Mathmatical Thory and Modling.iist.org ISSN 4-5804 (apr) ISSN 5-05 (Onlin) Vol., No.6, 01-Slctd from Intrnational Confrnc on Rcnt Trnds in Applid Scincs ith nginring Applications Analysis of Convction-iffusion roblms at Various clt Numbrs Using init Volum and init iffrnc Schms Anand Shukla Abstract partmnt of Mathmatics Motilal Nhru National Institut of Tchnology, Allahabad, 11 004, U.., India -mail: anandshukla86@gmail.com Surabhi Tiari (Corrsponding author) partmnt of Mathmatics Motilal Nhru National Institut of Tchnology, Allahabad, 11 004, U.., India -mail: surabhi@mnnit.ac.in; au.surabhi@gmail.com. Singh partmnt of Mathmatics Motilal Nhru National Institut of Tchnology, Allahabad, 11 004, U.., India -mail: psingh11@rdiffmail.com Convction-diffusion problms aris frquntly in many aras of applid scincs and nginring. In this papr, solv a convction-diffusion problm by cntral diffrncing schm, upinding diffrncing schm (hich ar spcial cass of finit volum schm) and finit diffrnc schm at various clt numbrs. It is obsrvd that hn cntral diffrncing schm is applid, th solution changs rapidly at high clt numbr bcaus hn vlocity is larg, th flo trm bcoms larg, and th convction trm dominats. Similarly, hn vlocity is lo, th diffusion trm dominats and th solution divrgs, i.., mathmatically th systm dos not satisfy th critria of consistncy. On applying upinding diffrncing schm, conclud that th critria of consistncy is satisfid bcaus in this schm th flo dirction is also considrd. To support our study, a tst xampl is takn and comparison of th numrical solutions ith th analytical solutions is don. Kyords: init volum mthod, artial diffrntial quation 1. Introduction Mathmatical modls of physical [, 7], chmical, biological and nvironmntal phnomna ar govrnd by various forms of diffrntial quations. Th partial diffrntial quations dscribing th transport phnomna in fluid dynamics ar difficult to solv, particularly, du to th convction trms. Such quations rprsnt th hyprbolic consrvation la for hich thir solutions alays contain discontinuity and high gradint. Thus accurat numrical solutions ar vry difficult to obtain. Spcial tratmnt must b applid to supprss spurious oscillations of th computd solutions for both th convction and convction-dominatd problms. In th prsnt scnario, bttr ays to approximat th convction trm ar still ndd, and thus dvlopmnt of accurat numrical modling for th convction-diffusion quations rmains a challnging task in computational fluid dynamics. In rcnt yars (s [7]), ith th rapid dvlopmnt of nrgy rsourcs and nvironmntal scinc, it is vry important to study th numrical computation of undrground fluid flo and th history of its changs undr hat. In actual numrical simulation, th nonlinar thr-dimnsional convction-dominatd diffusion problms nd to b considrd. hn vlocity is highr, that is, flo trm is largr, a simpl convction-diffusion problm is convrtd to convction-dominatd diffusion problm bcaus clt numbr is gratr than to. A clt numbr is a dimnsionlss numbr rlvant in th study of transport phnomna in fluid flos. It is dfind to b th ratio of th rat of advction of a physical quantity by th flo to th rat of diffusion of th sam quantity drivn by an appropriat gradint. In th contxt of th transport of hat, clt numbr is quivalnt to th product of Rynolds numbr and randtl numbr. In 16
