Three Dimensional Analysis and Application of A Line Solver for the Re-circulating Flows Using Multi-Grid Methods

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1 Tree Dmensonal nalyss and laton of Lne Soler for te e-rlatng los Usng Mlt-Grd Metods T. M. SHH N. TLT Deartment of Matemats r Unersty Ialamabad Pastan. bstrat Ts aer s an extenton of te earler blsed or W as a D lne soler for te Inomressble Naer-Stoes NS eqatons arred ot an analyss and alaton of a lne soler for te rerlatng flos sng te mltgrd metods. Te or as been desrbed a robst lne soler for te oled eqatons. Te allaton roedre alled SCGSLS Symmetral Coled Gass-SedelLne Soler as based on te SCGS seme. Ts tenqe s no aled to te 3d dren aty roblem for te rerlatng flos. 3D orer analyss of te mltgrd metod s also arred ot. Te seme s smle and easy to rogram as n te ase of to dmensons. Te rates of onergene and te omtatonal tmes are reorted for te test ase. Index Terms NS eqatons Mltgrd metods Lne soler orer analyss 3D dren aty. I. DIEENTIL ND DIEENCE EQUTIONS Te eqatons exressng onseraton of mass and momentm n tree dmensons for an deal nomressble Netonan fld s gen by: x y z x y z x x x y y x z z x y x z y y y 3 x y x z z y z x y z z z 4 x x z y y z Te dsrete eqatons are dered by frst ntegratng te dfferental eqatons oer ea ontrol olme srrondng te loaton of te arable. Tey are ten exressed for a nt olme for onenent se of te mltgrd tenqe. Te flxes oer ea ontrol olme srfae are onstant. Te resltng fnte-dfferene set of algebra eqatons for grd dmensons dx dy and dz at ea fnte dfferene node are as follos: Were for examle x y z Ma-tye staggered grd system s sed to loate te flo arables. Te elotes are stored on te ell faes and te ressre s stored at te ell enters. De to te staggerng of te mes tree dfferent tyes of ontrol olme are reqred for te momentm eqatons and te ontnty eqaton n te nteror regon t stragtforard modfatons near te bondares. detaled desrton s gen n

2 455 Te aboe eqatons an also be rtten n resdal form as II. DEIVTION O SCGSLS IN 3D Te resltng fnte-dfferene set of algebra eqatons n terms of orretons and resdals along a araxal lne say te y-axs n te xy-lane at ea fnte dfferene node are gen belo: Were s are orretons and = x = y = z s te nform mes sze. Elmnate and from -5 gng Sbsttte 7-3 ba nto 6 gng te reqred tr-dagonal system of eqatons n 3D to be soled 33 oeffents and resdals are gen as follos. 3 l e s n z x y x x

