Vortex particle intensified large eddy simulation - VπLES

Vo ex Pa icle In ensi ied La ge Eddy Simula ionN. Ko ne and S. Sama bakhsh VI In e na ional Con e ence on Pa icle-based Me hods – Fundamen als and Applica ions PARTICLES 2019 E. O˜na e, M. Bischoff, D.R.J. Owen, P. W igge s & T. Zohdi (Eds) VORTEX PARTICLE INTENSIFIED LARGE EDDY SIMULATION - VπLES S. Sama bakhsh1and N. Ko ne 2 Chai o modeling and simula ion Uni e si y o Ros ock Ros ock, Ge many 1e-mail: sina.sama bakhsh@uni- os ock.de 2e-mail: nikolai.ko ne @uni- os ock.de Key wo ds: LES, subg id model, meshless me hod, ee je , hyb id Eule ian/Lag angian me hod Abs ac . This pape p esen s a no el La ge Eddy Simula ion app oach wi h a di ec esolu ion o he subg id mo ion o fine concen a ed o ices. The me hod, p oposed fi s by [10], is based on combina ion o a g id based and he g id ee compu a ional o ex pa icle (VPM) me hods. The la ge scale flow s uc u es a e simula ed on he g id whe eas he concen a ed s uc u es a e modeled using VPM. Due o his combina ion he ad an ages o bo h me hods a e s eng hened whe eas he disad an ages a e diminished. The p ocedu e o he sepa a ion o small concen a ed o ices om he la ge scale ones is based on LES fil e ing idea. The flow dynamics is go e ned by wo coupled anspo equa ions aking wo-way in e ac ion be ween la ge and fine s uc u es in o accoun . The fine s uc u es a e mapped back o he g id i hei size g ows due o diffusion. Algo i h- mic aspec s specific o h ee dimensional flow simula ions a e discussed. Validi y and ad an ages o he new app oach a e illus a ed o a well ied benchma k es o he decaying homogeneous iso opic u bulence using he expe imen al da a o [4] and ee u bulen je flow using expe imen s o [8, 2, 15]. 1 INTRODUCTION Insufficien esolu ion o fine o ex s uc u es in u bulen flows is one o he key p oblems in Compu a ional Fluid Dynamics (CFD). The mos ad anced and popula echnique o esol e mul i scale flow s uc u es is he La ge Eddy Simula ion (LES) which is based on he idea o scale decomposi ion in o la ge and small ones. While he la ge eddies a e di ec ly esol ed on he g id, he effec o small o ices is aken in o accoun h ough a subg id s ess (SGS) model. The subg id mo ion is no esol ed in LES bu a he i is modelled using diffe en unc ional and s uc u al app oaches. Howe e , he e a e many p oblems which equi e di ec ep esen a ion o he subg id mo ion o simula e, o ins ance, mixing o pa icle 1 174 S. Sama bakhsh and N. Ko ne dynamics in u bulen flows. In ou p e ious pape s (see [9], [10], [11], [16] ]), we p oposed a simula ion echnique esembling LES wi h an effo o di ec ly ep oduce he subg id mo ion a leas in he s a is ical sense. I is sugges ed o apply a hyb id g id and pa icle based me hod, u ilizing a combina ion o he fini e olume and compu a ional o ex pa icle (VPM) (see [5]) me hods. The la ge scale field is ep esen ed on he g id like in LES, whe eas he small scale one (subg id field) is calcula ed using he VPM. The new me hod called VπLES is a pu ely Lag angian one o small s uc u es and pu ely g id based one o la ge scale s uc u es. The me hod is based on he decomposi ion o he eloci y uand o ici y fields ωin o he dis ibu ed la ge scale (uppe index ’g’) and concen a ed small scale (uppe index ’ ’) fields: u(x, )=ug(x, )+u (x, ),ω(x, )=ωg(x, )+ω (x, ) (1) The fine o ex de ec ion p ocedu e u ilizes he La ge Eddy Simula ion (LES) fil a ion applied o he g id based eloci y field ug: ug(x, )= ∞  −∞ ug(s, )F(x−s)ds(2) whe e F(x−s) is a ce ain fil e unc ion. The small scale eloci y field u�calcula ed as he diffe ence be ween he o iginal and fil e ed fields u�(x, )=ug(x, )−ug(x, ) (3) should be app oxima ed by o ex pa icles in egions o concen a ed o ices