Mathmatical Thory and Modling.iist.org ISSN 4-5804 (apr) ISSN 5-05 (Onlin) Vol., No.6, 01-Slctd from Intrnational Confrnc on Rcnt Trnds in Applid Scincs ith nginring Applications th contxt of spcis or mass disprsion, clt numbr is th product of Rynolds numbr and Schmidt numbr. or diffusion of hat (thrmal diffusion), clt numbr is dfind as, hr is th flo trm and is th diffusion trm. In most of th problms ith nginring applications (s [4]), clt numbr is oftn vry larg. In such situations, th dpndncy of th flo upon donstram locations is diminishd and variabls in th flo tnd to bcom 'on-ay' proprtis. Thus, hn modling crtain situations ith high clt numbrs, simplr computational modls can b adoptd. A flo can oftn hav diffrnt clt numbrs for hat and mass. This can lad to th phnomnon of doubl diffusiv convction. In th contxt of particulat motion, clt numbrs ar also calld Brnnr numbrs, ith symbol B, in honors of Hoard Brnnr. r. iscrtization Algorithms Using Cntral iffrncing Schm and Upinding iffrncing Schm Th gnral transport quation for any fluid proprty is givn by ( ) div( u) div( grad) S, (1) t hr is a gnral proprty of fluid, is th dnsity and u is th vlocity of th fluid. In quation (1), th rat of chang trm and th convctiv trm ar on th lft hand sid and th diffusiv trm (Γ is th diffusion cofficint) and th sourc trm S rspctivly ar on th right hand sid. In problms, hr fluid flo plays a significant rol [1], must account for th ffcts of convction. In natur, diffusion alays occurs alongsid convction, so hr xamin a mthod to prdict combind convction and diffusion. Th stady convction-diffusion quation can b drivd from transport quation (1) for a gnral proprty by dlting transint trm: div( u) div( grad) s () ormal intgration ovr control volum givs n.( u) da n( grad) da S dv. A A CV x x u u -------------------------------------------------- -------------------------------------------------- x igur 1: is a gnral point around hich discussions taks plac, and ar st and ast nodal points rspctivly 17
Mathmatical Thory and Modling.iist.org ISSN 4-5804 (apr) ISSN 5-05 (Onlin) Vol., No.6, 01-Slctd from Intrnational Confrnc on Rcnt Trnds in Applid Scincs ith nginring Applications In th absnc of sourcs (s [5]), th stady convction and diffusion of a proprty in a givn on dimnsional flo fild u is govrnd by: d d d dx dx dx u, and continuity quation bcoms d( u) 0. dx Intgrating quation (), hav ( ua) ( ua) A A, x x (4) and continuity quation bcoms ( u A ) ( u A ) 0. To obtain discrtizd quation, shall tak som assumptions: Lt A u rprsnt convctiv mass flux pr unit ara and rprsnt diffusion conductanc. No ar taking A A and applying th cntral diffrncing, th intgratd convction-diffusion bcoms. Hr u rprsnts vlocity of fluid. x () ( ) ( ), (5) and continuity quation bcoms 0. (6) Cntral iffrncing Schm Th cntral diffrncing approximation has bn usd to discrtiz th diffusion hich appars in quation (5). So / ; /. utting ths valus in quation (5), gt ( ) ( ) ( ) ( ) quation (7) can b rittn as a a, a ( ) (7) (8) hr a, a and a ar cofficints of, and rspctivly. 18
Mathmatical Thory and Modling.iist.org ISSN 4-5804 (apr) ISSN 5-05 (Onlin) Vol., No.6, 01-Slctd from Intrnational Confrnc on Rcnt Trnds in Applid Scincs ith nginring Applications. Tst xampl xampl 1: Th gnral proprty of a fluid is transportd by mans of convction and diffusion through th on dimnsional domain. Th govrning quation is givn blo. Th boundary conditions ar 1 at x=0 and 0 0 L at x=l. Using fiv qually spacd clls and th cntral diffrncing schm for convction-diffusion problms, calculat th distribution of as a function of x for th folloing cass: Cas (1): u=0.1 m/s,cas (): u=.5 m/s. Th folloing data ar applying to obtain th solution of abov problm by cntral diffrncing schm. 1.0 kg / m, 0.1 kg / m / s, Lngth L=1.0 m, x B 1 0 1 4 5 X=0 X=L x x igur : iscrtization using fiv nodal points for xampl (1) No solv th problm by cntral diffrncing schm and upind diffrncing schm for both th cass. Thn compar ths rsults by finit diffrnc mthod and obsrv th ffct of schms on th convrgnc rat of solution. Solution of xampl (1) by Cntral iffrncing Schm Nod istanc init volum solution Analytical solution iffrnc rcntag rror 1 0.1 0.941 0.987-0.00-0.6 0. 0.8006 0.796-0.004-0.5 0. 0.676 0.64-0.005-0.8 4 0.4 0.416 0.4100-0.006-1.5 5 0.5 0.1579 0.1505-0.007-4.91 Tabl 1: Th solution of problm by cntral diffrncing schm for u=0.1 Nod istanc init volum Solution Analytical solution iffrnc rcntag rror 1 0.1 1.056 1.0000-0.005 -.56 0. 0.8694 0.9999 0.11 1.0 0. 1.57 0.9999-0.57-5.74 4 0.4 0.51 0.9994 0.647 64.70 5 0.5.4644 0.9179-1.546-168.48 Tabl: Th solution of problm by cntral diffrncing schm for u=.5 19
