An axisymmetric incompressible lattice BGK model for simulation of the pulsatile ow in a circular pipe

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1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS In. J. Nume. Meh. Fluds 005; 49: Publshed onlne 3 June 005 n Wley IneScence DOI: /d.997 An axsymmec ncompessble lace BGK model fo smulaon of he pulsale ow n a ccula ppe T. S. Lee, Habo Huang ; and Chang Shu Mechancal Engneeng; Naonal Unvesy of Sngapoe; Sngapoe; Sngapoe SUMMARY Applyng he dea of Hallday e al., hough nseng he souce em no he wo-dmensonal lace Bolzmann equaon o ecove he ncompessble Nave Sokes equaon n he cylndcal coodnaes, an axsymmec ncompessble Lace-BGK DQ9 model was poposed hee o smulae he pulsale ows n a ccula ppe. The pulsale ows n a ccula ppe wh 1 Re 000 Reynolds numbe s based on ppe s damee), Womesley numbe 1 5 wee nvesgaed and compaed wh he exac analycal soluons. The excellen ageemens beween numecal and he analycal soluon valdae ou model. The eec of schemes o mplemen pessue gaden and he model s spaal accuacy wee also dscussed. To show he pefomance of he poposed model, he same poblems wee also smulaed by Hallday s axsymmec model whch deved fom sandad LBM and he hee-dmensonal ncompessble LBGK model. I s obseved ha he pesen model educes he compessbly eec n Hallday s model and s much moe ecen han he LBGK D3Q19 model fo an axsymmec pulsale ow poblem. Copygh? 005 John Wley & Sons, Ld. KEY WORDS: lace Bolzmann mehod; axsymmec; ncompessble Nave Sokes equaon 1. INTRODUCTION The lace Bolzmann mehod LBM) has been poposed as an alenave numecal scheme fo solvng he ncompessble Nave Sokes NS) equaons [1, ]. Among deen lace Bolzmann equaon LBE) models n applcaon, he lace Bhanaga Goss Kook LBGK) model s he smples one because only nvolves one scala elaxaon paamee and a smple equlbum momenum dsbuon funcon [3]. If he densy ucuaon can be negleced, he ncompessble Nave Sokes equaon can be ecoveed fom a LBE hough he Chapman Enskog pocedue [4]. Howeve, n LBM, he densy may ucuae o a gea exen n ows wh lage pessue gaden because he pessue and densy vaaons sasfy he equaon of Coespondence o: Habo Huang, Mechancal Engneeng, Naonal Unvesy of Sngapoe, Sngapoe, Sngapoe. E-mal: g @nus.edu.sg E-mal: mpelees@nus.edu.sg E-mal: mpeshuc@nus.edu.sg Receved 1 Januay 005 Revsed 8 Apl 005 Copygh? 005 John Wley & Sons, Ld. Acceped 3 Apl 005

