPLE Algorithm 11-8 A Rvisd Algorithm: SIMPLER 11-9 Th SIMPLEC Algorithm Convrgnc Critrion Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

2 11-1 Nd for Scil Procdur Th rl difficulty in th clcultion of th vlocity fild lis in th unknown rssur fild. Th rssur grdint forms rt of sourc trm for momntum qution Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

3 11-2 Som Rltd Difficultis (1) Rrsnttion of th Prssur-Grdint Trm 1. To intgrt t d/dxd ovr th control volum shown in Fig.1, w cn gt w -. W w P E x cv c.v. Fig Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

4 11-2 Som Rltd Difficultis (2) 2. To xrss w - in trms of th grid-oint rssur, w my ssum icwis-linr rofil for rssur. Thrfor, w cn gt W E w 2 2 W 2 E This mns tht th momntum qution will contin th rssur diffrnc btwn two ltrnt grid oints, nd not btwn djcnt ons Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

5 11-2 Som Rltd Difficultis (3) 3. Thr is nothr imliction tht is fr mor srious. It cn b sn from Fig.2, whr 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 - E cn b sn to zro, sinc th ltrnt rssur vlus r vrywhr qul Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

6 11-2 Som Rltd Difficultis (4) Rrsnttion of th continuity Eqution If w intgrt th continuity qution ovr th c.v. shown in Fig1, w hv u -u w Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

7 11-2 Som Rltd Difficultis (5) Onc gin, th us of icwis-linr rofil for u nd of th midwy loctions of th control-volum fcs lds to u u 2 E u W u 2 0 or u E u Thus, th discrtizd d continuity it qution dmnds th qulity of vlocitis t ltrnt grid oints nd not t djcnt ons W 0 Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

8 11-3 A Rmdy : Th stggrd Grid (1) Th difficultis dscribd so fr cn b rsolvd by rcognizing tht w do not hv to clcult ll th vribls for th sm grid oints. W cn, if w wish, mloy diffrnt grid for ch dndnt d vribl Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

9 11-3 A Rmdy : Th stggrd Grid (2) Stggrd grid: 1. Th vlocity comonnts r clcultd for th oints tht li on th fc of th control volum, s shown in Fig Othr vribls r clcultd l for th grid oints(smll circls). Fig.3 u i,, v i, 11-9 othr vribls Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

10 11-3 A Rmdy : Th stggrd Grid(3) 3. Th imortnt dvntgs r twofold: ) For tyicl c.v. shown in Fig.3, it is sy to s tht th discrtizd continuity qution would contin th diffrnc of djcnt vlocity comonnts, nd this would rvnt wvy vlocity fild. b) Th scond imortnt dvntg of th stggrd grid is tht th rssur diffrnc btwn two djcnt grid oints now bcoms th nturl driving forc for th vlocity comonnt loctd btwn ths grid oints Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

11 11-4 Th Momntum Equtions (1) Th discrtiztion qution (2-D) cn b writtn s n u v n similrly nb nb t u v w nb nb t b b ( P E ) A ( 1) ( P N ) An (1b ) w b ( ) A (1 c ) nb nb P t t Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

12 11-4 Th Momntum Equtions (2) n Fig.4 c.v. for u Fig.4b c.v. for v Th rssur grdint is not includd in th sourc-trm S C nd S P Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

13 11-4 Th Momntum Equtions (3) Th momntum qution cn b solvd whn th rssur fild is givn or is somwht stimtd. Unlss th corrct rssur fild is mloyd, th rsulting vlocity fild will not stisfy th continuity qution. Such n imrfct fild bsd on gussd rssur fild will b dnotd by u,v,w. This strrd vlocity fild will rsult from th solution of th following qutions: u u b ( ) nb nb P E A (2 ) ( ) nvn nbvnb b P N An (2b) w w b ( ) A (2 c ) t t Alid Comuttionl Fluid Dynmics Y.C. Shih Sring nb nb P t t Chtr 11: Clcultion of th Flow Fild

14 11-5 Th Prssur nd Vlocity Corrctions (1) On im is to find wy of imroving th gussd rssur such tht th rsulting strrd vlocity fild will rogrssivly gt closr to stisfying th continuity qution. Lt us roos tht th corrct rssur is obtind from (3) whr will b clld th rssur corrction Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

15 11-5 Th Prssur nd Vlocity Corrctions (2) Similrly, w cn gt u u u (4), v v v (4 b ), w w u, v, w w vlocity (4 c ), corrctions Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

