SIMPLE METHOD FOR THE SOLUTION OF INCOMPRESSIBLE FLOWS ON NON-STAGGERED GRIDS

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1 SIML MTHOD FOR TH SOLUTION OF INCOMRSSIBL FLOWS ON NON-STAGGRD GRIDS I. Szai - astr Mditrraa Uivrsity, Mchaical girig Dpartmt, Mrsi 10 -Trky 1. Itrdcti Thr ar sally t kids f grid arragmts sd t slv th flid fl prblms: staggrd grids ad -staggrd grids. Fr th staggrd grids, vctr variabls ad scalar variabls ar strd at th sam lcatis, hil fr th staggrd grids, vctr cmpts ad scalar variabls ar strd at diffrt lcatis, shiftd half ctrl vlm i ach crdiat dircti. Staggrd grids ar pplar bcas f thir ability t prvt chckrbard prssr i th fl slti as discssd i Chaptr 6. Hvr, prgrammig f th staggrd grid mthd is tdis sic ad v-mmtm qatis ar discrtizd at diffrt ctrl vlms shiftd i diffrt dirctis frm th mai ctrl vlms. Th prgrammig difficltis icras h dals ith crviliar r strctrd grids. As a rslt, arly all cds ritt crviliar r strctrd grids s -staggrd grid arragmt fr th slti f flid fl prblms. O th thr had -staggrd grids ar pr t prdc a fals prssr fild - chckrbard prssr if spcial prcatis is t tak. Fr this ras, i th arly 1980s ad bfr, -staggrd r rarly sd fr icmprssibl fl. Hvr, sic 1983 th staggrd grid (r cllcatd grid has b sd mr idly, aftr Rhi ad Ch (1983 prpsd a mmtm itrplati mthd t limiat th chckrbard prssr prblm. 1. Mathmatical Frmlati Th gvrig qatis fr t dimsial icmprssibl lamiar hat ad flid fl i Cartsia crdiats ith cstat prprtis ar: ctiity qati + v y = 0 (1 1

2 -mmtm qati ( ρ ( ρ ( ρ v p μ = + + μ + S t y y y v-mmtm qati ( ρ ( ρ ( ρ v v vv p v v μ = + + μ + S t y y y y rgy qati ( ρt ( ρt ( ρvt T T = k + k + ST t y y y qatis (2 (4 ca b prssd i a gral frm as: ( ( v ρφ ρ φ ρ φ φ φ = Γ + Γ + Sφ + f( p t y y y (5 hr ad v ar th vlcity cmpts, T is tmpratr ad φ is ay dpdt variabl (i.., v ad T, ad t, ρ, Γ, ad S φ ar tim, dsity, diffsi cfficit, ad src trm, rspctivly. Als f ( p = p/ fr -mmtm qati f ( p = p/ y fr y-mmtm qati Nt that fr th ctiity qati, φ = 1, Γ = 0, ad S φ = 0. Th gvrig qatis ar discrtizd by sig th fiit-vlm mthd a staggrd grid systm i hich all variabls ar strd at th ctr f th ctrl vlm (Figr 1. Itgratig q. (5 vr th ctrl vlm ith bdd cll facs,,, ad s srrdig ctr, hav ρδδy ( φ φ + ( ρφ ( ρφ Δ y+ ( ρvφ ( ρvφ Δ = s Δt Γ Γ Γ Γ s ( φ φ ( φ φw Δ y+ ( φn φ ( φ φs Δ (6 δ δ δ y δ ys + S ΔΔ y+ S φ ΔΔ y+ f( p ΔΔy c hr th sprscript rfrs t th prvis tim lvl ad Δ, Δ y, δ, δ, δy ad δ y s ar gmtric lgths as sh i Figr 1. v (2 (3 (4 2

3 Figr 1. Nstaggrd Grid Arragmt Th discrtizd ctiity qati fr icmprssibl fl is ( ρ Δy ( ρ Δ y+ ( ρv Δ ( ρvs Δ = 0 (7 Th cll fac vals φ, φ tc ar calclatd sig QUICK r ay thr high rdr schm i a dfrrd crrcti mar. Sbstittig ths vals it q. (6 ad rarragig, btai th fial discrtizd qati fr ay gral variabl φ as flls: aφ aφ awφw anφn asφs aφ b φ = (8 hr 3