2 100 T. S. LEE, H. HUANG AND C. SHU saes of an sohemal gas gven by p = c s [4], whee c s s a consan. In many pevous sudes [4 6], he compessbly eec of sandad LBGK model has been hghlghed. Some ncompessble models wee poposed o elmnae he compessbly eec of he sandad LBGK model [6, 7]. One of he mos successful ncompessble LBGK model was poposed by He and Luo n 1997 [6]. The ncompessble LBGK model was valdaed by seady plane poseulle ow and he unseady D Womesley ow. In he model, he compessbly eec of he ode om ) s explcly elmnaed [6]. In he pas, besdes he above sudy [6] on D Womesley ow, Cosgove e al. [8] also smulaed he D Womesley ow wh 400 Re 1000 and Aol e al. [9] suded he accuacy of D Womesley ow usng D 9-velocy DQ9) model wh deen bounday condons. Snce we ae neesed n sudyng blood ow n lage aees, we would lke o smulae he 3D oscllaoy ow n ppes. Howeve, when comes o he poblems of 3D axsymmec Womesley ow [5] o 3D seady ows [10] n ube, mos of pevous sudes [5, 10] sll ecouse o he 3D LBGK model and usng he 3D cubc laces wh pope cuvaue wall bounday eamen decly. Tha means a lage mesh sze and s no so ecen o smulae such an axsymmec poblem n ha way. To smulae he axsymmec ow moe ecenly, n 001, Hallday e al. [3] poposed an axsymmec DQ9 model fo he seady 3D axsymmec ow poblems and seems vey successful fo seady ube ow. Lae Nu e al. [11] fuhe deved an axsymmec model fo smulaon of he Taylo Couee ows. The man dea of he model s nseng seveal spaal and velocy-dependen souce ems no he adjused evaluaon equaon fo he lace ud s momenum dsbuon [3], whch s smla o he convenonal CFD mehods. Howeve, he axsymmec DQ9 model poposed by Hallday e al. [3] and Nu e al. [11] s deved fom he sandad DQ9 model. When we smulae he D Womesley ow wh he sandad DQ9 model, he soluon nvolves sgncan compessbly eec [6]. Smla ccumsances may appea when usng he axsymmec model poposed by Hallday e al. [3]. Tll oday, smulaon of he pulsale ow n ccula ppe usng ncompessble axsymmec DQ9 model has no been epoed. In hs pape an axsymmec ncompessble DQ9 model s poposed. To valdae and evaluae he pefomance of he model, 3D Womesley ow smulaons wee pefomed. The 3D Womesley ow pulsale ow n axsymmec ppe) s dven by peodc pessue gaden a he nle of he ppe. Snce a ypcal Reynolds numbe n he abdomnal aoa s 150 and a ypcal Womesley numbe = 8 [5], he paamees of mos cases n hs pape ae close o hese wo values. Fo 3D Womesley ow smulaons, wo deen schemes wee adoped o mplemen he oscllaoy pessue gaden. One s applyng an equvalen body foce n he pos-collson sep, he ohe s specfyng he oscllaoy o xed pessue a ends of he ppe decly. Mos of he pevous sudes [3, 8] adop he s scheme. Tha may noduce he nuence of he body foce and snce hee ae deen ow condons a he oule of aees hs scheme may be uneal fo he vascula ow smulaon [5]. The eecs of deen schemes o mplemen he oscllaoy pessue gaden and he model s spaal accuacy wee also dscussed hee. To show he pefomance of he poposed model, he same poblem was also smulaed by he ncompessble LBGK 3D 19-velocy D3Q19) model. Dealed compason of CPU me and accuacy of numecal esuls beween ou model and 3D model s gven. The compason demonsaes ha he pesen axsymmec

3 AN AXISYMMETRIC INCOMPRESSIBLE LBGK 101 model s much moe ecen han he LBGK D3Q19 model fo an axsymmec pulsale ow poblem n ems of CPU me and memoy.. NUMERICAL METHODS Fo he ncompessble nenal ow n a ccula ppe f we assume ha he ow s axsymmec, he azmuhal velocy u and coodnae devaves vansh fom he ncompessble connuy and Nave Sokes equaons. Hence he connuy equaon @ = u 1) The axal and adal veloces u x ;u sasfyng wo momenum equaons n he wo spaal coodnaes x and ae Du x D Du D = + u x = + u ) ) 3).1. Axsymmec ncompessble LBGK model Hee, n ou sudy, he ncompessble LBGK DQ9 model [6] was employed o fuhe deve ou axsymmec ncompessble model. Theoecally he LBE smulaed he compessble Nave Sokes equaon nsead of he ncompessble one, because he spaal densy vaaon s no zeo n LBE smulaons. In ode o coecly smulae ncompessble Nave Sokes equaon n pacce, one mus ensue ha he Mach numbe s low and he densy ucuaon ) s of ode om ) [6]. The devaon pocedue and addonal lm L x =c s T ) 1 s llusaed n Appendx A. Fom he devaon, Equaons 1) 3) can be ecoveed. Fo ou pesen axsymmec ncompessble DQ9 model, he nne dscee veloces ae as follows: 0; 0); =0 e = cos[ 1)=]; sn[ 1)=])c; =1; ; 3; 4 4) cos[ 5)=+=4]; sn[ 5)=+=4])c; =5; 6; 7; 8 whee c = x =, x and ae he lace spacng and me sep sze. In hs pape c = 1. The dscee veloces wee also llusaed n Fgue 1. In ou axsymmec DQ9 model, f x; ; ) s he dsbuon funcon fo pacles wh velocy e a poson x; ) and me. The macoscopc pessue p and momenum 0 u ae dened as 8 8 f = p=cs ; f e = 0 u 5) =0 whee 0 s he densy of ud. =0

