Lattice Boltzmann flux solver for simulation of hypersonic flows

La ice Bol zmann lux sol e o simula ion o iscous hype sonic lows V In e na ional Con e ence on Pa icle-based Me hods – Fundamen als and Applica ions PARTICLES 2017 P. W igge s, M. Bischo , E. Oña e, D.R.J. Owen, & T. Zohdi (Eds) LATTICE BOLTZMANN FLUX SOLVER FOR SIMULATION OF HYPERSONIC FLOWS Z. X. MENG¹, C. SHU², L. M. YANG3, W. H. ZHANG 4, F. HU5 AND S. Z. LI6 ¹ College o Ae ospace Science and Enginee ing, Na ional Uni e si y o De ense Technology Changsha, 410073, China Co esponding au ho : [email protected] ² Depa men o Mechanical Enginee ing, Na ional Uni e si y o Singapo e 10 Ken Ridge C escen , Singapo e 119260, Singapo e m[email p o ec ed] 3 Depa men o Mechanical Enginee ing, Na ional Uni e si y o Singapo e 10 Ken Ridge C escen , Singapo e 17576, Singapo e yang[email p o ec ed]m 4 College o Ae ospace Science and Enginee ing, Na ional Uni e si y o De ense Technology Changsha, 410073, China zhang[email p o ec ed] 5 College o Ae ospace Science and Enginee ing, Na ional Uni e si y o De ense Technology Changsha, 410073, China hu [email protected] 6 College o Ae ospace Science and Enginee ing, Na ional Uni e si y o De ense Technology Changsha, 410073, China [email protected] Key wo ds: Non- ee pa ame e D1Q4 model, la ice Bol zmann lux sol e , hype sonic lows, ini e olume me hod Abs ac : In his pape , a s able La ice Bol zmann Flux Sol e (LBFS) is p oposed o simula ion o hype sonic lows. In LBFS, he ini e olume me hod is applied o sol e he Na ie -S okes equa ions. One-dimensional La ice Bol zmann model is applied o econs uc he in iscid lux ac oss he cell in e ace, while he iscous lux is sol ed by con en ional smoo h app oxima ion unc ion. The p esen wo k ex ends he exis ing LBFS o calcula e hype sonic low ield on he leewa d, which is ha d o ge con e gen esul s due o ex emely low p essu e e ec s in his a ea. Simula ion o a biconics model is s udied. I is disco e ed ha he ail a ea o double cone is ela ed o he maximum Mach numbe ha could be con e gen . The la ge he diame e o ail a ea is, he smalle Mach numbe could be con e gen . Hence, he low p essu e a ea behind double cone ail will ha e la ge e ec s du ing he LBFS simula ion o hype sonic low. Two measu emen s a e applied in his pape o o e come he low p essu e p oblem. The i s one is o apply a local block g id e inemen me hod based on he low condi ions o imp o ing he s abili y. The second is o add a cons ain pa ame e o elimina e nega i e alue and gi e ou a p ope one. Hence, LBFS is able o ge con e gen esul o he hype sonic low ield on bo h windwa d and leewa d. Se e al nume ical examples a e es ed o compa e he pe o mance o me hod p esen ed in his pape . Simula ion esul s show ha me hod p esen in his pape is able o calcula e hype sonic low ield on he leewa d wi h bo h ine accu a e and e icien . 203 Z. X. Meng, C. Shu, L. M. Yang, W. H. Zhang and S. Z. Li 2 1 INTRODUCTION The hype sonic low ield is a esea ch ocus wi h i s special cha ac e is ics [1]: (1) s ong shock e ec will cause ie ce comp ession a e he shock wa e, (2) iscosi y g ea ly changed om wall su ace o a ield a ea, (3) o hype sonic ehicles lying in high al i ude, con inuous medium assump ion is no longe app op ia e because o low densi y e ec . The compu a ional luid dynamics (CFD) is becoming mo e and mo e popula in simula ing hype sonic low ield. The ini e olume me hod (FVM) is widely used in exis ing nume ical me hods [2]. This