4 ΓΔ y a = + ma ( ρ Δy, 0 Δ ΓΔ y aw = + ma ( ρ Δy, 0 Δ ΓΔ an = + ma ( ρv Δ, 0 Δy ΓΔ s as = + ma ( ρv Δ, 0 s Δy s ρδδ y a = Δt a = a + a + a + a + a S ΔΔ y+δf W N S ( ρ ( ρ ( ρ ( ρ Δ F = Δy Δ y+ v Δ v Δ b = S ΔΔ y+ b c dc s ( ρ ( φ φ + ma ρ Δy, 0 φ φ ( ρ y ( φ φ ( ρ y ( φ φ b = ma Δy, 0 dc ma Δ, 0 + ma Δ, 0 ( ρv ( φ φ ( ρv ( φ φ ma Δ, 0 + ma Δ, 0 ( ρv ( φ φ ( ρv ( φ φ ma Δ, 0 ma, 0 s + Δ s = ( p p Δy fr -mmtm qati φ φ = ( ps p Δ fr v-mmtm qati φ = 0 fr all thr qatis W N s s S (9 hr th trm b dc rslts frm th adpti f th dfrrd-crrcti prcdr i hich th fac vals f th idpdt variabl, φ, φ, φ, φ s ar calclatd frm a sitabl high rdr schm sch as QUICK. φ is th val f φ frm th prvis tim stp. It ca b bsrvd that th cfficits f th discrtizd - ad y-mmtm qatis ar th sam i cllcatd grid systm, prvidd that th diffsi cfficits ar th sam i ad y-dirctis. I rdr t sl d th chags f dpdt variabls i csctiv sltis, a drrlaati factr is itrdcd it th discrtizd qati as flls: α φ = φ + φ + φ + φ + φ + α φ ( a a a a b φ a ( 1 φ φ 1 W W N N S S a αφ = + a ( aφ awφw anφn asφs B φ (10 4

5 hr th sprscript -1 rfrs t th prvis itrati ad ( 1 αφ 1 B = b + a φ + a φ (11 αφ Nt that th trm b i qatis (8 ad (9 shld b rplacd by B if th qatis ar rlad. 4. Mmtm Itrplati 4.1. Mmtm itrplati fr stady-stat prblm B = b + 1 α α a [q.(11]. Fr th 1 Nt that fr th stady prblms vlcity cmpt at ds ad, q.(10 ca b ritt as α i a i i + Bp α Δy p p = (12 α i a i i + Bp α Δy p p = (13 Mimickig th frmlati f ad, ca btai fllig prssi fr th itrfac vlcity at th cll fac. α i a i i + Bp α Δy p p = (14 hr th trms th right-had sid ith sbscript shld b itrplatd i a apprpriat mar. Th itrfac vlcity at cll facs,, ad s ca b btaid similarly. I Rhi ad Ch s mmtm itrplati, th first trm ad 1 i scd trm f th q. (14 ar liarly itrplatd frm thir ctrparts i qs. (12 ad (13: a + B a + B a + B ( 1 f i i i p + i i i p + i i i p = f + a a a = f + (16 a a a ( 1 f (15 5