4 10 T. S. LEE, H. HUANG AND C. SHU e 6 e e 5 x e 3 e0 e 1 e 7 e 4 e 8 Fgue 1. The dscee velocy fo axsymmec DQ9 model. The wo man seps of LBGK DQ9 model ae collson and seamng. In he collson sep, a goup of calculaons fom 6a) o 6d) ae mplemened: f ne = f x; ; ) f eq x; ; ) 6a) f + h 1) h ) =! 0 u 3 =! x; ; ) =f eq x; ; )+ [@ yp + x u x u + u u + u x u )e x ] 1 1 ) f ne + h 1) + h ) 6b) 6c) 6d) whee f eq s he equlbum momenum dsbuon funcon and f ne pa of dsbuon funcon. h 1) and h ) The devaon pocedue of h 1) and h )! 0 =4=9,! =1=9, =1; ; 3; 4),! =1=36, =5; 6; 7; 8). f eq followng equaon [6]: [ f eq p e u x; ; )=! +! cs 0 + e u) cs cs 4 u cs s he non-equlbum ae he souce ems added no he collson sep. was llusaed n Appendx A. In above expessons, can be obaned hough he ] ; =0; 1; ;:::;8 7) I s noced ha he man deence beween he above ncompessble DQ9 model and he sandad DQ9 model s he fom of Equaon 7). In he above fomulas 6c) and 6d), he elaxaon me consan and he ud vscosy sasfy equaon = 1) x =6. In he seamng sep, he new dsbuon funcon value obaned fom 6d) would popagae o he neghboung egh laces. Tha can be mplemened by he followng Equaon 8), whch would be appled on all unfom laces: f x + e x ;+ e ;+ )=f + x; ; ) 8) Hee, n hs pape = x =1.

5 AN AXISYMMETRIC INCOMPRESSIBLE LBGK 103 Fo he velocy devaons n above Equaon 6c), he u x x x u x u can be obaned hough Equaon 9) wh = x; = ; = = x; = =, especvely: u u )= 1 1 ) 8 f 1) e e = 1 1 ) 8 f ne e e + o ) 9) =0 Fo he u x u n Equaon 6c), s equal u x +@ x u x u. u x +@ x u ) can be easly obaned by Equaon 9), only he value x u s lef unknown o u x u. Hee we ecouse o ne deence mehod o x u a lace node ; j), whch can be calculaed by Equaon x u ) ; j =u ) +1;j u ) 1;j )=x) 10) The values u x x x u u, x u fo he lace nodes whch ae jus on he wall bounday can also be calculaed fom Equaons 9) and 10). Obanng hese values fo lace nodes on he peodc bounday s also easy. Howeve, o oban hese values fo he nodes on he nle=oule pessue-speced bounday, hese values ae exapolaed fom hose of he nne nodes... Bounday condon To mplemen he oscllaoy pessue gaden, wo schemes can be appled. One s applyng an equvalen oscllaoy body foce [8] he ohe s smply specfyng oscllaoy pessue a he nle bounday and xng he oule pessue [6]. To apply an equvalen body foce, he peodc bounday condons ae mposed a he open ends of he ppe and n he collson sep, afe sep 6d) was mplemened, a fuhe pos-collson sep s necessay: f + x; ; )=f + x; ; )+! F e =cs ; =1; ;:::;8 11) whee F =p cos!); 0) s he body foce and p s he maxmum amplude of he oscllaoy pessue gaden. In hs pape, f s no menoned explcly, he oscllaoy pessue gaden was mplemened n hs way. On he ohe hand, n ou sudy, he nle=oule pessue bounday condon was also appled because he pessue gaden canno always be eplaced by an exenal foce n LBGK compuaons. Some pessue bounday condon eamens have been poposed n pevous LBM sudes [1, 13]. Hee he mehod poposed by Guo e al. [1] was adoped fo s smplcy. If specfyng oscllaoy pessue a nle bounday condon and xng he oule pessue, he coespondng velocy value n hese boundaes was exapolaed fom he nex nne nodes. Hence, he equlbum pa of dsbuon funcon can be deemned fom he above Equaon 7) and he non-equlbum pa of dsbuon funcon can be obaned hough exapolaon [1]. So, he collson sep fo bounday nodes can be mplemened nomally as nne nodes. Fo wall bounday condons, s well known ha he mos commonly appled n LBM s bounce back model [5]. Howeve, fo he 3D ccula ppe ow poblem, o ea he cuvaue wall bounday n unfom cubc laces, ognal bounce back model s no accuae enough fo cuvaue bounday [5]. Fo dec 3D smulaon, n hs sudy, he non-equlbum dsbuon funcon exapolaon mehod [14] was appled fo cuvaue wall bounday. In =0