3 Te 3D SCGSLS algortm soles eqaton 33 n a ay smlar to D and ges te follong ne aroxmatons III. MULTIGID METHODS Tere are seeral ays of mltgrd ylng. We sed S stable for non-lnear roblems. detaled ree of mltgrd metods and arant strateges an be fond n 6. or grd oarsenng e adoted a ontnty ontrol olme lmng 4 ensres tat te omatblty ontnty ondton beteen te fne and te oarse grd s satsfed. In sort ea oarse grd ontnty ontrol olme s omosed of for fne grd ontnty ontrol olmes. Ts follos te dsrete omatblty ondton on all grds s ral to obtanng otmal mltgrd onergene rates. In te resent stdy te restrted oarse grd elotes are defned to be te mean of ter to nearest negborng fne grd elotes. Coarse grd ressres are defned to be te mean of te for negborng fne grd ressres. Te rolongaton relatons are dered by a blnear nterolaton. Detals of restrtons and rolongatons an be fond n 4. IV. SMOOTHING POCEDUE In te resent stdy e deeloed a lne soler for te oled system of eqatons. We soled te dered Eqn. 33 for ressre orretons along a onstant lne ten a dret sbsttton nto 7-3 yelds ales of te elotes along te same lne. n nder-relaxaton arameter as mlemented by ddng te oeffents resetely. Te grd s sanned lne by lne n one dreton folloed by a smlar see n te oter dreton tang te z-lane onstant. Te sees are reeated n te mltgrd strategy ntl te desred onergene s obtaned on te fnest grd. Te onergene rteron s based on te smmed aeraged resdals n te tree eqatons. ere.x = x + y + 3z. Ten drng te smootng roess ts sngly orreted and te flly orreted errors are gen by and ere ;.. denote sngly and flly orreted ales. Note tat and are ea orreted te bt only one. fter some tenal manlaton t an be son n ere M 3d s no te amlfaton matrx n 3D. and o o o o are flly orreted and norreted ales resetely. Te smootng fator n 3D s ten gen by ere 3d 3 \ 3 freqenes and omonents. ex. X ex. ex. X M s 3d 3 s te set of g are te orer V. LOCL MODE NLYSIS To ge an dea of te beaor of te relaxaton semes n terate metods loal mode analyss s an effent tool. Sne te redton of g freqeny error omonents s essentally a loal roess te analyss of ts redton need not tae aont of dstant bondares. Consder an arbtrary loal seton of te mes t elotes staggered along te x y z dretons resetely and te ressre loated at nodes. ssme tat at te start of te smootng roess errors n and are gen n orer modes as follos: In te ase of oneton domnated flos te aboe smootng fator an be defned n terms of frozen elotes are n order to mantan te releant eynolds nmber s 3 d. VI. SMOOTHING TES To obtan smootng rates of te SCGSLS loal mode analyss as arred ot for te dren aty roblem n 3D. It s fond tat te nderlyng seme as exellent smootng 456