16 11-5 Th Prssur nd Vlocity Corrctions (3) If (1)-(2),(2), w hv ( ) A (5) u nbunb E At this oint, w shll boldly dcid to dro th trm nb u' nb from q.(5) nd th rsult is u ( ) or u d ( ) whr d A E E (6) Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

17 11-5 Th Prssur nd Vlocity Corrctions (4) Eq.(b) will b clld th vlocity-corrctioncorrction formul, which cn lso b writtn s u d ( ) (7 ) u E Similrly, v n w t v w E n t d n d t ( ) n (7b) ( ) (7c) t Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

18 11-6 Th Prssur-Corrction Eqution (1) Th discrtiztion q. of continuity q: Th discrtiztion q. of continuity q: ( ) ( ) ( ) 0 w v u ( ) ( ) ( ) [ ] 0 0 z y u u z y x z y x t ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) [ ] ) 0 (8 y x w w x z v v z y u u t b t s n w Chtr 11: Clcultion of th Flow Fild Alid Comuttionl Fluid Dynmics Y.C. Shih Sring

19 11-6 Th Prssur-Corrction Eqution (2) Substituting qs(7) into q(8), w cn obtin Substituting qs(7) into q(8), w cn obtin (9) b B B T T S S N N W W E E,,,, z y d z y d z y d z y d whr s s S n n N w w W E,, 0 z y d z y d B T S N W E b b B t t T ( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) [ ] ) (8 0 0 y x w w x z v v z y u u t z y x b w ( ) ( ) [ ] ( ) ( ) [ ] ) 0 (8 y x w w x z v v t b n s Chtr 11: Clcultion of th Flow Fild Alid Comuttionl Fluid Dynmics Y.C. Shih Sring

20 11-6 Th Prssur-Corrction Eqution (3) If b is zro, it mns tht th stind vlocity, in conjunction with th vilbl vlu of ( 0 - ), do stisfy th continuity qution nd no rssur corrction is ndd. Th trm b this rrsnts mss sourc, which th rssur corrctions must b nnihilt Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

21 11-7 Th SIMPLE Algorithm (1) SIMPLE stnds for Smi-Imlicit Mthod for Prssur-Linkd Equtions. Squnc of ortions: 1. Guss th rssur. 2. Solv th momntum qs, such s (2)~(2c), to obtin u,v,w. 3. Solv ' q. 4. Clcult ' 5. Cl Clcult lt u,v,w from (7)~(7c) (7)(7) 6. Solv othr Φ s. 7. Trt th corrctd rssur s nw gussd rssur, rturn to st (2) nd rt th whol rocdur until convrgd solution is obtin Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

22 11-7 Th SIMPLE Algorithm (2) Discussion of th Prssur-Corrction Eqution 1. If xrssions such s nb u' nb wr rtind, thy would hv to b xrssd in trms of th rssur corrctions nd th vlocity corrctions t th nighbors of u nb. Th 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, t tim solution rocdur Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

23 11-7 Th SIMPLE Algorithm (3) 2. Th words smi-imlicit in th nm SMPLE hv bn usd to cknowldg th omission of th trm nb u' nb. This 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 work with schm tht is only rtilly, nd not totlly, imlicit Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

24 11-7 Th SIMPLE Algorithm (4) 3. Th omission of ny trm, would of cours, b uncctbl if it mnt tht t th ultimt t solution would not b tru solution of th discrtizd forms of th momntum nd continuity qution. It lso hns tht th convrgd solution givn by SIMPLE dos not contin ny rror rsulting from th omission of nb u' nb Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

25 11-7 Th SIMPLE Algorithm (5) 4. Th mss sourc b srvs s usful indictor of th convrgnc of th fluid-flow solution. Th itrtions should b continud until th vlu of b vrywhr bcoms sufficintly smll. 5. Th 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 Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

26 11-7 Th SIMPLE Algorithm (6) 6. Th rssur-corrction qution is ron to divrgnc unlss som undr-rlxtion is usd. A gnrlly succssful rctic cn b dscribd s follows: w undr-rlx u,v,w whil solving th momntum qutions (with rlxtion fctor α0.5). Also, w mloy α, α 0.8 It is not imlid tht ths vlus r th otimum ons or will vn roduc divrgnc for ll roblms. Th otimum rlxtion fctor vlus r usully yroblm-dndnt Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

27 11-7 Th SIMPLE Algorithm (7) Boundry Conditions for th Prssur- Corrction Eqution 1. Thr r two kinds of conditions t boundry. Eithr th rssur t th boundry is givn (nd th vlocity is unknown) 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 will b zro Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