6 104 T. S. LEE, H. HUANG AND C. SHU he mehod, he velocy on wall nodes lace nodes ousde and mos nea o physcal bounday) s obaned fom exapolaon and p-value obaned fom he neaes ud node lace nodes nsde physcal bounday), hence he equlbum dsbuon funcon fo wall nodes can be obaned hough Equaon 7). Wh coespondng non-equlbum dsbuon funcon exapolaed fom he ud nodes, he collson sep on wall nodes can be fullled. Ths eamen s poved o be second ode n space [14]. When ou D axsymmec ncompessble model was appled o sudy he axsymmec ppe ows, he non-equlbum dsbuon funcon exapolaon mehod [14] fo wall bounday was also adoped. To apply hs wall bounday eamen, an addonal laye ousde he wall bounday s ncluded. 3. RESULTS AND DISCUSSION The 3D Womesley ow pulsale ow n axsymmec ppe) s dven by peodc pessue gaden a he nle of he ppe. In he followngs, p s he maxmum amplude of he snusodally vayng = p e! 1) R s dened as he adus of he ccula ppe.! s he angula fequency and s he knec vscosy of ud. The Reynolds numbe s dened as Re =U c R=, U c s he chaacesc velocy dened as U c = p 4! = p R 4 13) whch s he velocy ha would be obseved a he cene of he ube f a consan focng em p was appled n he lm of 0. The Womesley numbe s dened as = R!=. The Souhal numbe s dened as S = R=U c T ), whee T s he samplng peod. The analycal soluon fo axsymmec ppe pulsale ow [5] s { [ p u; )=Re 1 J 0[ 1 ] } +)=R)] e! 14)! 0 J 0 [ 1 +)] whee J 0 s he zeoh-ode Bessel funcon of he s ype. All he smulaons n hs pape began wh an nal condon of zeo velocy evey whee, and an nal un of 10T seps. I should be noced ha he maxmum velocy U max appeaed n ube axs dung a samplng peod would be less han he chaace velocy U c fo case 0. Fo case 1, he maxmum velocy U max would be much less han U c. Ths s llusaed n Fgue, whch shows he nomalzed maxmum velocy n ube axs U max =U c, and he phase lag of he velocy eld, nomalzed by ), as funcon of. In he gue, he numecal esuls agee well wh he analycal soluon. I seems ha when oscllaoy pessue gaden changes vey fas s mpossble fo velocy eld o each he fully developed velocy pole wh maxmum value U c.

7 AN AXISYMMETRIC INCOMPRESSIBLE LBGK 105 Fgue. Maxmum velocy n he axs of ube and he phase lag as funcon of Womesley numbe Convege ceon and spaal accuacy To evaluae he eo beween he numecal and analycal soluon, we noduce a velocy eo fomula whch s llusaed n 15). A each me sep he eo can be dened as = u ) u a ) u 15) a ) whee u ) s he numecal soluon, u a ) s he analycal velocy a n mddle ppe. The oveall aveage eo s aveaged ove he peod T. Fo all he cases n hs pape, he convegence ceon was se as follows: x ux;+ T) ux;) x ux;+ T ) ) whee was usually chosen as =1+nT n hs pape. Hee n all cases, he ppe lengh was chosen as N [8], whee N s he numbe of lace nodes n he damee usually N excludes he uppe and lowe exa layes ousde he wall bounday). The global eos behavou fo =7:97 and 3:171 s llusaed n Fgue 3. Fo a cean, asn was nceased was kep consan by vayng he peod T accodngly whn he ange T Fo he above, he coespondng was kep 0:6 and 1:0. In Fgue 3, he sold lnes epesen he lnea s, and he slope of he lnes ae 1:89 =7:97) and :0 =3:171). The gue demonsaes ha cuen LBGK model ncopoang he exapolaon wall bounday condon and he focng em s second ode n space.

