Numerical Solution of Non-equilibrium Hypersonic Flows of Diatomic Gases Using the Generalized Boltzmann Equation

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1 Recet Advaces Flud Mechacs, Heat & Mass asfe ad Bology Numecal Soluto of No-equlbum Hypesoc Flows of Datomc Gases Usg the Geealzed Boltzma Equato RAMESH K. AGARWAL Depatmet of Mechacal Egeeg ad Mateals Scece Washgto Uvesty St. Lous Bookgs Dve, St. Lous, MO 633 USA Abstact: - he pape descbes the methodology fo computg hypesoc o-equlbum shock wave flows of datomc gases usg the Geealzed Boltzma Equato (GBE) cludg both the vbatoal - taslatoal (V) ad otatoal - taslatoal (R) elaxatos. Fo the V elaxato, GBE s always solved. Fo the R elaxato, two appoaches ae employed. I the fst appoach, fo the R elaxato GBE s solved. hs appoach s computatoally vey tesve sce t eques solvg the complete GBE fo both vbatoal ad otatoal degees of feedom. I the secod appoach, a two-level BGK type model of R elaxato s employed. he secod appoach s much moe effcet tha the fst (about tmes faste). he pape descbes the two-level R elaxato model. he model s valdated by computg the shock stuctue at hgh Mach umbes by compag the esults wth the complete GBE soluto. Computatos ae the pefomed fo the shock stuctue accoutg fo both the vbatoal ad otatoal exctatos, usg the secod appoach fo the R elaxatos. Key-Wods: - : Wag-Chag Uhlebeck Equato, Hypesoc Flows, Rotatoal ad Vbatoal Relaxatos Itoducto Fo calculatg the o-equlbum flow of datomc gases, cuetly DSMC method s the most wdely used techque. I DSMC appoach, oe usually employs some empcal models [], whch vey much esemble a BGK type elaxato model. I ths pape, we descbe a computatoal methodology fo computg hypesoc oequlbum shock wave (SW) flows of datomc gases such as Ntoge ad Oxyge usg the Geealzed Boltzma Equato (GBE). I the GBE (smla to Wag-Chag Uhlebeck equato (WC-UE) []), the teal ad taslatoal degees of feedom ae cosdeed the famewok of quatum ad classcal mechacs espectvely. he GBE [3] ca be cosdeed as a moe geeal fom of WC-UE ad s also applcable whe the eegy levels ae degeeated. he computatoal famewok avalable fo the classcal Boltzma equato s exteded by cludg both the otatoal ad vbatoal degees of feedom the GBE [4]. he whole poblem that cludes both the vbatoal - taslatoal (V) ad otatoal - taslatoal (R) eegy tasfes s solved by applyg a thee-stage splttg pocedue to the GBE. he thee stages cosst of fee molecula taspot, V elaxato, ad R elaxato. Fo the V elaxato, GBE s always solved. Fo the R elaxato, two appoaches ae employed. I the fst appoach, fo the R elaxato GBE s solved. hs appoach s computatoally vey tesve sce t eques solvg the complete GBE fo both vbatoal ad otatoal degees of feedom. I the secod appoach, a two-level BGK type model of R elaxato that equlbates otatoal ad taslatoal eeges s employed. he secod appoach s computatoally less tesve tha the fst ad theefoe s much moe effcet (about tmes faste tha the fst appoach). he pape descbes the two-level R elaxato model wth a sgle elaxato tme. he model s valdated by computg the shock stuctue at hgh Mach umbes by compag the esults wth the complete GBE soluto fo R elaxatos. Computatos ae the pefomed fo the shock stuctue at hgh Mach umbes accoutg fo both the vbatoal ad otatoal exctatos; the secod appoach s employed fo computg the R elaxatos. echcal Appoach he shock wave stuctue datomc gases s studed usg the GBE [5] that eplaces the classcal Wag Chag Uhlebeck Equato (WC-UE) fo the case whe eegy levels ae degeeated. I ts mathematcal fom, the GBE s close to the WC-UE ad dffes fom the late by a facto elated to the ISBN:

2 Recet Advaces Flud Mechacs, Heat & Mass asfe ad Bology statstcal weghts of the levels. It should be oted that the GBE gves the equlbum spectum dffeet fom WC-UE. he GBE has the fom: f t f x R () he collso opeato s gve by: b m ( k l j j) j j j R f f f f P gbdbdd () Hee f s the dstbuto fucto fo the eegy level, P s the pobablty of the tasfe fom j levels, j to the levels,, ad the facto j ( qq k l )/( qq j ) ; q beg the eegy level degeeato. Fo smple levels, the GBE chages to the WC-UE, theefoe t ca also be cosdeed as a moe geeal fom of the late oe. I [3], the GBE equato was obtaed dectly fom WC-UE by goupg the q sgle levels that have the same eegy ad fom a degeeated level. We cosde molecula Ntoge havg the Lead- Joes potetal (6, ) wth the depth of the eegy hole 9K, degeeato of otatoal level q,,,.., ad the otatoal eegy of the level e ( ),.9K. he molecula teacto dug the collso cossts of two phases. I the fst phase, the molecules teact a elastc mae accodg to the molecula potetal. hs stage detemes the devato agle of the elatve velocty. I the secod stage, the modulus of the elatve velocty chages accodg to the eegy cosevato equato. Fo the tasto pobabltes P we apply the fomulae j gve [3] that ae obtaed by fttg the expemetal data of molecula dyamcs smulatos of teactos of gd otos that model N molecules. Pj P j [ exp( 3 4) exp( 3 4)] whee e e / e, t e e / etot 3 4 e /( et e ), 4 t 4 e /( e e ) e e ek, e ej el,.4 etot / et j, etot et e ej. et mg /4 he eegy cosevato law a collso selects vtual collsos wth o zeo pobablty. Fom the equato mg /4 e e mg /4e e, t ca be show that g, othewse P. j j k l Pj, f he elastc collso s a patcula case of the collso. he pobabltes obey the omalzato codto: P. he ketc equato () s k, l j solved by the splttg scheme. At a tme step τ < τ, whee τ s a mea te-collso tme, the equato () s eplaced by the sequece of equatos: f f (a) t x (b) f t R he collso opeato R s evaluated at the ufom gd S the velocty space by the cosevatve pojecto method poposed [4]. 3 wo-level Ketc Model fo R Relaxato a Gas he poposed model equato s amed at smplfyg the smulato of the otatoaltaslatoal (R) eegy exchage a gas. Such smplfcato s hghly eeded fo complex pocesses whch otatoal exctato s accompaed by the vbato - taslatoal (V) eegy tasfe. he model cossts of levels: the goud level wth the otatoal eegy ad the excted level wth some eegy max, whee max s the maxmum tempeatue the poblem ude cosdeato. We call the poposed model as LR model. he dstbuto fucto s also composed of two pats, f ad f wth coespodg populatos of the levels ad. he gas desty s ad the otatoal eegy s Eot. Let the desty of the gas at some pot be, the ketc eegy E k, ad the otatoal eegy E. Oe ca the deteme the ot j ISBN:

3 Recet Advaces Flud Mechacs, Heat & Mass asfe ad Bology populatos of the levels by the smple fomulas Eot / ad. Maxmal value of E ot s gve by Eot max, theefoe max /, ad oe obtas ad. Havg E k, oe ca deteme the equlbum tempeatue, ( E E ) / 5 eq k ot ad the equlbum otatoal populatos, eq eq /,, eq, eq. hese paametes deteme the equlbum dstbuto fuctos f,m ad f,m. Fo costucto of the model equato we beg wth the Wag Chag Uhlebeck equato (WC-UE) fo the cosdeed levels system ({, j, k}, ). (3) f / t p ( f f f f ) g bdbd d,, j k l j, j j j,, I equato (3), we eplace the collso opeato by a elastc collso opeato Qel ad the o-elastc opeato Q. hs eplacemet ca ot be stctly justfed, as t s assumed [5], because fo R exchage the puely elastc collsos peset a excepto, ad about all the collsos ae accompaed wth the tasfe of elatvely small pat of ketc eegy to/fom the otatoal eegy. O the othe had, because of small elastcty of teactos the ma collso elaxato pocess s close to the case of elastc collsos, except that oe should take to accout the elastc tasfe of the eegy. he elastc opeato s the same as the Boltzma collso tegal fo a two-compoet gas mxtue: Q ( f ' f ' f f ) g bdbdd (4) el, j j, j j j he o-elastc opeato s take a elaxato fom: Q ( f f ) (5), M, It was foud by a umbe of umecal expemets that the choce fo f M, equato (5) as the Maxwella dstbuto fuctos f M, s possble, but s ot the best. he fucto f