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1 SIENE HINA Technologcal Scences Acle Apl 4 Vol.57 No.4: 84 8 do:.7/s The andom walkng mehod fo he seady lnea convecondffuson equaon wh axsymmec dsc bounday HEN Ka, SONG MengXuan & ZHANG Xng * Depamen of Engneeng Mechancs, Key Laboaoy fo Themal Scence and Powe Engneeng of Mnsy of Educaon, Tsnghua Unvesy, Beng 84, hna Receved Augus 8, 3; acceped Decembe 8, 3; publshed onlne Febuay 8, 4 In hs acle, he andom walkng mehod s used o solve he seady lnea convecon-dffuson equaon (DE) wh dsc bounday condon. The negal soluon coespondng o he andom walkng mehod s deduced and he elaonshp beween he dffuson coeffcen of DE and he nensy of he andom dffuson moon s obaned. The andom numbe geneao fo abay axsymmec dsc bounday s deduced hough he polynomal fng and nvese ansfom samplng mehod. The poposed mehod s esed hough wo numecal cases. The esuls show ha he andom walkng mehod can solve he seady lnea DE effecvely. The nfluence of he paamees on he esuls s also suded. I s found ha he eo of he soluon can be deceased by nceasng he pacle eleasng ae and he oal walkng me. convecon-dffuson equaon, andom walkng mehod, dsc bounday, andom numbe geneaon aon: hen K, Song M X, Zhang X. The andom walkng mehod fo he seady lnea convecon-dffuson equaon wh axsymmec dsc bounday. Sc hna Tech Sc, 4, 57: 848, do:.7/s Inoducon Many physcal, bologcal, and bochemcal phenomena can be descbed by he convecon-dffuson equaon (DE) []: u x x x *oespondng auho (emal: x-zhang@snghua.edu.cn) whee s he concenaon of he vaable, u s he h velocy componen and ν s he dffuson coeffcen. Eq. () can be solved by vaous mehods, such as fne dffeence mehod, fne elemen mehod, and andom walkng mehod, among whch andom walkng mehod s he mos commonly used one. Random walkng mehod s ndependen of he meshes and can guaanee he accuacy of he, () soluon wh less calculaon. Meanwhle, hs mehod can epoduce he bounday condons wh some specal shape. Theefoe, andom walkng mehod has been gven much aenon by eseaches []. Many sudes have been done o vefy he effecveness of he andom walkng mehod o he dffuson sysem [3] and he convecon-dffuson sysem [4,5]. Random walkng mehod s also wdely appled o smulae he amosphec polluan dspeson [6 8]. In a ecen sudy, Song e al. eaed he wake flow of he wnd ubne as he convecon-dffuson moon wh axsymmec dsc bounday and used he andom wakng mehod o solve he DE, whch educed he calculaon effecvely compaed o he FD calculaon [9]. Howeve, he paamees of he mehod wee chosen empcally n Song s sudy. The pacle eleasng bounday was no vefed. The accuacy of mehod sll needs o be fuhe suded. In hs acle, he effecveness of he andom pacle walkng mehod o solve he seady lnea DE s suded. The negal soluon based on he mehod s deduced. The Scence hna Pess and Spnge-Velag Beln Hedelbeg 4 ech.scchna.com lnk.spnge.com

2 hen K, e al. Sc hna Tech Sc Apl (4) Vol.57 No.4 85 elaonshp beween he dffuson coeffcen (ν) of he DE and he nensy of he andom dffuson moon s obaned. The andom numbe geneao fo abay axsymmec dsc bounday s consuced hough polynomal fng and nvese ansfom samplng mehod. The nfluence of he paamees of andom walkng mehod on he esuls s also suded. Random walkng mehod The seady lnea dffuson-convecon equaon n ef. [8] s u, x x x whee u s unfom and ν s a consan. Assume ha he compuaonal doman s a cubc aea. The velocy nle s on he lef sde, whch s shown n Fgue. Thee s a dsc souce wh fxed axsymmec concenaon nea he enance. The doman s lage enough ha he concenaons of he suoundng boundaes can be se o. The oule bounday s se o be fully developed. The expessons of he bounday condons ae shown as follows: G,,, nle,op,boom,lef,gh n oule whee epesens he dsc souce aea n Fgue. s he dsance fom he dsc cene and G() s he axsymmec concenaon dsbuon a he dsc. The seps of he andom walkng mehod ae shown as follows [9]. ) Pe-calculaon: The flow feld of he doman whou he dsc s obaned hough he FD calculaon. ) Poducon: Pacles ae geneaed a a cean ae whn he dsc aea. The ae s denoed as m, whch sands fo he numbe of pacles geneaed n un me peod. 3) onvecon: The oal me s dscezed no nevals of lengh. In each me sep, he convecve dsplacemen n each coodnae componen s added o he pacle poson, expessed as Fgue () (3) x u, (4) c Schemac of he compuaonal doman (sde vew). whee u s he local h velocy componen a he poson of he pacle. 4) Dffuson: In each me sep, a Gaussan dsbued andom dsplacemen s added o each coodnae componen of he pacle poson, expessed as x logr cos π R, (5) d whee s he dffuson nensy, R and R ae wo ndependen unfomly dsbued andom numbes n [,]. 5) Sascs: Afe smulang fo some me, he dsbuon of he pacles n he compuaonal doman asympoes sable. The doman s meshed and he pacles n each mesh ae couned n each me sep. The numbe of pacles epesens he concenaon. To oban a sable sascal esul fo he densy of pacles, s necessay o smulae suffcen me seps. The concenaons of each me sep ae aveaged o educe he andomness and he fnal sable pacle concenaon dsbuon s obaned. 3 Devaon of he one-dmensonal andom pacle walkng 3. Pacle concenaon Fo smplfcaon, consde he one-dmensonal andom walkng fs. Pacles ae eleased fom a sngle pon. Pevous sudy has poved ha a Gaussan dsbued andom numbe can be geneaed hough he followng expesson []. x logr cos π R, (6) whee R and R ae wo ndependen unfomly dsbued andom numbes n [,]. The dffuson dsplacemen by eq. (5) sasfes he pobably densy funcon (PDF) as [] p x; x exp. π In each me sep, he numbe of pacles eleased s m. The numbe of pacles a an abay poson a he momen of s he sum of he numbe of pacles ha ae eleased a each me sep and each hs poson afe convecon and dffuson. Theefoe, he numbe of pacles locaed n [x, x+dx] a he momen of can be calculaed by / N x; ; d m p x u x / (7) ; d m p xu x. (8) The pacle concenaon a x a he momen of s N x; / x; ;. d m p x u x (9)

3 86 hen K, e al. Sc hna Tech Sc Apl (4) Vol.57 No.4 When, he Remann sum eq. (8) uns no negaon ; ; x mp x u d x u m exp π d. () I can be poved ha when,, and x, we have d. p p u m u () x x x x Tha s, he concenaon expessed by eq. () sasfes he one-dmensonal DE, excep x=. The value of a x= s deemned by he bounday condon, so ha () does no have o sasfy he equaon. onsde he hee-dmensonal DE. Smla o he one-dmensonal suaon, he pacle concenaon a he pon (x, y, z) can be calculaed as x, y, z; R, f mf x u exp 3 π, z sn w exp d dd cos π y v exp R () whee u, v, and w ae he hee velocy componens. R s he adus of he dsc. f() s he pacle eleasng PDF a he dsc, he ae of whch depends on he concenaon dsbuon of he dsc. I can be poved ha (x, y, z;r, f()) sasfes he sable lnea hee-dmensonal DE when x y z and a leas one vaable among x, y, z does no equal o. The pon x=y=z= s he bounday of he equaon, whch does no need o sasfy he equaon. The pacle concenaon expessed by eq. () depends on he pacle eleasng ae m a he dsc. To oban he pacle concenaon ndependen of m, needs o nomalze he pacle concenaon. Assume ha he chaacesc concenaon s. In he dsc aea, he nomalzed concenaon equals o he concenaon on he bounday, ha s,,, z; R, f G( z), (3) consde an aea wh he lengh of L downseam and akng he negaon of eq. (3) n hs aea, gves L L R R,, ; R, f πdd dl G πdd l, (4) whee he lef sde s he numbe of pacles eleased. Le L. I gves ml L R G πd, u (5) whee m s he pacle eleasng ae. The chaacesc concenaon s expessed as m. (6) u G πd R Theefoe, he nomalzed concenaon s,, ;,, x y z R f x, y z; R. (7) 3. PDF of pacle eleasng In eq. (), f() s a funcon o be deemned, whch depends on he concenaon dsbuon G() a he dsc. We have z G,, z; R, f mf u exp 3 π cos v exp, z sn w exp ddd π R (8) s que dffcul o oban f() hough solvng eq. (8) wh a known funcon G(). In hs acle, a polynomal fng mehod s used o solve f(), he seps of whch ae shown as follows. ) Le f ()= (=,, ), hen a sees of funcons can be obaned G z,, z; R, f. (9) ) G(z) can be appoxmaed hough he lnea combnaon of G (z) Gz ( ) G( z ), ()

4 hen K, e al. Sc hna Tech Sc Apl (4) Vol.57 No.4 87 he lnea egesson coeffcens ae denoed as. 3) f() can be appoxmaed as f () f(). () Table Paamees of andom walkng mehod Pacle eleasng ae (m) Lengh of me sep () Toal numbe of me seps (N) Random numbe geneao of pacle eleasng Table Paamees of he numecal cases In hs secon, andom numbe sasfyng he PDF f() s consuced hough nvese ansfom samplng mehod [] usng he unfomly dsbued andom numbe n [,]. The bounday a he dsc s axsymmec, and hence obans πx () whee x s he unfomly dsbued andom numbe n [,]. Fo he adal decon of he dsc, assume ha hee exss a ansfomaon = (x) whch makes he PDF of be f(), whee x s he unfomly dsbued andom numbe n [,]. We have Thus,, R= R π f x R x x R x R x R x R F P d P P ascends descends ascends, descends. R F R x F ascends, R x descends. (3) (4) Fom he deducon above, he andom numbe sasfyng he PDF f() can be geneaed by x, πx, (5) u (m/s) v (m/s) w (m/s) (m/s / ) ν (m /s) R (m) ) osne bounday condon: π (,, z) R. 5cos. 5. zr z R 4. Unfom bounday condon Fo unfom bounday condon, he appoxmaed expesson of f() s obaned hough polynomal fng, shown as f. (6) πr Though he nvese ansfom samplng mehod, he andom numbe geneang funcon s x R x, πx. (7) Pefom he andom walkng mehod usng he andom numbe geneaon ep. (7) and he dsc concenaon dsbuon s obaned. Fgue shows he compason of he andom walkng esul and he specfed exac concenaon dsbuon on he bounday. I can be seen ha boh of he concenaon dsbuons ae n good ageemen. Thee s a lle flucuaon fo he andom walkng esul, whch s manly due o he andomness. whee x and x ae wo ndependen unfomly dsbued andom numbes n [,]. 4 Numecal esuls and dscusson In hs secon, wo numecal cases ae used o es he poposed mehod. A cubc aea s consdeed as he compuaonal doman. The paamees ae shown n Fgue. Table lss he paamees of he andom walkng mehod and Table lss he paamees of he DEs. Two ypes of bounday condons ae consdeed. ) Unfom bounday condon: (,, z ). RzR Fgue ompason of he andom walkng concenaon dsbuon and he specfed exac dsbuon on he bounday (unfom bounday).

5 88 hen K, e al. Sc hna Tech Sc Apl (4) Vol.57 No.4 Fgue 3 shows he compason of he andom walkng soluon and he numecal soluon of he DE. Fgue 3(a) shows he nomalzed concenaon along he z-axs a 4 m downseam of he dsc. I can be seen ha he eo aound he dsc cene s a lle bg and he andom walkng soluon agees wh he numecal soluon of he DE fo ohe aeas. Thee ae moe pacles aound he cenelne of he dsc such ha he flucuaons of he numbe of pacles n hs aea ae lage caused by he andomness, whch n un causes a lage eo. Fgue 3(b) shows he nomalzed concenaon along he x-axs a he cenelne of he dsc. I can be seen ha he eos nea he dsc ae a lle lage and he ones of ohe aeas ae less. The aveage eo a he cenelne of he dsc s.5. I s found ha he funcon o be negaed n eq. (8) has a seep slope fo aound he dsc (whee x s small) when s small. Theefoe, he me scale should be small enough o educe he eo of he soluon down o an accepable level. Howeve, he oal calculaon nceases as deceases. Theefoe, should be chosen caefully o balance he me cos and he accuacy. In pesen sudy, s chosen o be. Alhough hee s some eo aound he dsc, fo he aea a lle fuhe fom he dsc (whee x s lage), he funcon o be negaed n eq. (8) has small slope fo. The chosen can dsngush changes of he funcon. Theefoe, he eos ae small fo he aea fa fom he dsc. 4. osne bounday condon Fo he cosne bounday condon, he appoxmaed expesson of f() s obaned hough polynomal fng of 7 odes, shown as f = , (8) he andom numbe geneao of f() s dffcul o be expessed by an explc fomula. Theefoe, (x) s solved numecally and he esul s shown n Fgue 4. I can be seen ha (x) nceases monooncally wh x. Pefom he andom walkng mehod usng he andom numbe geneaon funcon n Fgue 4. Fgue 5 shows he compason of he andom walkng esul and he specfed exac concenaon dsbuon on he bounday. I can be seen ha boh of he concenaon dsbuons ae n good ageemen. Fgue 6 shows he compason of he andom walkng soluon and he numecal soluon of he DE. Smla o he unfom bounday condon, fo he pofle along he z-axs downseam, he eo of he dsc cenelne s a lle lage and he ones n ohe aea ae small. Fo he dsc cenelne along he x-axs, he eos aound he dsc ae a lle lage and he ones fa fom he dsc ae small. 4.3 Influence of he paamees In hs secon, he unfom bounday condon s consdeed. Fgue 3 ompason of he andom walkng esul and he numecal soluon of he DE (unfom bounday). (a) Nomalzed concenaon along he z-axs a 4 m downseam of he dsc; (b) nomalzed concenaon along he x-axs a he cenelne of he dsc. Fgue 4 The andom numbe geneaon funcon of he adal decon of he dsc (cosne bounday).

6 hen K, e al. Sc hna Tech Sc Apl (4) Vol.57 No.4 89 Fgue 5 ompason of he andom walkng concenaon dsbuon and he specfed exac dsbuon on he bounday (cosne bounday). he paamees. Table 3 shows he esuls of vaous values of m, whee he aveage eo and he maxmum eo ae he ones along he x-axs a he dsc cenelne. I can be seen ha he aveage eos of he esuls decease as m nceases. The ncease of m educes he andomness of he andom walkng mehod, and so he eo s educed. Meanwhle, he maxmum eos fo vaous values of m ae almos he same. The ny dscepancy s manly due o he andomness. Table 4 shows he esuls of vaous values of. I can be seen ha he aveage eos decease as nceases. The ncease of nceases he oal smulaon me and educes he andomness. Hence, he aveage eo s educed. Howeve, epesens he calculaon esoluon, whch deemnes he accuacy of he esuls nea he pacle eleasng aea (dsc). The nceasng of wll ncease he eo nea he pacle eleasng aea. Theefoe, he maxmum eo nceases wh. Table 5 shows he esuls of vaous values of N. I can be seen ha he aveage eos end o decease wh N. When s fxed, he ncease of N nceases he oal smulaon me and educes he aveage eo. Meanwhle, he maxmum eos fo vaous N ae almos he same because of he fxed. The ny dscepancy s manly due o he andomness. Oveall, he aveage eo deceases as he pacle eleasng ae nceases and as he oal smulaon me nceases. Table 3 Resuls fo vaous pacle eleasng ae (unfom bounday) m N Aveage eo Maxmum eo Table 4 Resuls fo vaous lenghs of me sep (unfom bounday) m N Aveage eo Maxmum eo Fgue 6 ompason of he andom walkng esul and he numecal soluon of he DE (cosne bounday). (a) Nomalzed concenaon along he z-axs a 4 m downseam of he dsc; (b) nomalzed concenaon along he x-axs a he cenelne of he dsc. The nfluences of he pacle eleasng ae (m), me sep lengh (), and he numbe of oal me seps (N) on he numecal esuls ae suded by choosng vaous values of Table 5 Resuls fo vaous numbes of oal me seps (unfom bounday) m N Aveage eo Maxmum eo