which a e de ec ed using any o ex iden ifica ion c i e ia, o ins ance, λci ([1]). The cells wi h λi>λ ci,min con ain he o ices which in p inciple can be con e ed o o ex pa icles. Such cells a e ma ked as ac i e ones using he λi,ac i e field: λi,ac i e =1,i λci >λ ci,min 0,o he wise (4) whe e λci,min is a ce ain small alue in oduced in o de o limi he numbe o pa icles. To keep he equi ed compu a ional esou ces on an accep able le el, only small o ices wi h size p opo ional o he local cell size Δ a e o be con e ed o single o ex pa icles. Neighbo ing cells which all ha e λi,ac i e = 1 o m la ge o ices. Fo hem i is supposed ha he la ge o ices wi h scales o a ew Δ can accu a ely be ep esen ed on he g id. The e o e he nex ask is o de ec cells wi h fine o ices. Acco ding o he algo i hm, all neighbo ing cells o he i− h cell a e checked o he condi ion λci >λ ci,min. I all neighbo s ulfill his condi ion, we iden i y a cell clus e which emains on he g id and all i s cells become non- ac i e λi,ac i e = 0. Only o ices in cells wi h λi,ac i e = 1 a e o be eplaced by o ex pa icles. A each cell wi h λi,ac i e = 1 he new o ex pa icle is in oduced a he cell cen e i he pe missible numbe o o ex pa icles pe cell Np is no exceeded. O he wise, 2 175 S. Sama bakhsh and N. Ko ne he new o ex eplaces he cell’s weakes one. The numbe Np was in oduced o keep he o al numbe a a easonable le el. This es ic ion is con o m wi h he concep ha he la ges con ibu ion o he subg id kine ic ene gy is made by a small ac ion o he s onges o ices. The adius o he new o ex is se as σ=βV ol1/3 i, whe e βis he o e lapping a io which is aken as β= 2 and V oliis he olume o he i- h cell. A ho ough analysis o he influence o Np and λci,min o he je case is gi en in [11] in Sec. 4.3.5. The o ex pa icle s eng h is calcula ed as α=V oliω =V oli(∇×u) (5) The eloci y u (x, ), induced by he o ex pa icles, is calcula ed a g id poin s xusing he Bio -Sa a law u (x, )= 1 4π∇×FlowV olume ω (ξ, ) |x−ξ|dV (ξ) and sub ac ed om he g id eloci y ug,new =ug−u . Thus, he o al eloci y a g id poin s ug,new +u =ug emains cons an a e he o ex pa icle gene a ion p ocedu e. 2 GOVERNING EQUATION The e olu ion o o ex pa icles and la ge scale flow ep esen ed on he g id is desc ibed by a sys em o wo coupled anspo equa ions de i ed in [9, 10, 11] o incomp essible iso he mal flows: ∂ug ∂ +(ug·∇)ug=−1 ρ∇pg+νΔug+u ×ωg(6) dω d =(ω ·∇)(u +ug)+νΔω +∇×[u ×ωg−u ×ωg],(7) The sum o he cu l o he fi s equa ion and he second equa ion e ie es he o iginal Na ie S okes equa ion w i en in he o m o he o ici y anspo equa ion. The fi s equa ion (6) is coupled wi h he second one (7) h ough he addi ional e m u ×ωg whe eas he coupling o he second equa ion wi h he fi s one is due o he e ms (ug· ∇)ω ,(ω ·∇)ugand ∇×[u ×ωg−u ×ωg. The equa ions (6) and (7) a e sol ed sequen ially. The fi s equa ion is sol ed on he g id whe eas he second one uses he g id ee Vo ex Pa icle Me hod (VPM) [5]. The physical meaning o he coupling e m u ×ωgis explained in [10]. The o ex pa icle displacemen is calcula ed om he ajec o y equa ion d i d =ug i+u i,(8) whe e iis he pa icle numbe . Compu a ion o he eloci y induced by o ex pa icles u is pe o med wi h he di ec summa ion o he Bio-Sa a law aking in o accoun one o wo laye s o neighbo ing cells. Only induc ion o he neighbo ing poin s is aken in o accoun because he eloci y u is much less han ugand i is mos ly de e mined 3 176 S. Sama bakhsh and N. Ko ne by in e ac ion o neighbo ing pa icles lying a a sho dis ance. The jus ifica ion o his simplifica ion p esen ed in [11] is ha he co ela ion be ween neighbo ing small scale o ices is weak and hey a e well sepa a ed. This simplifica ion can be conside ed as a kind o model which esul s in a