M, epesets the ellptc dstbuto defed by the dagoal elemets of the tempeatue teso. m 3/ / fm, eq, ( ) ( yy zz) exp( mc x/ k mc y/ kyy mc z/ kzz ) k (6) whee cx x u, cy y v, cz z w, ad uvw,, ae the compoets of the bulk velocty vecto. he compoets aa of the tempeatue teso ae defed by self-smla tasfomato of the tal compoets as ( / ) (7) aa aa eq k he use of the fucto gve equato (6), stead of the Maxwella, meas that the elastc opeato Q peseves to some extet the shape of the dstbuto fucto the velocty space. he R elaxato fequecy ca be defed as a pat of the elaxato fequecy of the BGK model equato av. (8) he o-elastc opeato cotbutes to the evoluto of the velocty dstbuto fucto towad the equlbum state. o take to accout ts fluece oe should dmsh the elastc collso opeato by a facto ( a ), a. Fally, the poposed R elaxato model cotas two opeatos, the elastc opeato gve by equato (5) wth the fequecy gve by the equato (8), ad the elastc opeato Q, el ( a ) Q. he coeffcets, el a ad a ca be detemed fom compasos of the solutos of the poposed model wth solutos of the WC-UE equato. Note that some addtoal fttg paametes ca also be appled the model to obta good ageemet. 4 Results o Computato of Shock Wave (SW) Stuctue he esults o the computato of shock stuctue ae peseted thee pats: (a) the computatos ae peseted fo shock stuctue Ntoge cludg the otatoal degees of feedom up to Mach 5 by solvg the complete GBE wth a umbe of eegy levels, (b) the computatos ae peseted fo SW Ntoge cludg the otatoal degees of feedom usg the two-level ketc model ad ae compaed wth the soluto obtaed pat (a), ad (c) the computatos ae peseted fo SW ISBN:

4 Recet Advaces Flud Mechacs, Heat & Mass asfe ad Bology Ntoge wth both otatoal ad vbatoal degees of feedom by solvg the complete GBE fo V elaxatos ad employg the two-level ketc model fo R elaxatos. (a) he SW stuctue s fomed as a fal stage of the evoluto of dscotuty the tal dstbuto fucto. he poblem s cosdeed fo the teval L x L wth the dscotuty at x. he tal dstbuto fucto o both sdes of the dscotuty s descbed by the veloctes ad spectal levels. It has the fom,,,, 3/ m( u ) e f (, x) [ m/( )] exp[ ] exp( ),, Q whee Q deotes the statstcal sum. Paametes, (,, u ) ae defed by the Rake-Hugoot elatos wth 7/5. At the bouday, the tal dstbuto fucto s kept costat. Fo modeate Mach umbes, the SW stuctue Ntoge ca be computed wth eal value of the spectal eegy gap, but fo the hypesoc case the eeded umbe of levels becomes too hgh (up to 5-7 levels) ad theefoe the computatos become vey tme cosumg. o facltate ths poblem t s possble to cease ths eegy gap ad theeby educe the umbe of levels. By coductg umecal expemets at Mach umbe M = 4, we detemed that ths cease spectal eegy gap does ot fluece the esults of calculatos ay sgfcat way. Fgue shows a example of hypesoc SW stuctue Ntoge at Mach 5, computed wth 6 levels stead of ealy 7 atual levels. hese types of calculatos compae well wth the expemets of Alsmeye [6] fo desty fo epoted Mach umbes up to. Fgue shows the compaso fo desty dstbuto at Mach betwee the computato ad expemetal data; excellet ageemet s obtaed. (b) I pat (a), the shock stuctue Ntoge was computed fo R elaxato usg the complete collso tegal GBE wth a umbe of eegy levels. I ths pat, we show the SW calculatos usg a two-level ketc model fo R elaxatos. We choose the fxed coeffcets a ad a such a way that the soluto of the LR model s close to the soluto of the GBE fo lage umbe of Mach umbes Fg : Shock Wave Stuctue Ntoge fo M=5; = desty, = ketc temp, = logtudal temp, = otatoal temp (omalzed) , expemet t Fg. : Shock Wave Stuctue Ntoge fo M=; = computed desty,, expemet = expemet desty, t = ketc temp, = logtudal temp, = otatoal temp (omalzed). he compasos betwee the two set of solutos have bee made at M =.4 ad M =, ad also fo seveal temedate Mach umbes. he solutos of the GBE equato show pat (a) wee obtaed fo Lead-Joes molecula teacto potetal ad eal otatoal spectum. Fo LR model, we have appled the (-) vese powe epulsve potetal whch gves ν < ( k ).33. Afte a umbe of tests, the followg best fttg paametes wee detemed as a.36, a.9. he compaso of solutos of the LR model wth the soluto of GBE fo M =.4 ae gve Fgues 3 ad 4, ad fo M = Fgues 5 ad 6. he solutos of GBE ISBN:

5 Recet Advaces Flud Mechacs, Heat & Mass asfe ad Bology wee computed by the method descbed [6]. he gaphs fo flow vaables usg LR model ae maked by stas Fg. 3: Nomalzed Flow Vaables at M = Fg. 4: Nomalzed Desty at M =.4. (c) I ths pat we cosde the gas wth both otatoal ad vbatoal degees of feedom. he teacto potetal s aga the fom of the Lead-Joes potetal wth paametes of Ntoge take at oom tempeatue of 3K. he vbatoal spectum coespods to Ntoge wth eegy gap of 334K. Fo otatoal spectum the eegy quatum was ceased agast the actual (whch s equal to.9k) fo educg the umbe of the levels as descbed befoe the Itoducto ad pat (a). he pobabltes of R- tasfes ae take fom the data of mathematcal smulato by Fg. 5: Nomalzed Flow Vaables at M = Fg. 6: Nomalzed Desty at M =. Beylch [3]. he pobabltes of V- tasfes ae geeally accepted to be much smalle tha those of the R- tasfes ad ae the age of.5. of the elastc collso pobablty. As metoed above the techcal appoach, V tasfes ae computed usg GBE ad the R tasfe ae computed usg the two-level ketc model to smplfy the computatos. Computatos ae pefomed at Mach 6 ad. I computato of the shock stuctue, we calculate the vbato eegy as j j m vb vb j j E j. Assumg that the vbatos possess two degees of feedom, oe ca assocate the vbato eegy wth the classc ISBN:

6 Recet Advaces Flud Mechacs, Heat & Mass asfe ad Bology vbatoal tempeatue, E / k. hs vb cl tempeatue measues the amout of eegy stoed the vbatos. Fom the vewpot of quatum mechacs, the vbatos fom the Bose gas whch the tempeatue the themodyamc equlbum s elated to the vbatoal eegy by vb the fomula Evb [7]. I the exp( vb / kvb, q) classcal lmt, oe obtas vb, q vb, cl. I Fgues 7 ad 8, we peset both vbato tempeatues alog wth the desty, ketc taslatoal, logtudal taslatoal ad otatoal tempeatues. It should be oted that the quatum vbatoal tempeatue eaches the themodyamc equlbum value, but the classc vbatoal tempeatue s below ths lmt. At M = 6, the vbatoal spectum cossts of 4 levels whch ae suffcet to cove the ete eegy age. At M =, the umbe of eeded levels s eght. I Fgues 7 ad 8, the pat of the x axes s show. he computatos wee made fo x/λ = (-, 5) vb k vb,cl vb,q Fg. 7: SW Stuctue Ntoge at M = 6. 5 Coclusos A umecal methodology has bee developed fo computg hypesoc o-equlbum shock wave flows of datomc gases usg the Geealzed Boltzma Equato (GBE) cludg both the vbatoal - taslatoal (V) ad otatoal - taslatoal (R) elaxatos. I ode to speed up the calculatos fo R elaxatos, a two-level BGK type model has also bee developed whch has bee valdated by compag ts esults wth that of complete GBE. he method ca be easly exteded to compute -D ad 3-D o-equlbum hypesoc flows of datomc gases k vb,cl vb,q Fg. 8: SW Stuctue Ntoge at M =. Refeeces: [] E.S. Oa, C.K. Oh ad B.Z. Cybyk, Dect Smulato Mote Calo: Recet Advaces ad Applcatos Aual. Rev. Flud Mech., Vol. XXX, 998, pp [] F.G. Cheemsse, Soluto of the Wag Chag Uhlebeck Maste Equato, Doady Physcs, Vol. XXXXVII,, pp [3] A.A. Beylch, A Itelaced System fo Ntoge Gas, Poc. of CECAM Wokshop, ENS de Lyo, Face,. [4] F.G. cheemsse, Soluto of the Boltzma Ketc Equato fo Hgh Speed Flows, Computatoal Mathematcs ad Mathematcal Physcs, Vol. XXXVI, 6, pp [5].F. Mose, Ketc Model fo Gases wth Iteal Degees of Feedom, Phys. Fluds, Vol. VI, 964, p.59. [6] H. Alsmeye, Desty pofles Ago ad Ntoge Shock Waves Measued by the Absopto of a Electo Beam, J. Flud Mech.,Vol. LXX IV, 976, pp [7] L.D. Ladau ad E.M. Lfshtz, heoetcal Physcs: Statstcal Physcs, Nauka-Fzmathlt, Moscow, 995. ISBN: