An Alternative Strategy for the Solution of Heat and Incompressible Fluid Flow Problems via Finite Volume Method

Transcription

1 A Alteratve Strategy for the Solto of Heat ad Icompressble Fld Flow Problems va Fte Volme Method Masod Nckaee a, Al Ashrafzadeh b, Stefa Trek a a Isttte of Appled Mathematcs, Dortmd Uversty of Techology, Dortmd, Germay b Faclty of Mechacal Egeerg, K.N. Toos Uversty of Techology, Tehra, Ira rg head ttle: Characterstc-Based Splt Fte Volme Method Correspodg athor: M. Nckaee, Isttte of Appled Mathematcs, Dortmd Uversty of Techology, Vogelpothsweg 87, D-447 Dortmd, Germay. E-mal: masod.ckaee@math.t-dortmd.de

2 A Alteratve Strategy for the Solto of Heat ad Icompressble Fld Flow Problems va Fte Volme Method The characterstc-based splt (CBS) method has bee wdely sed the fte elemet commty to facltate the mercal solto of Naver-Stokes (NS) eqatos. However, ths comptatoal algorthm has rarely bee employed the fte volme cotet ad the stablzato of the mercal solto procedre has tradtoally bee addressed dfferetly volme-based mercal schemes. I ths paper, the CBS-based fte volme algorthm s employed to formlate ad solve a mber of lamar compressble flow ad covectve heat trasfer problems. Both eplct ad mplct versos of the algorthm are frst eplaed ad valdated the cotet of the solto of a ld-drve cavty problem ad a backward facg step (BFS) flow problem. The modfed algorthm, capable of modellg the coplg betwee the mometm ad eergy balace eqatos, s the trodced ad sed to solve a boyacy-drve cavty flow problem. Comptatoal reslts show that the CBS fte volme algorthm ca be relably sed the solto of lamar compressble heat ad fld flow problems. Keywords: fte volme, characterstc based splt, fractoal step, colocated grd, compressble flow, atral covecto

3 NOMENCLATURE c sod speed artfcal sod speed c P specfc heat at costat pressre thermal epaso coeffcet ER epaso rato BFS flow t tme step g gravtatoal accelerato, y grd spacg ad y drectos h cotrol volme legth scale tme step cotrol parameter k fld thermal codctvty fld vscosty L characterstc legth fld kematc vscosty N Nsselt mber fld desty p Pressre Pr Pradtl mber (= / ) Sbscrpts Ra Raylegh mber (= g TL 3 / ) P, N, E,W,S odal pot vales Re Reyolds mber, e, w, s tegrato pot vales t Tme 1,,... corer odes T temperatre cov covecto related vale -velocty compoet dff dffso related vale U mass fl (= ) h, c hot ad cold walls v y-velocty compoet far feld vale V volme horzotal coordate Sperscrpts y vertcal coordate, +1 old ad ew teratos X 1, X, X 3 characterstc legths BFS flow * termedate feld vale thermal dffsvty o-dmesoal vale

4 1. INTRODUCTION Over the last decade, Comptatoal Fld Dyamc (CFD) smlato of the NS eqatos has become a basc tool for dstral desg ad aalyss. Comptatoal Fld Dyamc aalyss s ow commo a dverse rage of applcatos sch as aerospace egeerg ad power geerato, chemcal ad materal egeerg processes, ad atomotve dstry st to ame a few. A eample of the applcato of CFD the real world egeerg problems s preseted [1], whch the atomoble pat crg process smlato s dscssed. The wde spread se of CFD the solto of scece ad egeerg problems s partly de to the developmet of robst ad accrate dscretzato schemes the cotet of fte volme method []. A commo symptom the mercal solto of compressble NS eqatos s the appearace of realstc oscllatos covecto domated flows. It s well-kow that the cetral dfferece-based dscretzato FVM ad stadard fte elemet Galerk method fal to properly model the advecto terms the fld flow goverg eqatos. The oscllatory mercal soltos ca be preveted by employg the pwdg strateges orgally developed the fte dfferece cotet [3, 4]. I the fte elemet framework, Petrov-Galerk ad Galerk least sqare schemes [5] are soltos wth smlartes to those sed the fte volme commty. Nmercal schemes sch as characterstc Galerk or Taylor-Galerk [5] are other remedes for the problem of advecto mercal modellg. I the preset stdy the cocept of characterstc-galerk procedre s employed to stablze the solto of covecto dffso eqatos. We perform the temporal