8 106 T. S. LEE, H. HUANG AND C. SHU Fgue 3. The global eo as a funcon of he ppe damee N fo =7:97 and =3: Eec of Womesley numbe on ow paen As a ypcal Reynolds numbe n he abdomnal aoa s abou 150 and a ypcal Womesley numbe = 8 [5], n ou smulaon, sly he case of Re = 100, =7:97, T = 100, =0:6 was pefomed, N = 41 and he coespondng U c =1:0. The exac analycal soluons of Equaon 14) wee compaed o numecally evaluaed velocy poles along he damee n Fgues 4a) and b). The velocy s nomalzed by U c. The -axs s non-dmensonalzed by dvdng by he adus of he ube as ndcaed n Fgue 4. Alhough n hs case U c =1:0, he U max obseved n whole oscllaoy peod s only abou 0:063, M =0: :109 1, whch s conssen wh he lm of LBM. In hs case, paamees T = 100, =0:6, U c =1:0 wee chosen o avod numecal sably and save compuaonal me because f U c =0:1, o x he paamee Re and and he same mesh sysem, he should be 1=300, and hen s 0.51, whch s vey close o 0.5, numecal nsably may appea. On he ohe hand, he coespondng T value should be whle no jus 100. Hence much moe CPU me s equed. Anyway, U c =1:0 n hs case s coec because M 1. Hee, n he above case, he oveall numecal aveage eo s abou 1.3%. Whle Aol e al. [5] menoned ha he oveall aveage eo fo almos he same 3D case s aound 7% usng he cuve bounday condon poposed by Bouzd e al. [15]. The pesen bee pefomance may be due o ou axsymmec ncompessble DQ9 model elmnang he compessbly eec whle he D3Q19 model appled by Aol e al. [5] nvolves he eec and ou second-ode exapolaon wall bounday eamens may also accoun fo he bee pefomance. Fgues 5a) and b) show he velocy evoluon of an oscllaon ove half a peod fo =1:373 and 4:56, especvely. Fo he case llusaed n Fgue 5a), Re =1:, =1:5,

9 AN AXISYMMETRIC INCOMPRESSIBLE LBGK 107 a) b) Fgue 4. Poles of: a) deceasng; and b) nceasng veloces along he damee of a ube fo =7:97, T = 100, Re = 100, =0:6, a = nt=16 n =0;:::;15) U c =1:0, acually U max 0:07). a) b) Fgue 5. Velocy pole along he damee ove half a peod: a) =1:373; and b) =4:560. T = 4000, U c =0:01, N = 41, whch s a vscous-domnaed sysem [8]. Fo case llusaed n Fgue 5b), Re = 190, =0:7, T = 1000, U c =0:8, N = 161, whch s a momenumdomnaed sysem n he lamna egme [8]. Snce shea sess plays a cucal ole n he pogesson of aheoscleoss. The shea sess enso compuaon s mpoan. In ou 3D axsymmec poblem, he shea sess enso x can be obaned convenenly fom Equaon 9). The calculaon s usually mplemened dung he collson pocess. I s convenen o ge shea sess enso n LBM. Hee we gve an

10 108 T. S. LEE, H. HUANG AND C. SHU Fgue 6. Shea sess, Re = 100, =0:6, =7:97, T = 100 n a oscllaoy ube ow, he measuemens ae aken a = nt=16, n =0; 1;:::;16. Table I. The oveall aveage eo compason fo wo schemes o mplemen he pessue gaden. Scheme Scheme of addon Scheme of specfy pessue Cases foce em on nle=oule BC Re =1; p = e e-3 Re = 10; p =0: e e- Re = 600; p =0: e e- example wh =7:97, T = 100, =0:6, Re = 100, N = 41. In Fgue 6, he analycal soluons wee compaed o numecally evaluaed shea sess along he damee a me = nt=16, n =1;:::;16. The esuls of numecal soluon and analycal soluon agee vey well Schemes o mplemen pessue gaden All he above accuae esuls wee acheved hough addng foce em no he pos-collson sep. Hee, we would also lke o make fuhe nvesgaon on he wo schemes o mplemen pulsale pessue gaden. In Table I, he pefomance of wo schemes was compaed. Hee, n all of he cases, =3:963, mesh sze N x N =41 41, T = 4800 and he convegence ceon s Equaon 16).