e y as compu a ional p ocedu e o a local cha ac e sui able o pa allel calcula ions. The eloci y induced by a o ex pa icle can be calcula ed om he o mula u p=1 4π α×ξ ξ3(1 −e−ξ3/σ3) (9) p oposed by [14]. He e αand σa e he s eng h and he adius o a o ex pa icle, which a e defined below. The eloci y induced by a se o pa icles u is calcula ed as he sum o u p. I should be no ed ha he echnique p esen ed he e is independen o any specific choice o o ex pa icles. Pa icula ly, a se o unc ions in oduced in [12] can be used wi hin he p esen me hod. 2.1 Nume ical solu ion o he equa ions (7) and (8) using he VPM Ins abili y o nume ical solu ion o he equa ion (7) caused by he s e ching e m (ω·∇)uis he mos impo an p oblem o he VPM along wi h he compu a ion o he eloci y u . In g id based me hods wi h low and mode a e o de schemes, he ac ion o he s e ching is effec i ely coun e balanced by he nume ical iscosi y which is e y low in Lang angian o ex pa icle me hods. Theo e ically, a s able VPM solu ion can be ob ained by inc easing he accu acy o he s e ching and diffusion simula ion which can be a ained by a high numbe o o ex pa icles and high empo al esolu ion. Bo h make he me hod imp ac ical a leas o high Reynolds numbe s. A e many effo s he au ho s se led on he algo i hm which was o iginally p oposed by [7] and modified in [11]. This algo i hm consis s o he ollowing subs eps: •Calcula ion o he change o he o ici y s eng h magni ude d|ω | d =d√ω ·ω d =ω |ω |·dω d (10) whe e dω d is calcula ed om (7) wi hou he iscous diffusion e m. The e m (ω ·∇)u is calcula ed aking in o accoun adjacen o ex pa icles loca ed only wi hin one o wo laye s o neighbo ing cells. •Calcula ion o he pa icle leng h om he equa ion o he elemen a y sec ion dl ans- po ed in in iscid flow dl d =l |ω | d|ω | d (11) •Calcula ion o he pa icle co e adius om he equa ion desc ibing he anspo o an elemen a y ube wi h leng h land adius σin in iscid flow dσ d =−σ 2l dl d (12) 4 177 S. Sama bakhsh and N. Ko ne •Conside a ion o he iscosi y influence using he co e sp eading me hod (CSM) (see [5]). The pa icle co e adius is inc eased by Δσ: Δσ=√4νΔ (13) •Calcula ion o he new pa icle o ien a ion ω∗=ω ( )+dω d Δ (14) •Calcula ion o he pa icle s eng h magni ude om [7] |α( +Δ )|=|α( )|σ( +Δ ) σ( )(15) •Calcula ion o he new s eng h ec o α( +Δ )=|α( +Δ )|ω∗ |ω∗|(16) In he o iginal e sion p oposed by [7] he nex s ep should be he edis ibu ion o pa icles whose leng h has doubled. Acco ding o ou expe ience he edis ibu ion esul s in an a alanche-like inc ease o he o ex pa icles numbe in a eas o s ong s e ching. To p e en his [7] p oposed a special elimina ion p ocedu e based on a knowledge o a h eshold o he dissipa ion a e which is difficul o se in a gene al flow case. To de elop a obus code, o ob ain a s able solu ion and o keep he pa icle numbe in a easonable ange we a oid he edis ibu ion p ocedu e in ou compu a ions. Thus, he smalles o ices a e emo ed. This educes he ange o scales ha mus be esol ed in a nume ical calcula ion. Such a educ ion, as poin ed ou by [3], is an immanen pa o e e y u bulence model. The e is a pe manen exchange be ween he small o ices and la ge scale ones ep e- sen ed on he g id. La ge scale o ices become small due o s e ching and a e con e ed o pa icles. I pa icles g ow due o iscosi y and flow s agna ion and exceed some size hey a e mapped back o he g id. The simple Eule me hod is used o he in eg a ion o he diffe en ial equa ions. The flowcha o he whole algo i hm is p esen ed in [11]. The p esen me hod has he same e o sou ces as e e y LES model [6]. Two commen s should be made on he fil e ing e o s. Fi s , he applican s unde s and ha he models elying on he small scales compa able wi h cell sizes can suffe om he fil e aliasing e o s inhe en ly p esen ed in each nume ical me hod. Fo ins ance, such e o s mos s ongly affec dynamic ype models which ely hea ily on he smalles scales o de e mine SGS p ope ies [6].Second, since ou algo i hm does no use commu a ion o diffe encing and fil e ing ope a o s which is he big difficul y in LES o malism and ep esen s he second pa o he fil e ing e o s, he commu a ion e o is no p esen . 