6 hr f + is a liar itrplati factr dfis as f = (17 2δ + Δ I rdr t hav a bttr drstadig f q. (14, sbstittig ( a B a q. (15 ad ( a + B a, ( a B a i i i p (14 ad mittig th trm a, btai i i i p + frm i i i p + frm qs. (12 ad (13 it q. ( p + f ( a α y( p p + ( p αδy p αδy p + = f ( 1 f + + Δ + ( 1 f qatis (14 ad (18 ar sstially qivalt. Hvr, q. (18 sparats th itrfacial vlcity it t parts: a liar itrplati part ad th additial. Th trm i first st f brackts f q. (18 is th arithmtic-avragd vals (liar itrplati mthd f t ighbr dal vlcitis. Th trm i bracs ca b rgardd as a crrcti trm, hich has th fcti f smthig th prssr fild, ad it is this trm that may rmv th ralistic prssr fild. qatis (14 ith (15 ad (16 r qati (18 cstitt th Rhi ad Ch s Origial Mmtm Itrplati Mthd (OMIM. Majmdar (1988 rprtd that sltis f stady-stat prblms frm Rhi ad Ch OMIM ar dpdt th drralati factr. T limiat this drrlaati factr dpdcy, a itrati algrithm as prpsd by him t calclat th cll-fac vlcity fr stady-stat prblm as flls: α a + B α Δy p p f f ( 1 α ( 1 i i i p = + ( a + hr sprscript -1 rfr t prvis itrati vals. This itrativ implmtati algrithm ca achiv a iq slti that is idpdt f drrlaati factrs Mmtm itrplati fr stady prblms Chi (1999 rprtd that, th slti sig th rigial Rhi ad Ch schm is tim stp siz dpdt. H prpsd a mdifid Rhi ad Ch schm fr a (18 (19 6

7 stady prblm as flls, hich is qit similar t Majmdar s schm fr a stady prblm: a + b α Δy p p α a i i i p 1 = α + ( 1 α + a a ( a ith a ρδ Δy = Δt (20 (21 It is t b td that bdy-frc trm is glctd fr simplicity f prstati. By a similar sbstitti prcss as th fr th Majmdar s itrplati q. (19, q. (20 ca b ritt qivaltly as ( 1 f f ( p + f ( a α Δy( p p + α Δy p + + ( 1 f ( p α Δy p = ( 1 α f ( 1 f α α ( a α ( a f f + a a (22 Accrdig t Y t al. (2002, sltis by sig this schm ar still tim stp siz dpdt, thgh th dpdc is qit small. Thy prpsd a diffrt itrplati tchiq fr th trms apparig i qati (20 hich appars t b bth dr rlaati factr ad tim stp siz idpdt. I this mthd th first trm th right-had sid f q. (20 is itrplatd as flls: a + B ( ( 1 ( + 1 f ( Sc ( 1 f ( Sc δ y ( + ( 1 ( f a b f a b i i i i i i + + Δ i i i p = a f i ai f i ai hr b 1 is dfid i q. (9. ( 1 f S + f S δ Δ y+ a p p Als, th dmiatr f th scd ad third trms i q. (20 is itrplatd as flls: (23 7

8 = f ( a + ( 1 f ( a i i i i ( 1 f S + f S δ Δ y+ a p p qati (20 cmbid ith q. (23 ad q. (24 is Y t al. s (2002 schm. Sbstittig q. (23 it q. (20 th fllig qati is btaid: (24 f ( a + ( 1 f ( a + ( f ( Sc ( 1 f ( Sc δ y f ( Sc y + Δ Δ Δ α + + ( 1 f ( Sc y Δ Δ 1 = + α Δy p p + f Δy p p + f Δy p p ( a + + α ( ( 1 α + + ( a f ( a ( 1 f ( a + + a f ( a ( 1 f ( a hr a is fd frm qati (24. It shld b td that th cll fac vlcitis fd frm th mmtm itrplati mthd ar sd t dtrmi th mass fls acrss th cll facs. Thy shld t b sd fr th cll fac val f th idpdt variabl φ i th dfrrd crrcti trm b dc i q. (9 i th cas φ stads fr r v i th - r y- mmtm qatis. Th fac vals f th idpdt variabl φ ar calclatd sig a sitabl cvcti schm, sch as UWIND r QUICK. Th SIML Algrithm Fr a gssd prssr fild p * th crrspdig fac vlcity q. (20 as * * * * i i i p 1 ( 1 α ( a ( a ( a (25 * ca b ritt sig a b y p p a α + α Δ = + + α (26 8