5 dscretzato of the modfed mometm eqatos sg the characterstc cocept. Ths leads to the trodcto of addtoal secod order dffso-lke terms that prevet the mercal oscllatos ormally occrred wth cetral appromato of covecto terms covecto domated flows. Therefore, we take advatage of secod order appromatos for both covecto ad dffso modellg wthot sg ay addtoal pwdg methods to cotrol the mercal oscllatos. The ecessty of proper pressre-velocty coplg compressble flows (pressre stablzato), has bee aother maor dffclty the mercal solto of compressble NS eqatos. Proper remedes ad algorthms ths regard have bee devsed both fte volme [] ad fte elemet [5] cotets. Oe partclar famly of techqes, also sed ths stdy, s the class of splttg methods. The splttg process was tally trodced by Chor [6] for compressble flow problems the fte dfferece framework. Afterwards, the splttg method was employed the fte elemet cotet ad sed for the solto of dfferet compressble flow problems [7-9]. Characterstc Based Splt (CBS) s a varat of splttg methods whch was frst trodced 1995 by Zekewcz ad Coda [10] to solve the fld flow problems. The foremost advatage of ths method s the capablty of solvg ether compressble or compressble sbsoc ad spersoc flows by the same algorthm [5]. Frther developmets ad applcatos of the CBS fte elemet method ca be fod the revew papers by Zekewcz et al. [11] ad Ntharas et al. [1] ad recetly by Ntharas [13]. Ths method has bee also developed the mesh free cotet for the solto of compressble NS eqatos [14]. The CBS has bee recetly eteded ad employed the fte volme cotet for the solto of compressble NS eqatos. I the sem-mplct verso of the CBS,

6 trodced [15], a Posso pressre eqato eeds to be solved for the kow pressre feld. I the flly eplct verso of the scheme, trodced [16], artfcal compressblty approach s employed. Coceptally, the CBS fte volme scheme ca be cosdered as a varat of the well establshed famly of fractoal step methods. The dea s to obta the solto of the NS eqatos throgh the soltos of a mber of easer eqatos. The mercal algorthm s ofte mplemeted three steps;.e. the calclato of a termedate velocty feld, the calclato of the pressre feld correspodg to a dvergece-free velocty feld ad, fally, the proecto of the termedate velocty o a dvergece-free velocty feld. A comparso betwee the sem-mplct ad flly eplct varats of the CBS fte volme method, whch defer oly the pressre pdate step, s preseted [17]. I ths paper the above-metoed varats of the CBS fte volme method are revewed ad the employed to ot oly smlate the coplg betwee the mass ad mometm eqatos, bt also the coplg betwee the mometm ad eergy balaces a free covecto heat trasfer problem. The paper s orgazed as follows: the et secto, secto, presets the goverg eqatos ad a descrpto of the CBS fte volme method. The dffereces betwee the two versos of the CBS scheme are also preseted ths secto. I the et secto, performace of the CBS fte volme method s vestgated throgh the solto of three steady compressble flow problems,.e., ld-drve cavty flow, backward facg step flow, ad boyacy-drve flow. Fally, some coclsos are draw the last secto.

7 . NUMERICAL SOLUTION OF THE NS EQUATIONS.1. Goverg Eqatos The goverg eqatos for the vscos compressble flow are epressos for the coservato of mass, mometm compoets ad eergy as follows: Mass coservato U t 0 (1) Mometm coservato U U t p g () Eergy coservato c T c T p t p T k (3) I the above eqatos U s the th mass fl compoet. It shold be oted that the traset desty term the cotty eqato, Eq. 1, ca be replaced by the followg relato der setropc flow assmpto 1 t c I ths Eqato c s the wave (sod) speed. p t (4).. The CBS Method The CBS s developed here for the solto of compressble NS eqatos the fte volme cotet.

8 .3. Temporal Dscretzato alog Characterstcs The frst aspect of the CBS scheme s the dscretzato alog the characterstcs [5]. I partclar, the temporal dscretzato of the mometm eqato, Eq., alog the characterstcs, sg a backward Eler dscretzato method, reslts the followg tme dscrete eqato [5]: k k g p U t g p U t U U 1 (5) Note that addtoal dffso-lke terms are trodced the mometm eqatos. These terms are reqred to stablze the velocty solto covecto-domated flows, ad hece the dscretzato step, cetral dfferece scheme ca be sed to dscretze the mass fles ad the velocty gradets. Pressre stablzato s obtaed va fractoal step method, throgh the solto of a appromate mometm eqato whch may be obtaed wthot cosderg the pressre term. Eqato (5) ca be modfed as follows: k k g U t g U t U U * (6) I the absece of pressre gradet, the velocty feld * U from Eq. (6) does ot satsfy the dvergece free costrat of the compressble flows. Ths s called termedate velocty ad the solto to Eq. (6) s the frst step of the fractoal step algorthm. It shold be oted that Eq. (6) wll be solved by a eplct tme step appled to the flly dscretzed