Mathmatical Thory and Modling.iist.org ISSN 4-5804 (apr) ISSN 5-05 (Onlin) Vol., No.6, 01-Slctd from Intrnational Confrnc on Rcnt Trnds in Applid Scincs ith nginring Applications Solution of roblm by Upind iffrncing Schm On of th major inadquacis of th cntral diffrncing schm is its inability to idntify th flo dirction. Th valu of proprty at st cll fac is alays influncd by both and in cntral diffrncing. In a strongly convctiv flo from st to ast, th abov tratmnt is not suitabl bcaus th st cll fac should rciv much strongr influncing from nod than from nod. Th upind diffrncing and donor cll diffrncing schms taks into account th flo dirction hn dtrmining th valu at cll fac: th convictd valu of at a cll fac is takn to b qual to th valu at upstram nod. sho th nodal valus usd to calculat cll fac valus hn th flo is in th positiv dirction in igur thos for th ngativ dirction. U U x igur : iscrtization tchniqu for upinding diffrncing schm hn th flo is in positiv dirction, u 0, u 0( 0, 0), th upind schms sts. and 8 And th discrtizd quation (5) bcoms ( ) ( ). This may b rittn as, hich givs [ ( )] (9) hn flo is in th ngativ dirction, u <0, u <0 ( <0, <0), and 0
Mathmatical Thory and Modling.iist.org ISSN 4-5804 (apr) ISSN 5-05 (Onlin) Vol., No.6, 01-Slctd from Intrnational Confrnc on Rcnt Trnds in Applid Scincs ith nginring Applications U U x No th discrtizd quation is ( ) ( ) igur 4: iscrtization Tchniqu for Upinding iffrncing Schm for xampl (1) Or [ ( ) ( )] (10) Idntifying cofficints of and as a and a, th quations (9) and (10) can b rittn in th gnral form as a a, a (11) ith cntral cofficints a a a ( ). Th grid shon in abov figur is usd for discrtization. Th discrtization quation at intrnal nods, and 4 and rlvant nighbor cofficint ar givn by (11). Not that in this xampl u and vryhr. x At th boundary nod (1), th us of upind diffrncing schm for th convctiv trms givs ( ) ( ), A A A A and at nod (5), ( ) ( ). B B B At th boundary nods hav A B and A B and as usual boundary condition ntr th discrtizd quation x as sourc contribution: a a a Su ith a a a ( ) S. 1
Mathmatical Thory and Modling.iist.org ISSN 4-5804 (apr) ISSN 5-05 (Onlin) Vol., No.6, 01-Slctd from Intrnational Confrnc on Rcnt Trnds in Applid Scincs ith nginring Applications Nod a a S S u 1 0 -(+),,4 + 0 0 5 + 0 - ( ) A B Nod a a S S u a 1 0 0.5-1.1 1.1 1.6,,4 0.6 0.5 0 0 1.6 5 0.6 0-1 0 1.6 Tabl : trmination of cofficints a, a, Sp and Su Tabl 4: trmination of cofficints a, a, Sp, Su and a for U=0.1 m/s Cas 1: U=0.1 m/s: u 0.1, 0.1/ 0. 0.5 so 0. hr x is knon as clt numbr. Th cofficints ar givn in Tabl 4. 1.6 0.5 1.1 1 1.1 0.6 0.5 1 1.1 0.6 0.5 4 1.1 0.6 0.5 4 5 1.6 0.6 1.6 5 4 1.6000 0.5000 0 0 0 1 1.1000 or, 0.6000 1.1000 0.5000 0 0 0 0 0.6000 1.1000 0.5000 0 0 0 0 0.6000 1.1000 0.5000 0 4 0 0 0 0.6000 1.6000 0 5, or, 1 0.9 7 0.7 8 7 9 0.6 1 0 0.4 0 1 4 0.1 5 1 5 Nod istanc Upind solution Analytical solution Cntral iffrncing solution init diffrnc solution 1 0.1 0.97 0.987 0.941 0.9048 0. 0.7879 0.796 0.8006 0.7884 0. 0.610 0.64 0.676 0.6461 4 0.4 0.401 0.4100 0.416 0.47 5 0.5 0.151 0.1505 0.1579 0.597 Tabl 4: Comparison for various rsults for U=0.1 m/s Cas : U=.5 m/s: u.5, 0.1/ 0. 0.5 so 5 hr x >. u to this rason upinding diffrncing schm provid bttr convrgnc. In this cas, th cofficints ar givn blo: Nod a a S S u a 1 0 0.5 -.5.5 4.0,,4.0 0.5 0 0.5 5.0 0-1.0 0 4.0 Tabl 7: trmination of cofficints a, a, Sp, Su and a for U=.5 m/s