la gely a ibu es o i s spa ial disc e iza ion is ca ied ou di ec ly in he physical space. Hence, s uc u al g id is sui able o disc e iza ion, which p o ides an e ec i e way o sol ing complex geome y p oblems. In he p ocess o sol ing N-S equa ions by FVM, lux sol e is he key o e alua e iscous and in icid luxes a cell in e ace. The iscous lux is e alua ed by applying a smoo h unc ion app oxima ion and in iscid lux is calcula ed by a ious upwind schemes such as Roe scheme [3], an Lee scheme [4] and AUSM (Ad ec ion Ups eam Spli ing Me hod) scheme [5]. Bol zmann equa ion-based me hod is an al e na i e app oach o simula ing comp essible low ield, including DVBE [6][8] (disc e e eloci y Bol zmann equa ion) and gas-kine ic scheme [9]-[13]. Gas-kine ic scheme ha e be e e iciency han DVBE me hod. Howe e , bo h o hem a e less e icien and mo e complica ed han con en ional Na ie - S ocks slo e s. La ice Bol zmann Flux Sol e (LBFS) is a mo e e icien me hod which is p oposed by Ji e al. [14] and o simula ing in iscid lows. LBFS is imp o ed by Shu and Yang e al. [15]- [19] and has been p o ed ca ching s ong shock wa e and expansion wa e e y well. In LBFS, Eule equa ions a e disc e ized by FVM and he in iscid lux a cell in e ace is econs uc ed by local solu ion o 1-D comp essible La ice Bol zmann model. LBFS is mo e e icien and easy o apply han DVBE and gas-kine ic me hod due o only 1-D La ice Bol zmann model is applied and mac oscopic go e ning equa ions a e sol ed. Howe e , because o 1-D model is applied along no mal di ec ion a cell in e ace, he angen ial e ec canno be p ope ly conside ed. Viscous lux should be aken in o accoun o sol e iscous p oblems. F om chapman-Enskog expansion analysis [19], he in iscid lux can be ully de e mined by equilib ium non-equilib ium dis ibu ion unc ion a cell in e ace, and he non-equilib ium pa can be ea ed as nume ical dissipa ion. Fo comp essible iscous lows, especially o hype sonic lows, he nume ical dissipa ion should be con olled. A swi ch unc ion is p oposed by Yang e al. [20] o weigh iscous nume ical dissipa ion and has been p o ed pe o med well in iscous simula ion. Fo hype sonic low ield, physical alues such as p essu e in leewa d side can be ex emely low. Hence i is e y easy o ge a nega i e alue in calcula ing p ocess, which make i e y di icul o ge a con e gen esul . In his wo k, a s able LBFS is p oposed o simula ion o 2-D comp essible hype sonic iscous lows. The in iscid lux is calcula ed by LBFS and iscous lux is compu ed by smoo h unc ion app oxima ion. Mul iblock g ids and local g id e inemen me hod a e applied o imp o e calcula ion s abili y. Also a cons ain pa ame e is added o elimina e nega i e alue and gi e ou a p ope one in calcula ing p ocess. Besides, he implici LU-SGS me hod is used o speed up he con e gence a e. In he end, a biconics model in Ma=9.86 low ield is simula ed o alida e he de eloped sol e . The esul s o sol e p esen ed in his wo k is compa ed wi h an Lee and Roe scheme o e alua e he p ecision p oposed in his a icle. 204 Z. X. Meng, C. Shu, L. M. Yang, W. H. Zhang and S. Z. Li 3 2 METHODOLOGY In LBFS, Na ie -S ockes equa ions a e sol ed in mac oscopic scale and local solu ion o La ice Bol zmann equa ions a e applied in o cons uc in iscid lux sol e a he cell in e ace. I has been p o ed ha pa icle po en ial ene gy is independen om la ice eloci y [15]. La ice eloci y would be decided by highe momen um conse a ion ela ion in his wo k ins ead o