9 form. Reformlatg the mometm coservato Eq. () sg termedate veloctes Eq. (6), oe ca obta the followg eqato for the fal veloctes: k k p t p t U U * 1 (7) Sce the pressre s stll kow, ths eqato ca be sed after the solto of a approprate pressre eqato to obta fal veloctes. I fact, solto of Eq. (7) s the thrd step of the proposed fractoal step algorthm. To eforce the dvergece free costrat ad to fd the pressre, the dvergece operator s sed to operate o Eq. (7): k k p t p t U U * 1 (8) At ths pot, two versos of the CBS method stded ths work ca be obtaed from the above eqato. I order to eforce the compressblty costrat at each tme step, the kow velocty dvergece,.e. U 1, s assmed to be zero for compressble flows, ad the fal eqato for pressre pdate reads:. 1 * 1 U t p (9) Eqato (9) s a Posso pressre eqato whch ca be solved mplctly for the kow pressre. The mplct solto of Eq. (9) combed wth the eplct solto of Eq. (6) leads to the sem-mplct verso of the proposed algorthm [15].

10 To devse the flly eplct verso of the CBS method [16], the kow velocty dvergece term Eq. (8) s obtaed from a cotty-lke eqato. The traset desty term s retaed the mass coservato Eq. (1), ad s related to the tme varato of pressre throgh the trodcto of a artfcal sod speed, whch has a fte vale, as follows t p t 1 (10) Ths the flly eplct verso of the algorthm employs the followg eplct formla for the pressre pdate:. 1 * 1 p t U t p p (11) Ths eqato has o physcal meag at each tme step, bt reslts the satsfacto of the compressblty costrat at steady state. The solto of ay of the pressre eqatos (9) or (11) s doe the secod step of the fractoal step method. The eergy eqato, as a covecto dffso eqato for the temperatre kow, s also dscretzed temporally alog the characterstcs [5] P k k P P T k T c t T k T c t T T c 1 (1) The solto of eergy eqato ca be cosdered as the forth step of or fractoal step CBS scheme o-sothermal flows. I ths paper, however, the atral covecto problem s smlated throgh the se of Bossesq appromato ad therefore, temperatre ad velocty felds are copled. To mata proper velocty-temperatre

11 coplg, the eergy eqato s solved after each velocty pdate drg the CBS teratos. The above eqatos, Eqs. (6)-(1), represet the basc steps of the CBS algorthm. Fte volme method s the employed to dscretze the space dervatves o a colocated strctred grd arragemet. It s mportat to ote that cetral dfferece scheme has bee sed for all the velocty ad pressre appromatos. Ths s allowed here becase the stablzato reqred for the covectve terms s doe va trodcto of some addtoal dffso terms by sg the characterstcs method..4. Spatal Dscretzato Usg FVM Colocated Grd To start the fte volme spatal dscretzato, the solto doma mst be dvded to cotrol volmes. I ths stdy, colocated, strctred grds are sed a Cartesa coordate system. The comptatoal grd s geerated based o the cell-cetered scheme ad employs commo fte volme otatos. A eample for a grd geerated a rectaglar doma s show Fgre 1. I Fgre 1, captal ad small letters represet the cotrol volme ceters ad the Itegrato Pots (IP), respectvely. Nmbers from 1 to 4 are the for corers of a typcal cotrol volme. Fte Volme Dscretzato: Step 1 To carry ot the frst dscretzato step of the CBS scheme, we tegrate Eq. (6) over a cotrol volme Vp assocated wth cell p show Fgre 1. Usg the Dvergece

12 theorem, the volme tegrals chage to srface tegrals ad ths trodces the IP qattes to the eqato. For eample, the spatal sem-dscrete form of the covecto terms the -mometm eqato (=1 Eq. (6)) s wrtte as follows: U U Uv dvp dvp Vp Vp y (13) y p U e U w p Uv Uv s The IP veloctes are related to odal vales va lear appromato. For eample, the velocty o the rght face of the cell p Fgre 1 s appromated as follows: e E P P E. (14) 1 1 P E I Eq. (14), ad y represet the cell-face areas Fgre 1. All other IP vales are treated smlarly wth smple geometrcal terpolatos. To complete the frst step of the dscretzato procedre, we have to appromate the dffso fles. These fles are also appromated sg the cetral dfferece scheme. Dscretzato of addtoal dffso terms Eq. (6) reqres the vales of the veloctes at the cotrol volme corers. These corer vales are appromated sg the weghted average of the srrodg odal vales. For eample, the -velocty of pot 1 Fgre 1 s appromated as follows: N NE E P VN VNE VE VP 1. (15) VN VNE VE VP