9 * A similar qati ca b ritt fr th fac vlcity v. N dfi th crrcti p' as th diffrc bt th crrct prssr fild p ad th gssd prssr fild p * s that * p = p + p (27 Similarly dfi vlcity crrctis ' ad v' as = + (28 * v = v + v (29 * Sbtractig q. (26 frm q. (20 givs ( ( α i a i i + bp α Δy p p = (30 As a apprimati, i SIML mthd th first trm i th abv qati is glctd givig = d p p (31 hr d α A = (32 ( a hr A = Δy is th ara f th ctrl vlm at th ast fac. Similarly, v = d p p (33 v N hr d α A = (34 v v ( a Th th crrctd vlcitis bcm ( = + d p p (35 * ( v = v + d p p (36 * v N Sbstittig th crrctd fac vlcitis sch as that giv by q.s (35ad (36 it th discrtizd ctiity qati (7 givs a p = a p + a p + a p + a p + b (37 W W S S N N hr 9

10 a = ( ρ Ad a = ( ρad a = ( ρad a = ( ρad W N S s * * * * ( ρ ( ρ ( ρ ( ρ b= A A + v A A s Aftr slvig th p' fild frm q. (37 th fac vlcitis ar crrctd sig q.'s (35 ad (36 ad th prssr fild is crrctd by sig = + α (39 * p p p p hr α p is th prssr dr-rlaati factr hich is chs t b bt 0 ad 1. Similarly th dal vlcitis ar crrctd sig ( = + d p p (40 * ( v = v + d p p (41 * v s hr d (38 αa v αva = ad d = (42 Th prssr crrctis at th cll facs i qs. (40 ad (41 ar calclatd by liar itrplati frm th dal vals as p = f p + (1 f p (43 W p = f p + (1 f p (44 p = f p + (1 f p (45 s s s S p = f p + (1 f p (46 N Bdary Cditis fr rssr Sic thr is qati fr th prssr, bdary cditis ar dd fr th prssr at th bdary pits. Th prssr vals at th bdary pits ca b calclatd by liar traplati sig th t ar-bdary d prssrs. Bdary Cditis fr rssr Crrcti qati Wh th vlcitis at th bdaris ar k, thr is d t crrct th vlcitis at th bdaris i th drivati f th prssr crrcti qati. Fr ampl if th vlcity at th st bdary is k th fr a ctrl vlm ar th st bdary: ( = + d p p (47 * 10

11 = (48 all ( v = v + d p p (49 * v N ( v = v + d p p (50 * v s s s S Sbstittig qatis (47-(50 it th discrtizd ctiity qati (7 btai th fllig prssr crrcti qati fr a ctrl vlm ar th st bdary ap = aw p W + ap + asp S + anp N + b (51 hr a = ( ρ Ad aw = 0 an = ( ρad as = ( ρad s * * * (52 b= ( ρa all ( ρ A + ( ρv A ( ρ A s Cmparig q.'s (51-(52 ith (37-(38 fr a ar bdary ctrl vlm th sam dfiiti f th cfficits as sd fr th itrir pits ca b sd fr a ar bdary ctrl vlm by sttig th crrspdig cfficit (a i this cas t zr ad sig all i th b trm. As a rslt val f prssr crrcti at th bdary ( p is ivlvd i this frmlati. Hvr, th val f th prssr crrcti is dd fr crrctig th dal vlcitis ar bdaris. Fr ampl, fr crrctig th -vlcity at a dal pit ar a st bdary, p at th st bdary is dd i accrdac ith qati (40. This val ca b btaid by liar traplati bt th t dal vals f prssr crrctis ar th bdary, that is sig p (2, j ad p (3, j. Rfrcs Chi S. K., "Nt th Us f Mmtm Itrplati Mthd fr Ustady Fls", Nmrical. Hat Trasfr art, A, vl. 36, pp , Majmdar S., "Rl f Udrrlaati i Mmtm Itrplati fr Calclati f Fl ith Nstaggrd Grids", Nmrical Hat Trasfr, art B, vl. 13, pp , Rhi C. M. ad Ch W. L., "Nmrical Stdy f th Trblt Fl ast a Airfil ith Trailig dg Sparati", AIAA Jral, vl. 21, 11, pp , Y B., Ta W., ad Wi J., "Discssi Mmtm Itrplati Mthd fr Cllcatd Grids f Icmprssibl Fl", Nmrical. Hat Trasfr art B, vl. 42, pp ,