5 178 S. Sama bakhsh and N. Ko ne 3 RESULT AND DISCUSSION 3.1 Summa y o p e ious esul Valida ion and e ifica ion is pe o med o wall ee flows including decaying iso opic u bulence (DIT) [4] and ee u bulen je [8, 2] es cases. Resul s e ealed ha he effec o he e m u ×ωgin he equa ion (7) is simila o ha o a LES subg id model. The e m u ×ωgbeha es as an ene gy d ain ans e ing he ene gy o he g id based mo ion in o he fine scale ene gy. A coa se esolu ions, i ac s as a diffusi e La ge Eddy Simula ion subg id model esul ing in a LES-like beha io o he whole me hod. The addi ional e m u ×ωgis au oma ically swi ched off when he esolu ion in- c eases, he p esen me hod is consis en and con e ges o he Di ec Nume ical Simula- ion. The ene gy back sca e ing is also cap u ed by he p esen me hod. As men ioned in [16], he in ensifica ion o he u bulen kine ic ene gy due o back sca e ing is p o ed o be e y impo an o p ope ly ep oduce he je b eakdown and ansi ion o u bulence close o he nozzle wi hou any a ificial u bulence o cing a he nozzle. The Reynolds s esses o he eloci y field induced by pa icles possess he p onounced aniso opy which is space dependen . I was also shown ha he model o he je case can be sufficien ly educed by neglec ing he inne in e ac ion be ween pa icles. This esul s in a d as ic educ ion o he compu a ional ime. Some addi ional esul on he model educ ion and aniso opy Reynolds s esses a e gi en in he nex sec ion. The esul a e ob ain o he ee je a he Reynolds numbe Re = 104. Mo e de ails abou g id p ope ies and se up o he simula ion a e ho oughly desc ibed in [16]. 3.2 Model educ ion As shown in [11] he equa ions 7 and 8 can sufficien ly be educed by neglec ing inne in e ac ion be ween o ex pa icles wi hou a significan loss o he simula ion accu acy. The educed equa ions ake he o m: ∂ω ∂ +(ug·∇)ωg=(ω ·∇)ug+νΔω ,(17) and he .h.s o he ajec o y equa ion (18) con ains only he g id based eloci y d i d =ug i(18) As seen in [11], he influence o inne in e ac ions on spa ially a e aged kine ic ene gy and scala dissipa ion a e is ela i ely weak and can be neglec ed in he calcula ion. The eby he compu a ions can be done sufficien ly as e .This model is u he e e ed o as he passi e o ices model. Wi hin he nex simplifica ion s ep he influence o he g id based solu ion on he e olu ion o o ex pa icles s eng hs is neglec ed. The equa ions desc ibing he o ex pa icle e olu ion ake he simples o m: ∂ω ∂ +(ug·∇)ωg=νΔω ,(19) 6 179 S. Sama bakhsh and N. Ko ne (a) influence o igno ing inne in e ac ion be ween o ex pa icles (b) influence o conside ing o ex pa icles only in he cu e n ime s ep Figu e 1: Influence o he in e ac ion be ween o ex pa icles and g id on he e olu ion o u ms d i d =ug i(20) Figu e 1-a demons a es esul s o he .m.s. o he axial eloci y ob ained using he ull model equa ions 7 and 8, passi e o ices model equa ions 17 and 18 and he model wi hou influence o he g id based flow on o ex pa icles s eng hs equa ions 19 and 20. The diffe ence be ween esul s is negligible poin ing ou ha o ex pa icles se e jus as igge s o in ensifie s o u bulence and hei inne in e ac ion doesn’ con ibu e sufficien ly o he flow e olu ion. Hence he name o he me hod is he LES in ensified by he o ex pa icles o VπLES . In he nex s ep o model educ ion only he influence o o ex