13 As t s metoed earler, we se cetral dffereces to appromate both covectve ad dffsve fles. Therefore, the dscretzato procedre yelds a eplct eqato relatg the cell-cetered vale of termedate veloctes to that of the cellcetered veloctes ad ther eghbors (E, W, N, S, NE, NW, SE ad SW Fgre 1) from the prevos tme step. Fte Volme Dscretzato: Step I the secod step of the CBS method, the pressre s obtaed from ether Eq. (9) or (11) for sem-mplct ad flly eplct versos, respectvely. The fte volme dscretzato of these eqatos s straghtforward. Aga, after tegratg over cotrol volmes ad trodcg the srface tegrals to the eqato, the cetral dfferece scheme s sed to dscretze pressre gradets o the IPs. Fte Volme Dscretzato: Step 3 From steps 1 ad we have the termedate veloctes ad the pressre, respectvely. Usg these vales, oe ca proceed wth the fte volme dscretzato of Eq. (7). Itegratg Eq. (7) over the cotrol volme, the followg eqato ca be obtaed: Vp 1 * p t p U U dvp t k dvp Vp k (16) A eplct solto of ths eqato s carred ot to obta the fal veloctes. Fte Volme Dscretzato: Step 4 Fte volme dscretzato of Eq. (1) s the same as what we have doe for termedate veloctes. Here, the IP temperatre vales ad temperatre gradets are appromated

14 sg the cetral dfferece scheme. Usg a appromato lke Eq. (15), the reqred temperatre vales at cotrol volme corers are also obtaed..5. Bodary Codtos I the preset stdy, the velocty bodary codtos are appled the frst step of the algorthm. Ths, the real velocty bodary codtos have bee sed to obta the termedate veloctes of the frst step. The solto of the secod step reqres the mplemetato of approprate pressre bodary codtos. I the case of kow pressre vales o the bodary, otflow bodary codto a chael flow, the mplemetato s smple ad straghtforward. I other cases wth veloctes or ther gradets as kow bodary codtos, the proper bodary codto for the pressre eqato s appled sg velocty compoets ormal to the physcal bodary of terest. Sce the cell-cetered grd s employed, the reqred pressre vales o the cotrol srfaces are estmated sg lear etrapolato from the er eghborg odes. I the thrd step, the etrapolated IP pressre vales of the secod step are aga sed to obta the correct veloctes. The applcato of dfferet bodary codtos to the eergy eqato s also smple o the strctred fte volme grds ad ca be doe the forth step of the CBS scheme..6. Tme Step Calclato Iteratve solto of the eplct modfed mometm eqatos of the frst step, Eq. (6), s sbected to tme step lmtatos ad so s the overall CBS scheme. The permssble tme

15 step s obtaed based o the tme scales of the covecto ad dffso terms [15, 16]. The crtcal tme step s calclated as follows: crt cov dff t m t, t (17) I Eq. (17), ad tdff are the covecto ad dffso tme steps respectvely, ad tcov are calclated as follows: Sem-mplct method h h tcov, tdff (18) cov dff Flly eplct method h tcov, cov t dff h dff (19) I Eq. (19), ma(, cov, ) s the artfcal sod speed ad s a costat (take as dff 0.5 here), cov ad dff are the covecto ad dffso veloctes, respectvely. For sothermal flows, these veloctes are calclated as follows: cov, dff (0) h Ad, for o-sothermal flows as cov, dff m,. h h (1) I the above eqatos, h s a cotrol volme legth scale. I a two dmesoal form grd, h s smply the wdth of the cotrol volme. O o-form grds; however, dfferet legth scales ca be sed. Frther formato regardg the defto of artfcal sod speed ca be fod [16].

16 fracto of To esre the stablty of the method, the fal global tme step s chose as a tcrt. I other words, a safety factor varyg betwee 0 ad 1 s mposed o the calclated tme step Eq. (17)..7. CBS Iteratve Algorthm The proposed CBS fte volme method ca be smmarzed as follows (1) Italze, p ad T ad calclate tcrt from Eq. (17) () Solve Eq. (6) to obta the termedate veloctes * (3) Solve ether Eq. (9) or Eq. (11) to obta the ew pressre 1 p (4) Use the termedate veloctes from step () ad the pressre feld from step (3) to calclate the ew veloctes 1 from Eq. (7) (5) Solve Eq. (1) to obta the ew temperatre 1 T (6) Check for covergece of p ad 1 1, 1 T. If solto s coverged termate the CBS loop, otherwse pdate tcrt from Eq. 17 ad cote to step () It shold be oted that for the problems wth depedet velocty ad temperatre felds, step (5) ca be also doe otsde the ma CBS terato loop after the coverged veloctes are determed. 3. RESULTS AND DISCUSSION I ths secto, some bechmarkg has bee doe for compressble lamar flow problems to vestgate the performace of the proposed flly eplct ad sem-mplct versos of the CBS scheme. I the frst test case, the ld-drve cavty flow has bee cosdered. Nmercal soltos at dfferet Reyolds mbers are preseted ad compared