by a i icial selec ion in adi ional me hod. N-S equa ions o in e g al o m wi hou sou ce e m can be w i en as ollowing. ( )0 c d ds       W FF (1) In which he conse a i e a iables W , in iscid lux c F and iscous lux F a e gi en by Eq. (2) in 2-D low ield. u E           W , () n nx c ny n U uU n p U n p E pU               F , 0 x xx y xy x yx y yy xx yy nn nn nn              F (2) The exp essions o no mal eloci y n U and o al ene gy E a e shown as Eq. (3). Whe e / [( 1) ]ep    is he po en ial ene gy o mean low. 22 1() 2 nx y U nu n Ee u    (3) The in eg al o luxes in Eq. (1) can be disc e ized by FVM and he app oxima ed summa ion o m can be w i en as 1 1() N I ci i i i I dF FS d     W (4) In which I ep esen s he con ol olume index, I  is he olume and N is numbe o aces o con ol olume I. In his pape , he in iscid lux c F in Eq. (4) will be sol ed by LBFS wi h non- ee pa ame e D1Q4 model [18] and iscous lux F will be sol ed by cen al di e ence me hod. 2.1 Non- ee pa ame e D1Q4 la ice Bol zmann model The dis ibu ion o disc e e la ice eloci ies o D1Q4 model is shown in Fig. 1. This model con ains 4 equilib ium dis ibu ion unc ions 1 eq , 2 eq , 3 eq , 4 eq and 2 la ice eloci ies 1 d , 2 d (shown as Eq. (5)). The de i a ion p ocess de ails a e shown in e e ences [18] . 2 43 -d 1 -d 2 d 1 d 2 1 205 Z. X. Meng, C. Shu, L. M. Yang, W. H. Zhang and S. Z. Li 4 Figu e 1: Dis ibu ion o disc e e la ice eloci ies o D1Q4 model 2 2 2 23 2 12 2 1 1 122 11 2 2 2 2 23 2 12 2 1 1 222 11 2 2 2 2 23 2 12 1 2 1 322 21 2 2 2 2 23 2 12 1 2 2 422 21 2 2 1 ( 3) 2( ) ( 3) 2( ) ( 3) 2( ) ( 3) 2( ) eq eq eq eq d d d u d u d c u uc dd d d d d u d u d c u uc dd d d d d u d u d c u uc dd d d d d u d u d c u uc dd d du                             2 22 4 2 2 22 4 2 34 6 34 6 c uc c d u c uc c     (5) In which /c Dp   ep esen s pa icula eloci y o pa icles and D is space dimension. I has been p o ed by Yang [18] ha physical conse a ion laws (Eq. (6)) can e i i y he N-S equa ions by applying ela ions in Eq. (5). i  means pa icle eloci y in i-di ec ion, o example 1 12 13 2 ,,d dd     and 42 d    . Fo highe dimensional p oblems, he D1Q4 model should be applied along he no mal di ec ion o cell in e ace [18] as is shown in Fig. 2 o he 2D case. The no mal eloci y n U (Eq. (3)) and angen ial eloci y (,) xy n U uu U     un in Fig. 2 will eplace u in Eq. (6). Finally, we will ge nx x u Un u    . 4 1 4 1 4 22 1 4 32 1 3 eq i i eq ii i eq i ii i eq i iii i u uc u uc                     (6) in e ace U τ Un u Figu e 2: Applica ion o 1D model o 2D case By applying Eq. (6) o Eq. (2), we will ge he conse a ion a iable W and in iscid lux c F 206 Z. X. Meng, C. Shu, L. M. Yang, W. H. Zhang and S. Z. Li 5 o 2-D low ield as ollowing. 2 2 () () ( /2 ) /2 nx x ny y n Un u Un u U eU                  W , 2 2 2 2 () () ( ( /2 ) ) /2 n n x nx c n y ny nn U U pn Uu U pn Uu U e pU U                   F (7) 2.2 In iscid lux model Suppose ha cell in e ace is loca ed a , 1/2 0 cj x , hen he in iscid lux a in e ace c F is decided by no mal eloci y and can be w i en as Eq. (8). In which he momen s 2 1, , / 2 T a ii p e      φ and (0, ) i is he dis ibu ion unc ion a cell in e ace. Gene ally speaking. (0, ) i is summa ion o equilib ium pa (0, ) eq i and non equilib ium pa (0, ) neq i . 4 2 1 2 (0, ) ( ( /2 ) ) n c n i ai i nn U Up U e pU              Fφ (8) To eco e N-S equa ions by Bol zmann equa ion om Chapman-Enskog analysis [19][21][22][23][24], he non-equilib ium pa (0, ) neq i can be w i en as ollowing. (0, ) (0, ) neq ii ii x        (9) Then applying Taylo se ies expansion in ome and physics space, Eq. (9) can be simpli ied as ollowing.   (0, ) (0, ) ( , ) ( ) neq i i ii          (10) A he cell in e ace he equillib ium dis ibu ion unc ion is (0, ) (0, ) eq ii  . (,) ii    is equilib ium dis ibu ion unc ion a su ouding poin o he cell in e ace. Subs i u ing Eq. (9) in o Eq. (8) as ollowing.   0 (0, ) (0, ) (0, ) ( , ) ( ) eq i i i ii           (11) In which 0/    is he dimensionless collision ime and  d is he s eaming ime s ep and ep esen s he physical iscous o N-S equa ions. The con ibu ion o non-equlib ium pa is always ea ed as nume ical dissipa ion in LBFS. The e o e, 0  can be ega ded as he weigh o nume ical dissipa ion. We ha e   0max , LR    in his wo k. In which     1, 1, max , max L R LR jj jN jN        and , anh LR jLR pp Cpp        . L N and R N a e he numbe o con ol olume on le and igh side o cell in e ace. The de ails can be ound in [25]. Subs i u ing Eq. (11) in o Eq. (8) we will ge in iscid lux c F which is decided by angen ial eloci y as ollowing. 207 Z. X. Meng, C. Shu, L. M. Yang, W. H. Zhang and S. Z. Li 6 44 4 0 11 1 0 (0, ) ( , ) (0, ) ) ci ai i ai i i ai ii i c cc                        Fφ φ φ F (F F Ⅰ Ⅱ Ⅰ (12) To al in isicid lux a cell in e ace conside ing angen ial eloci y con ibu ion is shown as ollowing. 0 ) cc c c     F F (F F Ⅰ Ⅱ Ⅰ (13) c  F Ⅰ ep esen s con ibu ion o equilib ium dis ibu ion unc ion (0, ) eq i a cell in e ace and c F Ⅱ means equilib ium dis ibu ion unc ion (,) ii    a su ouding poin o he cell in e ace. g 4L in e ace R L g 4R g 2Rg1L g 3L in e ace R L g 2L g 1L g 3L g2Rg1Rg3R g4R s eaming Figu e 3: S eaming p ocess based on D1Q4 model a cell in e ace Supposing ha a local Riemann p oblem is o med a cell in e ace, which is shown in Fig. 3. Hence, equilib ium dis ibu ion unc ion (,) ii    a su ouding poin o cell in e ace can be decided by he loca ion o i   , which is shown as Eq. (14). 0 (,) 0 L ii ii R ii            (14) Mo e speci ically, Eq. (14) can be w i en as ollowing o D1Q4 model. 1, 3 (,) 2,4 L i ii R i i i          (15) By applying he ela ionship o (0, ) (0, ) (0, ) eq neq ii i   , he conse a ion lux which is decided by angen ial eloci y can be w i en as ollowing. 44 11 2 (0, )+ (0, ) ( /2 ) Meq neq n ai ai ii n U Ue               Wφ φ (16) 208 Z. X. Meng, C. Shu, L. M. Yang, W. H. Zhang and S. Z. Li 7 The subsc ip M means pa ame e on cell in e ace. The non-equilib ium pa has no con ibu ion o calcula e conse a ion a iables acco ding o he compa ibili y condi ion [19], which is shown as ollowing.   44 0 11 (0, ) (0, ) ( , ) 0 neq ai a i i i ii           φ φ (17) Subs i u ing Eq.( 17) in o Eq. (16) we will ge 44 11 (0, ) ( , ) M ai ai i ii         Wφ φ (18) By applying Eq. (15) in o Eq. (18), he conse a ion lux which is decided by angen ial eloci y M W can be w i en as 1,3 2,4 MLR ai ai ii      Wφ φ (19) The e o e, densi y, no mal eloci y and p essu e a cell in e ace a e ob ained om Eq. (19). Then he angen ial eloci y a cell in e ace can be calcula ed by he ollowing equa ion. 1,3 2,4 M LL RR ii ii U U U         () (20) In which M U  , L U  and R U  a e angen ial eloci y a cell in e ace, on he le and igh side o cell in e ace espec i ely. Based on Eq. (19) and Eq. (20), all a iables a cell in e ace such as M  , M p M U  and M n U can be ob ained. Subs i u ing hese a iables o Eq. (5) and we can ge he equlib ium dis