4 aabltes eseally at ger eynolds nmbers e. Te SCGSLS seme s ndeed far more effent tan oter semes le SIMPLE. Ts s bease for SCGSLS te ratal smootng rate remans ell belo te teoretal maxmm at all eynolds nmbers. omarson of SCGS and SIMPLE as been reorted n 5. more detaled aont of te teoretal smootng aabltes of ea of te semes and lots of te redton fators an be fond n and 3. Here e ge to lots of te amlfaton fators of te SCGSLS seme at e = and e =. g. sos te amlfaton fator at e = for flo dreton o o o =. Te smootng fator n ts ase s eqal to.56 omared t te D smootng fator.57. t s a lo eynolds nmber te amlfaton fator s ndeendent of flo dreton assres good smootng aabltes for dffson-domnated flos n 3D. Hene t aldates te D assmtons tat te 3D dren aty an be ell aroxmated n to dmensons. g. sos te amlfaton at e = for flo dreton o o o =. Te smootng rate n ts ase s also.56. Ts s te ase of gly onete flos and te sad seme s roed effent bt at e 5 t ges a large smootng fator. Ts s bease te dsrete elltty along s lnes erendlar to te flo dreton s almost lost. detal of s flos an be fond n. It s noted tat te orst art of te aty s fond at 3 = ere te smootng fator as beome large. Ts s bease at g eynolds nmbers e 5 te strong seondary ortes aear at a lane 3 =. VII. TEST POBLEM or te rose of demonstratng te alablty and te robstness of te roosed seme and to ealate ts erformanes relate to te earler roedres 5 te dren aty roblem as soled. Ts test roblem as been dely sed for aldatng solton roedres for te Naer-Stoes eqatons. Te nderlyng seme s treated as a to-stage roess. fter solng a trle dagonal system of eqatons for ressres elotes an be allated by a dret sbsttton. Te elotes are dated te and te ressres one. In order to get a grd ndeendent onergene rate te grd s set lne by lne n alternatng dretons. Ts s aeed ealy at lo eynolds nmbers bt t s more exense at ger eynolds nmbers as e ae to emloy symmetr alternatng dretons. Nmeral exerments so tat symmetr alternatng dretons do not aelerate te onergene m at lo eynolds nmbers e. Te onergene rteron s based on te smmed aerage resdals n te tree eqatons.e. on sqrt 4 max max max ere and are resdals n te momentm eqatons and n te ontnty eqaton. max max max are te nmber of nternal mes onts oer te smmaton s made. on as set to -4 and en te fne grd resdal dereased belo ts ale te allatons ere termnated. It s fond to be te most effent to arry ot st one teraton n ea yle on eery grd exet te oarsest one. We soled te eqatons on te oarsest grd exatly by erformng more tan one loal teraton. VIII. ESULTS Te flo n a b aty s an examle of a real roblem as soled by sng te nderlyng mltgrd metod. Loal mode analyss of ts metod as also arred ot redted te rate of onergene of te mltgrd metod by smootng fator. Te onergene obtaned by bot loal mode analyss and te nmeral alaton of te mltgrd metod ere ery lose agreement at lo eynolds nmbers and slgtly dffered at g eynolds nmbers g. 3. Ts s bease te loal mode analyss as erformed on lnear eqatons ere as te atal alaton nlded all te nonlnearty of te Naer Stoes eqatons. Table smmarzes te onergene aratersts of te mltgrd metod togeter t CPU tme on VX785. It s seen tat te relaxaton arameters are qte g een for g eynolds nmbers as not te ase n D. Te ase of lo relaxaton arameter n D s manly de to an aroxmaton of tree-dmensonal roblem n to-dmensonal. Table also ndates tat te grd ndeendent onergene rates are beng aroaed at all eynolds nmbers. Unle D tree-dmensonal soltons are dfflt to resent n graal format e old ardly rode te artal tras n 3D one an salze te flo atterns n te ld-dren aty. gs. 4-5 are s tres of te flo atterns and te eloty dstrbton at e = and e = nsde te aty. It s seen tat at g eynolds nmbers te entre flo sread toards te all of te aty and generates g ressre at te entre of te aty ereas at lo eynolds nmbers te flo flls te entre aty and te entre of te rmary ortex to gro near te bottom front all. IX. SUMMY Te teoretal and ratal smootng aabltes of te roosed seme SCGSLS ae been resented. Te teoretal analyss soed tat te nderlyng seme as better -ndeendent onergene rates tan SCGS at all eynolds nmbers and n rate s onergene rates ae been aeed. Nmeral exerments demonstrated 457

5 tat s a soler s omette n terms of exeton tme and smlty of eqatons and odng en omared to SIMPLE SCGS and oter semes aled to omlex nomressble flos. Te seme s robst n ts nsenstty to te relaxaton fators eseally at te lo eynolds nmbers. Nmeral exerments ae son tat te nderlyng seme onerged monotonally een at e =. Ts soed te effeny of te seme; ene e frter aled ts metod to sole roblems le retanglar ates t baard and forard stes. EEENCES T. M. Sa. nalyss of a mltgrd metod. P.D tess Oxford Unersty Oxford Unersty Comtng Laboratory Deember 989. T. M. Sa D.. Mayers and J. S. ollett. nalyss and laton of Lne Soler for te erlatng los Usng Mltgrd Metods. In. annaer W. Habs edtor Nmeral Treatment of te Naer-Stoes Eqatons ages Branseg Janary 99. Veeg. 3 G. J. Sa and S. Saloganatan. On te smootng roertes of te SIMPLE ressre orreton algortm. Inter. J. Nm. Metods for lds 8: S. Saloganatan and G. J. Sa. mltgrd metod for erlatng flos. Inter. J. Nm. Metods for lds 8: S. Saloganatan G. J. Sa T. M. Sa and D. Mayers. Nmeral Metods for ld Dynams age. Oxford Unersty Press Mar K. Stben and U. Trottenberg. Mltgrd Metods: olme 96 ages -76. Letre Notes n Matemats 98. PPENDIX 458

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