17 wth avalable lteratre. I the secod eample, steady state Backward Facg Step (BFS) flow has bee smlated at dfferet Reyolds mbers. Reattachmet ad recrclato legths are take as a measre of valdatg accracy of the methods ths problem. Fally, boyacy-drve cavty flow s solved sg the CBS fte volme method. Ths s a stadard atral covecto problem whch, Bossesq appromato s employed to take to accot boyacy forces de to the desty dffereces the flow. I order to cote wth the reslts, dmesoless forms of the goverg eqatos are employed. I the frst two test cases, the gravtatoal forces are eglected wth respect to other terms of the mometm eqato ad the followg o-dmesoal parameters are sed:,,,, c c p p L t t L () Ad the Reyolds mber s defed as. Re L (3) For atral covecto problem, whch gravtatoal forces are mportat, the followg o-dmesoal parameters are sed: P P P C H C c c c k k k T T T T T pl p L t t L,,,,, (4) Ad the o-dmesoal mbers of terest are Raylegh ad Pradtl:. Pr, 3 TL g Ra (5) For frther detals regardg the o-dmesoal parameters oe shold refer to [18].

18 3.1. Test Case I (Ld-Drve Cavty) Steady state ld-drve cavty flow s cosdered here as the frst test case. Ths two dmesoal sqare cavty s show Fgre a. Ths problem s solved sg two dfferet o-form strctred meshes as show Fgres b ad c. These comptatoal grds whch cosst of (Mesh 1) ad (Mesh ) cotrol volmes are clstered ear the corers of the solto doma. Mesh 1 s sed to solve the cavty flow at Reyolds mbers 400 ad I Fgres 3 ad 4, the velocty dstrbtos at varos Reyolds mbers are compared wth the bechmark solto of Gha et al. [19]. The vertcal velocty (v) compoets alog the md-horzotal le are compared Fgre 3 ad the horzotal velocty () compoets alog the md-vertcal le are compared Fgre 4. The reslts of both CBS fte volme methods are close agreemet wth the bechmark solto. To obta accrate soltos at Reyolds mber 5000, the fer mesh, mesh, s sed. Aga ths case the horzotal ad vertcal md-plae veloctes are compared wth those of Gha et al. [19] Fgre 5. I Table 1, the vales of mamm ad mmm veloctes ad ther correspodg postos alog the md les the cavty are also compared wth [19]. These reslts are obtaed from the flly eplct verso of the CBS fte volme method. Table 1 llstrates the approprate covergece behavor of the method wth respect to grd refemet ad also provdes the reslts for valdato of or proposed method. 3.. Test Case II (Backward Facg Step) I the secod test case, solto of Backward Facg Step (BFS) flow, show Fgre 6, s vestgated. It s a D chael flow wth the let wdth h ad the otlet wdth H. The

19 legth of the flow passage s eqal to 6H. Depedg o the vale of the flow Reyolds mber, dfferet recrclato zoes occr the flow as show schematcally Fgre 6. I ths problem, a flly-developed parabolc -velocty profle s prescrbed at the let of the chael ad zero-ormal velocty dervatves are mposed at the et. The locato of the otflow bodary s chose to be sffcetly far dowstream of the step so that t does ot affect the posto of the recrclato zoes. All other walls are sbected to the o-slp bodary codto. Epaso rato ER H / h s set to 1.94 to be able to compare the comptatoal reslts of CBS fte volme method wth the epermetal reslts of Armaly et al. [0]. A o-form strctred mesh, Fgre 7, s employed here to vestgate the reattachmet ad separato legths for Reyolds mbers from 100 to 800. Reyolds mber for the flow cofgrato Fgre 6 s defed as follows U h Re. (6) Grd depedet reslts, o-dmesoal separato ad reattachmet legths, of the CBS method are compared wth the epermetal data of Armaly et al. [0] Fgre 8. Sce the comptatoal reslts of both CBS methods are very smlar, here we have oly show the reslts from the flly eplct verso. The reslts of the flly eplct CBS method are close agreemet wth the epermetal data p to Reyolds 500. For flows wth Reyolds mber above 500, the reattachmet ad the separato legths X 1 ad X devate from the epermetal reslts. These devatos for the Reyolds mber above 500 are probably de to the three-dmesoal effects that are eglected the preset D aalyss.