11 AN AXISYMMETRIC INCOMPRESSIBLE LBGK 109 Fo he scheme of specfyng nle=oule pessue, fom he devaon n Appendx A, we noce ha he value L x =c s T) should be vey small so as o neglec he compessbly eec due o he me vaaon of pessue eld [6]. In hese cases, T = 4800 L x =c s =40 3, whch sasfy he lm. The physcal meanng s ha n he ange of he dsance L x, he me, T, dung whch he ow eld undegoes a macoscopc change mus be geae han he me, L x =c s [6]. Fom Table I, we can see ha fo he scheme of specfyng pessue on nle=oule bounday, he oveall aveage eo ncease wh p beween wo ends o Reynolds numbe, whch s conssen wh he concluson n pape [6]. Fo he scheme of addng focng em, he oveall aveage eo decease slghly wh p. They also agee wh Aol e al. s [5] esuls. I seems ha he scheme of applyng addonal focng em has moe advanages han he scheme of specc pessue on nle=oule bounday fo he 3D Womesley ow, whch has unfom pessue gaden a any me Compessbly eec and compason wh Hallday s model Hee, he compessbly eec was nvesgaed n deal. The quany ha epesens compessbly s he mean vaaon of densy. I s dened as = 1 0 x; x; ) 0 ) )=N whee he mean densy s 0 and N s he oal numbe of nodes. Fo cases of Re = 100, =7:97, he mean densy ucuaon was calculaed fo M max =0:18, 0:109, 0:055 and 0:0. The M max s he maxmum Mach numbe n ube axs. In all smulaons, N x N =81 81, he Re and wee kep hough vayng T, p and value. The scheme of specfyng nle=oule pessue was chosen o mplemen pessue gaden. The esuls of densy ucuaon wee lsed n Table II. The able shows ha fo boh models of Hallday and pesen, and M max =0:055) 1 4 M max =0:109) M max =0:0) 1 5 M max =0:109) These esuls agee wh he known elaonshp [16] ha s popoonal o M. Hee we also nd ha when M max 0:10, he of Hallday s model s slghly lage han ha of pesen model. I s also neesng o nd some clues of compessbly eec hough nvesgang he velocy eld eo. Hee o nvesgae he velocy eld eo, fou cases wh =3:963, N x N =41 41, T = 4800 wee smulaed usng boh he pesen model and Hallday s model. The scheme of specfyng nle=oule pessue was chosen o mplemen pessue gaden. Table III shows he velocy eld eo measued by and. a me s dened as = u ;) u a ;)) u a ;) 17)

12 110 T. S. LEE, H. HUANG AND C. SHU Table II. Mean densy ucuaon. Models Cases Hallday s model %) Pesen model %) M max =0:18; =0:9; p =0:00; T = M max =0:109; =0:7; p =0:0005; T = M max =0:055; =0:6; p =0:00015; T = M max =0:0; =0:54; p =0:0000; T = Table III. The eo of velocy eld n 3D Womesley ow. Models Hallday s model Pesen model Cases max %) max %) 1 Re = 10; p =0: Re = 40; p =0: Re = 600; p =0: Re = 100; p =0: whee he summaon s ove he damee n mddle ppe and he oveall aveage eo s aveaged ove he peod T. The max means he maxmum value of n a samplng peod. In Table III, he M max n ube axs fo cases 1 4 ae 0:054, 0:108, 0:7 and 0:544, especvely. Compang he maxmum pacula velocy eo and he oveall numecal aveage eos of wo models n Table III, we obseved ha as M max n ube axs nceases, he coespondng eos of Hallday nceases much fase han he pesen ncompessble model. The obsevaon s conssen wh concluson go fo he sandad and ncompessble DQ9 models [6]. Hence, compang wh Hallday s model, he pesen model can elmnae he compessbly eec Compason wh 3D LBGK To show he pefomance of he poposed model, some cases wee also smulaed by he 3D ncompessble LBGK model. The 3D smulaon s based on he D3Q19 lace velocy model. The mesh sze used fo he axsymmec model s N x N =81 41, whle n he 3D LBM smulaon, he mesh sze used s N x N y N z = Noce n 3D smulaons he cuvaue wall bounday eamen [14] was appled. In hs compason, we used wo cases wh paamees of =7:97, T = 100, =0:6 as examples. Table IV lsed he oveall numecal aveage eo, peod numbe o each convegence ceon 16) and he CPU me equed by he pesen axsymmec model and 3D LBM. All he compuaons wee caed ou on a supecompue Compaq ES40: oal pefomance of 5300 Mops) n he Naonal Unvesy of Sngapoe. I can be obseved fom Table IV ha he peodc numbe of eaon equed by 3D LBM s slghly less han o equal o ha of he axsymmec model. Howeve, he 3D LBM smulaon akes abou 110 mes moe compung me o oban soluons han he pesen axsymmec model. Hence, ou poposed