7 8 hen K, e al. Sc hna Tech Sc Apl (4) Vol.57 No.4 The maxmum eo nceases wh he me sep lengh, ndependen of he pacle eleasng ae and he numbe of oal me seps. 5 oncluson In hs acle, he andom walkng mehod s suded fo he seady lnea DE wh axsymmec dsc bounday. The negal soluon coespondng o he andom walkng mehod s deduced. I s poved ha he negal soluon sasfes he equaon when. Though polynomal fng and nvese ansfom samplng mehod, he andom numbe geneao fo abay axsymmec dsc bounday s developed. The esuls of wo ypes of bounday condons (unfom bounday and cosne bounday) ndcae ha he soluons by he poposed mehod ae n good ageemen wh he numecal esuls of he DE. The nfluence of he andom walkng paamees ae also suded. The esuls show ha he eo of he soluon can be educed hough choosng lage pacle eleasng ae and lage numbe of oal me seps. Random walkng mehod s a knd of meshless mehod. The eamen of bounday s no nfluenced by he sze of he meshes, whch can educe he eo caused by he meshng. When he pacle dsbuon s obaned afe smulaon, he pacle concenaon can be calculaed hough sascs. The sze of he meshes used n pacle sascs does no nfluence he oal compuaonal me much. The poposed andom numbe geneaon mehod s applcable o he abay axsymmec dsc bounday. ombnng wh andom walkng mehod, can solve he seady lnea DE wh abay axsymmec dsc bounday. Ths mehod s applcable o abay sze of meshes, especally o he suaon ha he mesh sze s lage han he dsc sze, such as he smulaon of he wnd ubne wake flow. Fo aflow smulaon of wnd fam, he sze of he mesh s usually lage, whch s dffcul o dsngush he wnd oo. The poposed mehod can sll guaanee he accuacy of he wake flow smulaon. Howeve, he above deducon s based on he lnea equaon. Tha s, he velocy componens n he equaon ae unfomly dsbued houghou he doman. When he flow feld s no unfom, he poposed mehod wll bng addonal eo. The suaon of non-unfom flow feld has scope fo fuue wok. Ths wok was suppoed by he Inenaonal Scenfc and Technologcal oopeaon Pogam of hna (Gan No. DFG3), he hna Posdocoal Scence Foundaon (Gan No. 3M5343), and he Naonal H-Tech Reseach and Developmen Pogam of hna ( 863 Poec) (Gan No. 7AA5Z46). Snvasan G, Taakovsky D M, Denz M, e al. Random walk pacle ackng smulaons of non-fckan anspo n heeogeneous meda. J ompu Phys,, 9: Jang J G, Wu J. Lace-walk mehod fo he convecon-dffuson equaon (n hnese). J Nanng Unv (Na Sc),, 47: oulbaly R, Leco. Smulaon of dffuson usng quas-andom walk mehods. Mah ompu Smula, 998, 47: Masgallc S. Pacle appoxmaon of a lnea convecon-dffuson poblem wh Neumann bounday condons. Sam J Nume Anal, 995, 3: Leco, Schmd W. Pacle appoxmaon of convecon-dffuson equaons. Mah ompu Smula,, 55: Yu H B, Jang W M. Random walk modelng of dspeson n wake aea of exhaus owe (n hnese). Aca Aeodynamca Sn, 996, : Holmes N S, Moawska L. A evew of dspeson modelng and s applcaon o he dspeson of pacles: An ovevew of dffeen dspeson models avalable. Amos Envon, 6, 4: Jn T S, Yang J, Han S Q, e al. Paccal dscusson on he andom floang model wh he amosphec polluan dspeson (n hnese). J Saf Envon, 8, 8: Song M X, hen K, He Z Y, e al. Wake flow model of wnd ubne usng pacle smulaon. Renew Eneg,, 4: 85 9 Box G, Mulle M E. A noe on he geneaon of andom nomal devaes. Ann Mah Sa, 958, 9: 6 6 Ln Y L. Appled Sochasc Pocesses. Beng: Tsnghua Unv Pess, Luc D. Non-Unfom Random Vaae Geneaon. New Yok: Spnge-Velag, 986