20 The comptatoal cost (tme) ad mber of teratos for dfferet Reyolds mbers are preseted Table for both CBS procedres. To esre that the grd depedet solto s obtaed, mesh refemet s doe for each Reyolds mber. The o-dmesoal reattachmet legth s the calclated ad compared wth the epermetal data of Armaly et al. [0]. The reslts of Table show that both methods provde accrate soltos. The sem-mplct solver s faster tha the flly eplct oe wth respect to comptatoal tme ad the mber of teratos reqred to reach the same desred accracy both schemes s larger the flly eplct verso Test Case III (Boyacy-Drve Cavty) Solto of boyacy-drve cavty flow s preseted here for the frst tme sg CBS fte volme method. The fld flls a sqare cavty wth the vertcal walls beg kept at dfferet temperatres, ad two horzotal adabatc walls as show Fgre 9. Fld s assmed to be vscos, compressble ad Bossesq-appromated. No-slp velocty bodary codtos are appled o all walls. A local temperatre dfferece creates a local desty dfferece wth the fld ad reslts fld moto becase of the boyacy forces. These forces have to be clded the mometm eqato, ad the odmesoal parameter ths case becomes the Raylegh mber: 3 g Th Tc L Ra (7) I Eq. (7), g s the magtde of the gravtatoal accelerato, the coeffcet of thermal epaso ad, T 1, T 0 h c are the hot ad cold wall temperatres, respectvely. The Pradtl mber s also sed as t s defed Eq. (5).

21 The seres of meshes sed for ths problem have the same o-form strctre as those the ld-drve cavty flow (Fgre ). The fld the cavty s ar ad the problem s solved for for dfferet Raylegh mbers,.e., 10,10,10 ad10. Fgre 10 llstrates a set of plots for each Raylegh mber. These plots are streamles, ad v velocty cotor ad temperatre cotors. Sce the reslts of both flly eplct ad sem-mplct algorthms are qte smlar, the reslts are preseted Fgre 10 oly oce for each Ra mber. I the et step, grd depedet behavor of the CBS fte volme methods for the solto of boyacy-drve flow s stded. A seres of for o-form strctred grds, 1 1, 41 41, 8181, , have bee employed at 3 Ra 10 ad the mamm vales (ma) of vertcal ad horzotal veloctes alog the md-heght ad mdwdth les are smmarzed Table 3, respectvely. Both sem-mplct ad flly eplct algorthms perform well ad eve a grd sze of 1 1 ca prodce reasoable reslts whch compare well wth those of other refed grds. I Table 4, the flly eplct verso s sed to compare a mber of mportat qattes for boyacy-drve flow wth the mercal reslts of [1] ad []. These qattes are the mamm horzotal velocty ad ts assocated y alog the md-wdth, the mamm vertcal velocty ad ts assocated alog the md-heght ad mamm (Ma.) ad mmm (M.) Nsselt mbers (N) alog the hot wall. A grd s sed for Raylegh mbers p to 5 10 ad a grd for 6 Ra 10. Comparso of the CBS reslts wth the bechmark reslts shows a close agreemet for 3 5 Ra For Ra 10 however, the CBS method der predcts the mamm ad mmm vales

22 of the flow qattes, whle stll matas the rght predcto for the posto of these qattes. 4. CONCLUSION A fte volme method based o the characterstc based splt approach, commoly employed fte elemet commtes, s trodced ths paper. The proposed method employs characterstc based stablzato for mercal treatmet of the covecto terms mometm ad eergy eqatos. Coseqetly, all tegrato pot vales are appromated wth smple geometrcal terpolato formlas wthot sg ay pwdg methods for covecto modellg. The pressre stablzato the proposed method s acheved va a fractoal step method. The algorthm s preseted two dfferet flly eplct ad sem-mplct forms whch dffer oly the pressre pdate step. The proposed CBS fte volme method s the sed to solve lamar compressble fld flow ad heat trasfer problems o strctred colocated grds. Ld-drve cavty, backward facg step ad the boyacy-drve cavty flows are sccessflly smlated the lamar regme to valdate the proposed algorthm. Comparso wth the bechmark mercal ad epermetal reslts shows that the predctos all test cases are satsfactory.