ibu ion unc ion (0, ) i a cell in e ace. The c  F Ⅰ in Eq. (13) including wo pa s, which a e c  F Ⅰ decided by no mal eloci y and he pa decided by angen ial eloci y. Then c  F Ⅰ can be w i en as ollowing by applying a iables a cell in e ace. 2 2 2 2 () () ( ( /2 ) ) /2 n n x nx c n y ny n nn U U pn Uu U pn Uu U e pU U U                   F Ⅰ (21) Also c F Ⅱ is consis s o c F Ⅱ decided by no mal eloci y and he pa decided by angen ial eloci y. Subs i u ing equilib ium dis ibu ion unc ion (,) ii    a su ouding poin o cell in e ace o Eq. (8) we will ge c F Ⅱ as ollowing. 1,3 2,4 LR c ii ii ii        F Ⅱ   (22) Then he angen ial eloci y a cell in e ace can be calcula ed by he ollowing equa ion. 209 Z. X. Meng, C. Shu, L. M. Yang, W. H. Zhang and S. Z. Li 8 1,3 2,4 2 22 1,3 2,4 M LL RR n ii ii ii M LL RR n ii ii ii UU U U UU U U                   () () (23) Now c FⅡ can be exp essed as ollowing. In which (,) i ii    . 4 1 44 11 44 11 44 2 11 11 () 22 ii i M iii x ii x ii c M iii y ii y ii M i ii p i ii ii n U n U e U                                      F Ⅱ (24) By subs i u ing Eq. (21) and Eq. (24) o Eq. (13) we ge in iscid lux c F on cell in e ace. 2.3 Viscous lux model In his wo k, cen al di e ence scheme is applied o sol e he isous lux F . S ess enso  and lux ec o  in Eq. (2) a e gi en by       2 =+ 3 T kT              u u uI u (25) In which  is dynamic iscosi y and decided by Su he land’s law and u bulence model. k is he mal conduc i i y and I is uni ma ix. Fo physical a iables on cell in e ace, such as andu, , w k , can be calcula ed by a i hme ic mean alue on le and igh side o cell in e ace, shown as ollowing. 1() 2 M LR     (26) Whe e  ep esen s any physical a iables. The de i a i es in Eq. (25) a e calcula ed by ini e di e ence scheme and de ils a e shwon in e e ence [26]. 2.4 Me hod o imp o e nume ical s abili y Mul iblock g ids a e applied o e ine local g id quali y. G ids local e inemen me hod is used nea wall su ace and block in e ace. Hence, scheme s abli y has been imp o ed by educing spa ial s ep, especially in he leewa d side o low ield. Besides, a cons ain judgemen is added in ou code o a oid nega i e alue in calcula ing p ocess. In his wo k, implici LU-SGS me hod [27] is applied o speed up he con e gence a e and keep calcula ion obus . 210 Z. X. Meng, C. Shu, L. M. Yang, W. H. Zhang and S. Z. Li 9 3 NUMERICAL SIMULATION Biconics model is used o e i y he p esen sol e . The shape o biconics model is shown as Fig. 4. The head cu a u e adius is 3.84mm, semi-cone angle is 12.84 deg ee in on and 7 deg ee in back, while he leng h o on cone is 69.55mm and 122.24mm o ally. The ee s eam has a p essu e o 59.92PaP   ， empe a u e o 48.88KT   and Mach numbe 9.86Ma   . Fig. 5 shows mach numbe dis ibu ion ob ained by p esen sol e in his a icle. I can be seen clea ly ha LBFS can cap u e s ong shock wa es and low inside he bounda y laye exac ly. The maximum Mach numbe is 16.93Ma. Fig. 6 shows he p essu e con ou s bibonics model wi hou ail low ield, and Fig. 7 is he one has ail ield. Fig. 8 and Fig. 9 show empe a u e dis ibu ions. One block g ids a e applied o low ield wi hou ail simula ion. I is easie o calcula ion con e gen wi hou low p essu e e ec in back ail low. Figu e 4: Measu emen s o biconics model Figu e 5: Mach numbe con ou s ob ained by LBFS a Ma=9.86 Figu e 6: P essu e con ou s ob ained by LBFS wi hou back ail low ield Figu e 7: P essu e con ou s ob ained by LBFS wi h back ail low ield 211