Mathmatical Thory and Modling.iist.org ISSN 4-5804 (apr) ISSN 5-05 (Onlin) Vol., No.6, 01-Slctd from Intrnational Confrnc on Rcnt Trnds in Applid Scincs ith nginring Applications 4 0.5.5 1.5 0.5 1.5 0.5 4.5 0.5 4 4 5 5 4, or 4.0000-0.5000 0 0 0 1.5000 -.0000.5000-0.5000 0 0 0 0 -.0000.5000-0.5000 0 0 0 0 -.0000.5000-0.5000 4 0 0 0 0 -.0000 4.0000 0 5, or 1 0.9998 0.9987 0.991 0.954 4 0.714 5 Nod istanc Upinding solution Cntral iffrncing Solution Analytical solution init diffrnc solution 1 0.1 0.9998 0. 0.9987 0. 0.991 4 0.4 0.954 5 0.5 0.714 1.056 1.0000 1.008 0.8694 0.9999 0.97 1.57 0.9999 1.0854 0.51 0.9994 0.814.4644 0.9179 1.475 Tabl 8: Comparison for various rsults for U=.5 m/s 4. iscrtization Algorithm for init iffrnc Mthod or solving any givn boundary valu problms, can divid th rang x 0, xn in to n qual subintrvals of idth h so that xi x0 ih, i=1, n. Th finit diffrnc approximations of drivativs of any fluid proprty at x xi ar givn by i 1 i 1 i 1 i i 1 ' i O( h ) and '' i o( h ) h h On th basis of abov discussion, no solv th givn convction dominatd diffusion problm in xampl (1) by finit diffrnc mthod for qual spacing. Cas1: U=0.1 Cas: U=.5 Nod istanc init volum Solution Analytical solution init diffrnc solution Nod istanc init volum Solution Analytical solution init diffrnc solution 1 0.1 0.941 0.987 0.9048 1 0.1 1.056 1.0000 1.008 0. 0.8006 0.796 0.7884 0. 0.8694 0.9999 0.97 0. 0.676 0.64 0.6461 0. 1.57 0.9999 1.0854 4 0.4 0.416 0.4100 0.47 4 0.4 0.51 0.9994 0.814 5 0.5 0.1579 0.1505 0.597 5 0.5.4644 0.9179 1.475 Tabl 9: Comparison for various rsults for U=0.1 m/s Tabl 10: Comparison for various rsults for U=.5 m/s
Mathmatical Thory and Modling.iist.org ISSN 4-5804 (apr) ISSN 5-05 (Onlin) Vol., No.6, 01-Slctd from Intrnational Confrnc on Rcnt Trnds in Applid Scincs ith nginring Applications 5. Graphical Rprsntation of Convrgnc for xampl (1) Th graphical rsult of Tabl 9 and Tabl 10 ar shon in fig. 5 hich dscribs convrgnc critria for various mthods Cas 1: U = 0.1 Cas: U =.5 igur 5: Convrgnc critria for various mthods for U=0.1 igur 6: Convrgnc critria for various mthods for U=.5 6. Concluding Rmark hn U=.5, obsrv that rsult sho bttr convrgnc sinc a and th convctiv contribution to th ast cofficint is ngativ. Thus, th solution convrgs fastr toards th xact solution for larg clt numbr. If th convction dominats, a can b ngativ. Givn that >0 and >0 (i.., flo is unidirctional), for a to b positiv and must satisfy th condition: p. If is gratr than th ast cofficint ill b ngativ. This violats on of th rquirmnts for bounddnss and may lad to physically impossibl solution. Rfrncs [1] J.. Andrson, Computational luid ynamics: Th Basics ith Applications, McgraHil, dition 6, 1995. [] M. Kumar, S Tiari, An initial valu tchniqu to solv third-ordr raction diffusion singularly prturbd boundary valu problm, Intrnat. J. Comp. Math., 89 (17) (01), 45-5, doi: 10.1080/0007160.01.70680 [] G. Manzini, A. Russo, A finit volum mthod for advction diffusion problms in convction-dominatd rgims, Comput. Mthods Appl. Mch. ngrg. 197 (008), 14 161. [4] A. Rodrıguz-rran, M. L. Sandoval, Numrical prformanc of incomplt factorizations for transint convction diffusion problms, Advancs in nginring Softar 8 (007) 49 450. [5] N. Sanın, G. Montro, A finit diffrnc modl for air pollution simulation, Advancs in nginring Softar 8 (007) 58 65. [6] H. Vrstg,. Malalaskra, An Introduction to Computational luid ynamics: Th init Volum Mthod, 1995. [7] X. ang, Vlocity/prssur mixd finit lmnt and finit volum formulation ith, AL dscriptions for nonlinar fluid-structur intraction problms, Advancs in nginring Softar 1 (000) 5 44. 4