23 REFERENCES 1. A. Ashrafzadeh, R. Mehdpor, ad M. Rezva, A Effcet ad Accrate Nmercal Smlato Method for the Pat Crg Process Ato Idstres, Proc. Iteratoal Coferece o Applcatos ad Desg Mechacal Egeerg (ICADME), Peag, Malaysa, S. Acharya, B. R. Balga, K. Kark, J. Y. Mrthy, C. Prakash, ad S. P. Vaka, Pressre- Based Fte-Volme Methods Comptatoal Fld Dyamcs, Joral of heat trasfer, vol. 19, pp , E. Isaacso, R. Corat, ad M. Rees, O the Solto of No-lear Hyperbolc Dfferetal Eqatos by Fte Dffereces, Comm. Pre Appl. Math., vol. 5, pp , D.B. Spaldg, A Novel Fte Dfferece Formlato for Dfferetal Eqatos Ivolvg both Frst ad Secod Dervatves, It. J. Nmer. Meth. Egrg., vol. 4, pp , O. C. Zekewcz, R. L. Taylor, ad P. Ntharas, The Fte Elemet Method for Fld Dyamcs, 6 th ed., Btterworth ad Heema, Oford, A.J. Chor, Nmercal solto of Naver Stokes Eqatos, Mathematcs of Comptato, vol., pp , G.E. Scheder, G.D. Rathby, ad M.M. Yovaovch, Fte Elemet Aalyss of Icompressble Flow Icorporatg Eqal Order Pressre ad Velocty Iterpolato, I C. Taylor, K. Morga ad C.A. rebba, edtors, Nmercal Methods Lamar ad Trblet Flow. Plymoth, Petech Press, B. Ramaswamy, Fte Elemet Solto for Advecto ad Natral Covecto Flows, Comp. Flds, vol. 16, pp , R. Raacher, O Chor Proecto Method for the Icompressble Naver Stokes Eqatos, Lectre Notes Mathematcs, vol. 1530, pp , O.C. Zekewcz, ad R. Coda, A Geeral Algorthm for Compressble ad Icompressble Flow, Part I. The splt Characterstc Based Scheme, It. J. Nmer. Meth. Flds, vol, 0, pp , 1995.

24 11. O.C. Zekewcz, P. Ntharas, R. Coda, M. Vázqe, ad P. Ortz, The Characterstc Based Splt Procedre: a Effcet ad Accrate Algorthm for Fld Problems, It. J. Nmer. Meth. Flds, vol. 31, pp , P. Ntharas, R. Coda, ad O.C. Zekewcz, The Characterstc-Based Splt (CBS) Scheme-a Ufed Approach to Fld Dyamcs, It. J. Nmer. Meth. Egg, vol. 66, pp , P. Ntharas, A Ufed Fractoal Step Method for Compressble ad Icompressble Flows, Heat Trasfer ad Icompressble Sold Mechacs, vol. 18, pp , A. Shamekh, ad K. Sadeghy, O the Use of Characterstc-Based Splt Mesh-free Method for Solvg Flow Problems, It. J. Nmer. Meth. Flds, vol. 56, pp , M. Nckaee, ad A. Ashrafzadeh, A sem-mplct CBS Fte Volme Algorthm for the Solto of Icompressble Flow Problems, Proc. 18 th aal coferece, vol. 1, CFD socety of Caada, Lodo, Caada, M. Nckaee, ad A. Ashrafzadeh, A Characterstc-Based Splt Fte Volme Algorthm for the Solto of Icompressble Flow Problems, Proc. Ffth Eropea Coferece o Comptatoal Fld Dyamcs, vol. 1, pp. 177, ECCOMAS CFD, Lsbo, Portgal, M. Nckaee, ad A. Ashrafzadeh, Comparso betwee two Characterstc Based Splt Fte Volme Methods for the Solto of Icompressble Flow Problems, Proc. 13 th Aal & d Iteratoal Fld Dyamcs Coferece, vol. 1, pp. 33, FD010, Shraz, Ira, RW. Lews, P. Ntharas, ad KN. Seetharam, Fdametals of the Fte Elemet Method for Heat ad Fld Flow, 1 st ed., Wley, NJ, U. Gha, K.N. Gha, ad C.T. Sh, Hgh-resolto for Icompressble Flow sg the Naver Stokes Eqatos ad Mltgrd Method, J. Comp. Phys., vol. 48, pp , B. F. Armaly, F. Drst, J. C. F. Perera, ad B. Schoeg. Epermetal ad Theoretcal Ivestgato of Backward-Facg Step Flow, J. Fld Mech., vol. 17, pp , 1983.

25 1. D. de Vahl Davs, Natral Covecto of Ar a Sqare Cavty: A Bech Mark Solto, It. J. Nmer. Meth. Flds, vol. 3, pp , DC. Wa, BSV. Patak, ad GW. We, A New Bechmark Qalty Solto for the Boyacy-drve Cavty by Dscrete Sglar Covolto, Nmercal Heat Trasfer: Part B, vol. 40, pp , 001.