13 AN AXISYMMETRIC INCOMPRESSIBLE LBGK 111 Table IV. Compason of CPU me and eo beween wo LBGK model fo 3D Womesley ow. To sasfy convegence ceon, Cases Models oal eae peod T ) CPU mn) Re = 100 3D D3Q19) e-00 D pesen model) e-00 Re = 10 3D D3Q19) e- D pesen model) e- model s much moe ecen fo such an axsymmec pulsale ow poblem. Accodng o he oveall aveage eo, he axsymmec LBM esul s slghly bee han he 3D LBM esul. One possble eason s ha he axsymmec model dd no nvolve he eo n he ccumfeenal decon. 4. CONCLUSION An axsymmec ncompessble LBGK model was deved n hs pape by noducng an addonal souce em o he ncompessble LBGK model [6]. Wh lm of Mach numbe M 1 and L x =c s T) 1, hs axsymmec LBGK model successfully ecoveed he Nave Sokes equaon n he cylndcal coodnaes hough Chapman Enskog expanson efe o Appendx A). Fo he addonal souce em n ou model, mos velocy gaden ems can be obaned fom hgh-ode momenum of dsbuon funcon, whch s conssen wh he phlosophy of he LBM. The axsymmec ncompessble LBGK model was successfully appled o smulae he 3D pulsale ow n ccula ube and he spaal accuacy of he numecal soluons s abou second ode. Though he smulaons wh 1 Re 000 Reynolds numbe s based on ppe s damee), Womesley numbe 1 5, he axsymmec ncompessble LBGK model gves ou vey accuae esuls fo 3D Womesley ow. Ou model s vald fo unseady axsymmec ows. In ou nvesgaon, cuen LBGK model ncopoang he exapolaon bounday condon and he focng em s second ode n space. Though compason of he wo schemes o mplemen pulsale pessue gaden, seems ha he scheme of applyng addonal focng em has advanages han he scheme of specfy pessue on nle=oule bounday fo he 3D Womesley ow, whch has unfom pessue gaden a any me. Though compason wh Hallday s axsymmec model, s obseved ha he pesen model can educe he compessbly eec n Hallday s model because pesen model deved fom he ncompessble model [6]. Compang wh 3D LBGK model, pesen axsymmec LBGK model gves ou slghly bee esuls han he 3D LBM and s much moe ecen han he 3D LBM n ems of compuaonal me and memoy. APPENDIX A: CHAPMAN ENSKOG DERIVATION OF THE AXISYMMETRIC INCOMPRESSIBLE DQ9 MODEL Hee we show ha he followng connuy and Nave Sokes equaons may be obaned fom a lace Bolzmann DQ9 @ = u A1)

14 11 T. S. LEE, H. HUANG AND @u + u + @ u ) A) whee and epesen ndex x o and x means x o. In he followng equaons, f x; ; ) s he dsbuon funcons of DQ9 model, he souce em h x; ; ) was ncopoaed no an adjused evaluaon equaon fo he lace ud s momenum dsbuon A3): f x + e x ;+ e ;+ ) f x; ; )= 1 [f0) x; ; ) f x; ; )] + h x; ; ) A3) Apply he Taylo expanson A4) and expandng he me and space devaves n ems of he Knudson numbe [17] n he followng A5): f x + e x ;+ e ;+ )= n=0 f = f 0) + f 1) + f ) h = h 1) + h ) + n n! Dn f x; ; ) A4) A5) In A5), hee s no equlbum h em, whee = and can epesen x o and + ), we also noce ha = 1 = e 1x + 1. Incopoae A4) and A5) no A3), n he zeoh, s and second odes of ae A6), A7) and A8), especvely: E 0 =f 0 f eq )= A6) E 1 =@ )f 0) + 1 f 1) h1) E f 0) )f 0) + h 1) ] + 1 f ) h) A7) A8) The equlbum f 0) elaonshps: s dened by 6) and he dsbuon funcon f sased he followng 8 =0 f 0) = p ; cs 8 =0 e f 0) = 0 u A9) 8 =0 f m) =0; 8 =0 e f m) = 0 fo m 0 A10)