pa icles gene a ed in he cu en ime s ep a e conside ed and o ex pa icles gene a ed in he p e ious ime s ep we e mapped back o he g id. In his case he ime o compu a ions is less han ull model since we only deal wi h a o ex pa icles in he cu en ime s ep. As can be seen in Figu e 1-b he esul shows ha he o ex pa icles igge he u bulence and ha e a good ag eemen wi h expe imen in he nea je exi egion while in a field egion decay o kine ic ene gy is no physical. I can be in e p e ed in his way ha he ene gy d ain o fine scale mo ion om la ge scale mo ion is no high enough and accumula ion o kine ic ene gy on he g id flow mo ion happens. Concluding, no only new gene a ed o ices bu also he whole se o o ices including hose gene a ed ups eam in p e ious ime s eps ha e a significan influence on he u bulence de elopmen downs eam. Wi h o he wo ds, he fine scale o ices model (7) and (8) can be educes bu no neglec ed. 3.3 Aniso opy o fine scale s uc u es induced by o ex pa icles Since a de e minis ic p edic ion o a u bulen flow as men ioned in he defini ion o he u bulen mo ion by Lesieu [13] is p ac ically impossible, he ask o e e y SGS model is o ep oduce he subg id mo ion only in he s a is ical sense. The ollowing ea u es o he 7 180 S. Sama bakhsh and N. Ko ne (a) o al eloci y (ug+u )(b) fine scale eloci y u Figu e 2: Dis ibu ion o he diagonal Reynolds s esses componen s o he o al and fine scale eloci y along he je axis subg id mo ion should be cap u ed by a p ope subg id model: non-equilib ium effec s including lamina - u bulen zone, ene gy backsca e and aniso opy o fine scale mo ion. In his subsec ion he aniso opy o eloci ies induced by fine o ices is discussed. The o al flow shows a well p onounced aniso opy wi h he dominance o he axial fluc ua ions on he je cen e line (see Fig. 2-a). Reynolds s esses R ii o he eloci y field u shows also a clea aniso opy which is space-dependen . On he cen e line a x/D > 5 wo diagonal s esses a e equal o each o he R 22 ≈R 33 and domina e o e R 11(see Fig. 2-b). To explain his effec , we conside s ochas ic dis ibu ion o he s a is ically independen axis-symme ic o ex pa icles on he cen e line wi h s eng hs aligned wi h he x-axis. They induce eloci ies u x= 0 and u y�= 0, u z�= 0. Due o axis symme ical cha ac e o each o ex he spa ially a e aged squa es o eloci ies u yand u za e equal, i.e. (u y)2= (u z)2. P ecession o o ices a ound hei spins causes he appea ance o he longi udinal eloci ies u xwhich a e much smalle han u yand u z. Thus, he je axis a ea a x/D > 5 is popula ed by o ex pa icles wi h axes p edominan ly o ien ed along he je p opaga ion o mean flow di ec ion. A x/D < 5 on he cen e line and /D =0.25 he fine scale u bulence is nea ly iso opic R 11 ≈R 22 ≈R 33 in he beginning o he je de elopmen (see Fig. 3-a), i.e. his a ea is popula ed wi h o ex pa icles wi h o ien a ions uni o mly dis ibu ed a ound a sphe e. Fu he downs eam he same aniso opy akes place as ha on he je axis. A he je bounda y he fine scale u bulence becomes aniso opic wi h a clea dominance o he adial fluc ua ions R 22 >R 11 ≈R 33 (see Fig. 3-b), i.e. he domina ing fluc ua ions a e in he di ec ion o he domina ing la ge scale en ainmen mo ion. Figu es 4 shows he p.d. o o ex axes o ien a ion whe e Lx, Lyand Lza e defined as Lx =αx |α|,Ly =αy |α|,Lz =αz |α|. 4 CONCLUSION The pape p esen s alida ion and e ifica ion s udy o a no el VπLES me hod which is based on he decomposi ion o he flow s uc u es in la ge scale ones, esol ed on he g id, 8 181 S. Sama bakhsh and N. Ko ne (a) (b) Figu e 3: Dis ibu ion o he diagonal Reynolds s esses componen s o he fine scale eloci y u along he line /D =0.25 and /D =0.5 (a) (b) Figu e 4: p.d. o o ex axes o ien a ion line /D =0 9 182