28 11-7 Th SIMPLE Algorithm (8) 3. Givn norml vlocity t th boundry: As shown in Fig.1, th vlocity u is givn. It th drivtion of th ' qution for th c.v. shown, th flow rt cross th boundry fc should not xrssd in trms of u nd Not: corrction, but in trms of u itslf N u u u, E E E W W S S N N b u ( (givn) Thn, ' E will not r or E will b zro in th ' qution. Thus no informtion bout ' E will b ndd. d S Fig Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

29 11-7 Th SIMPLE Algorithm (9) Th Rltiv Ntur of Prssur 1. Sinc no boundry rssur is scifid nd ll th boundry cofficints i such s E will b zro, th ' qution is lft without ny mns of stblishing th bsolut vlu of '. Th cofficints of th ' qution r such tht P nb ; this mns tht t ' nd ' c (c is n rbitrry constnt) would both stisfy th ' qution Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

30 11-7 Th SIMPLE 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 Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

31 11-7 Th SIMPLE Algorithm (11) 3. In mny y roblms, th vlu of th bsolut rssur is much lrgr thn th locl diffrncs in rssur tht r ncountrd. If th bsolut vlus of rssur wr for, round-off rrors would ris in clculting diffrncs lik - E. 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 Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

32 11-8 A Rvisd Algorithm: SIMPLER (1) Motivtion 1. SIMPLER stnds for SIMPLE 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 Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

33 11-8 A Rvisd Algorithm: SIMPLER (2) Th Prssur Eqution 1. Th momntum qution is first writtn s nbunb b u d ( ) (1) Now, w dfin sudovlocity û uˆ u nbu uˆ nb d b (2) ( ) similrly, w cn writ E E (3) v ˆ n vn d n w wˆ d t first writtn s ( N ) (3b) ( ) (3c ) substituting (3) ~ (3c) into continuity q., w cn gt b t t (4) E E W W N N S S T T B B T Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

34 11-8 A Rvisd Algorithm: SIMPLER (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. Thus, if corrct vlocity fild wr usd to clcult th sudo-vlocitis, th rssur qution would t onc giv th corrct rssur Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

35 11-8 A Rvisd Algorithm: SIMPLER (4) Th SIMPLER Algorithm Th rvisd lgorithm consists of solving th rssur qution to obtin th rssur fild nd solving th rssur-corrction qution only to corrct th vlocitis Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

36 11-8 A Rvisd Algorithm: SIMPLER (5) Th squnc of ortions cn b sttd s: 1. Strt with 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. Trt this rssur fild s, solv momntum q to obtin u,v,w 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 Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

37 11-8 A Rvisd Algorithm: SIMPLER (6) Discussion 1. In gnrl, sinc th rssur-corrction qution roducs rsonbl vlocity filds, nd th rssur qution works out th dirct consqunc of givn vlocity fild, convrgnc to th finl solution should b much fstr Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

38 11-8 A Rvisd Algorithm: SIMPLER (7) 2. In SIMPLE, gussd rssur fild lys n imortnt rol. On th othr hnd, SIMPLER dos not us gussd rssurs, but xtrcts rssur fild from givn vlocity fild. 3. Bcus of th clos similrity il it btwn th rssur qution nd th rssur-corrction qution, th discussion i in rvious sction bout boundry conditions for q. is lso rlvnt tto th rssur qution Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

39 11-8 A Rvisd Algorithm: SIMPLER (8) 4. Although SIMPLER hs bn found to giv fstr convrgnc thn SIMPLE, it should b rcognizd tht on itrtions of SIMPLER involvs mor comuttionl ti ffort. Sinc SIMPLER rquirs fwr itrtions for convrgnc, th dditionl ffort r itrtion is mor thn comnstd by th ovrll sving of ffort Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

40 11-9 Th SIMPLEC Algorithm SIMPLEC stnds for SIMPLE-Consistnt. It follows th sm sts s th SIMPLE lgorithm. Th u-vlocity corrction qution of SIMPLEC is givn by ( ) u d, d A ( not : u u ( ) A ) E nb nb nb Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

41 11-10 Convrgnc Critrion Φ nb Φ nb b Rsidul R nb Φ nb b- Φ Obviously, whn th discrtiztion q is stisfid, R will b zro. A suitbl convrgnc critrion is to rquir tht th lrgst vlu of R b lss thn crtin smll numbr Alid Comuttionl Fluid Dynmics Y.C. Shih Sring 2009 Chtr 11: Clcultion of th Flow Fild

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