26 Table 1. Comparso of mamm (Ma) ad mmm (M) veloctes ad ther correspodg postos alog the md les ld-drve cavty flow Re Referece Mesh M y M v Ma v M M CBS CBS Gha [19] CBS CBS Gha [19] CBS CBS Gha [19]

27 Table. Comparso of flly eplct ad sem-mplct schemes BFS flow at dfferet Re Re flly eplct sem-mplct 100 X 1 /(H-h) [0] = X 1 /(H-h) [0] = X 1 /(H-h) [0] = 8.46 Iterato 57,079 13,663 Tme (s) 1,86 58 X 1 /(H-h) Iterato 77,890 5,976 Tme (s) 11,051 5,66 X 1 /(H-h) Iterato 87,105 7,333 Tme (s) 1, X 1 /(H-h) Table 3. Grd stdes, mamm vales of vertcal ad horzotal veloctes alog the md-wdth ad md-heght les at Grd ma() flly eplct sem-mplct 3 Ra 10 ma(v) flly eplct sem-mplct

28 Table 4. Comparso of boyacy-drve flow qattes ad ther correspodg postos paretheses wth the 3 bechmark solto, 10 Ra 10 Ra Qatty Ref. [] Ref. [1] Preset Ma. () (0.813) Ma. (v) (0.188) Ma. (N) (0.08) M. (N) (1.0) Ma. () 16.1 (0.815) Ma. (v) (0.1) Ma. (N) (0.13) M. (N) (1.0) Ma. () (0.835) Ma. (v) (0.07) Ma. (N) (0.08) M. (N) (1.0) Ma. () (0.86) Ma. (v) 7.11 (0.040) Ma. (N) (0.03) M. (N) (1.0) (0.813) 3.70 (0.178) 1.50 (0.09) 0.69 (1.0) 16. (0.83) 19.6 (0.119) 3.53 (0.143) (1.0) (0.855) 68.6 (0.066) 7.71 (0.08) 0.79 (1.0) 64.3 (0.850) (0.038) 17.9 (0.038) (1.0) (0.8161) (0.1761) (0.0865) (1.0) (0.883) ) (0.1145) (0.1481) (1.0) (0.8519) (0.0639) ) (0.0834) (1.0) (0.8479) (0.0391) (0.039) (1.0)

29 Fgre 1. A cell-cetered fte volme grd. Fgre. (a) Ld-drve cavty flow, geometry ad bodary codtos, (b) Mesh 1, 85 85, (c) Mesh, Fgre 3. Ld-drve cavty flow, comparso wth Gha et al. [19] for the v-velocty profle alog the horzotal ceter le. (a) Re = 400, (b) Re = Fgre 4. Ld-drve cavty flow, comparso wth Gha et al. [19] for the -velocty profle alog the vertcal ceter le. (a) Re = 400, (b) Re = Fgre 5. Ld-drve cavty flow at Re = 5000 o mesh, comparso wth Gha et al. [19] of velocty profles alog the ceter les. (a) velocty profle, (b) v velocty profle. Fgre 6. BFS flow, geometry ad bodary codtos. Fgre 7. Sample fte volme grd BFS flow, step rego grd dstrbto. Fgre 8. BFS flow, comparso of reattachmet ad separato legths wth epermetal reslts [0]. Fgre 9. Boyacy-drve cavty flow, geometry ad bodary codtos. Fgre 10. Natral-covecto patters smlated by FV CBS, streamles, (b) so- cotors, (c) so-v cotors, (d) sotherms. Ra (a)

30 Fgre 1

31 = ld, v = 0 = 0 v = 0 = 0 v = 0 = 0, v = 0 (a) (b) (c) Fgre

32 Vertcal Velocty Sem-Implct Flly-Eplct Gha et al. Vertcal Velocty Sem-Implct Flly-Eplct Gha et al Horzotal Dstace Horzotal Dstace (a) Fgre 3 (b)

33 1 1 Vertcal Dstace Sem-Implct Flly-Eplct Gha et al. Vertcal Dstace Sem-Implct Flly-Eplct Gha et al Horzotal Velocty Horzotal Velocty (a) Fgre 4 (b)

34 Vertcal Dstace Flly-Eplct Gha et al. Vertcal Velocty Flly-Eplct Gha et al Horzotal Velocty (a) Fgre Horzotal Dstace (b)

35 Fgre 6

36 Y X Fgre 7

37 X 1 X X 3 X 1 X X 3 Ep. Ep. Ep. FV CBS FV CBS FV CBS X/(H - h) Re Nmber Fgre 8

38 Fgre 9

39 3 Ra 10 4 Ra 10 5 Ra 10 6 Ra 10 (a) (b) (c) (d) Fgre 10