15 AN AXISYMMETRIC INCOMPRESSIBLE LBGK 113 A.1. Mass consevaon and h 1) Summaon of Equaon A7) ove leads o 1 p + 0@ u = h 1) A11) By ewng A11) n a dmensonless fom, fo he s em o be neglgble, L x =c s T ) 1 should be sased [6]. Tha s an addonal lm of ou devaon besdes condon Mach numbe M 1. Summaon of Equaon A8) ove leads o p = 0 and fo smplcy, we assume )h 1) A1) h ) = 0 A13) Compae A11) wh A1) and noce! = 1, followng he selecon of h 1) h 1) =! 0 u s easonable. A14) Hence, a he same me, due o ou assumpon A13), we oban he followng equaon A15): h ) = )h 1) = 1 1 )[ ] u! 0 u ) ] A15) Noce ha o deve A15), we used equaon! e =0. A.. Momenum consevaon and h ) Mulple Equaon A7) wh e and summaon ove leads o 1 u 1 0 = whee 0 = 8 =0 e e f 0) = 0 u u + cs. So, Incopoae A17) no A15), we can ge VIP1) 1 u 0 c s + 0 u u ) h ) = h 1) e = 0 A16) 1 ) c s + 0 u u )= c s + 0 u u ) A17)

16 114 T. S. LEE, H. HUANG AND C. SHU Mulply Equaon A8) wh e and summaon ove lead o u ) = 1 ) 1 e h 1) 1 e e h 1) + To fuhe deve Equaon A18), sly we obaned A19) and A0): 1) = e e f 1) = e e D 1 f 0) + e e h 1) [ 0) k e e e k f 0) + e e h 1) ) e e e k f 0) = 0 k jk + j k + j k )u j = 0 k h ) e A18) j u j In Equaon A18), he 0 s of ode M,Ms he Mach numbe), because f we assume he velocy of ode u 0, lengh L, 0 = L=u 0, 0 u 0 = 0, whle he em =@x =@x ) s of ode cs u 0 =L, hence, compae wh em 0 / ema0) = O u 0 0 / c s u 0 L ) = O u0 c s ) = OM ) egadng he 0 s vey small, hs em can be omed. 1 1 ) 1) 1 1 )[ 0 j u j @x +cs 1 1 ) ) 0 u = 0 ))] ju j = 0 )] A1) n he above fomula we used e e h 1) = e e [! 0 u =]=cs 0 u =). The RHS em n A18) s RHS = c s 1 ) ) h ) e A) Hence ncopoae A16), A18), A1) and A), we can ) ) 0 u 1 + h ) @x Compae momenum equaon A3) wh Equaon A), o ecove he Nave Sokes momenum equaon VIP) should be esablshed: 0 u ) 1 + h e 0 u 1 ) 0u

17 AN AXISYMMETRIC INCOMPRESSIBLE LBGK 115 Solvng equaon sysem VIP1) and VIP), we can oban he expesson of h ) as follows: h ) 3 =! [@ cs u u )] 0 0 u )e ) ] 3 1 =! [@ 3 + x u x u + u u + u x u )e x A4) In he above devaon, we used Equaon A4): 0 u 0 u ) 0 u ) = 0 u A5) We successfully deved he expesson of h 1) A14), h ) A4) and ecoveed he connuy equaon A1) and Nave Sokes equaon A). APPENDIX B: NOMENCLATURE c = x= c s = c= 3 e f h M N p p Re R S T U max M max U c u u x u u he ao beween lace sze and me sep he speed of sound he pacle velocy veco along decon he pacle dsbuon funcon he souce ems Mach numbe numbe of lace nodes n damee ud pessue he maxmum amplude of he oscllaoy pessue gaden Reynolds numbe he adus of he ccula ppe Souhal numbe of Womesley ow me samplng peod he maxmum velocy appea n ube axs dung a samplng peod U max =c s he chaacesc velocy, whch s equal o 0) o much lage han 1) U max ud velocy veco x componen of he velocy componen of he velocy componen of he velocy, can epesen x o

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