Numerical simulation of a binary gas flow inside a rotating cylinder

Transcription

1 Journal of Mechancal Scence and Technology 3 (9) 848~86 Journal of Mechancal Scence and Technology wwwsprngerlnkcom/content/ x DOI 7/s6-8-- Numercal smulaton of a bnary gas flow nsde a rotatng cylnder A A Ganae and S S Nourazar * Mechancal Engneerng Department, Amrkabr Unversty of Technology, Tehran, Iran (Manuscrpt Receved December, 7; Revsed August, 8; Accepted December 4, 8) Abstract A new approach to calculate the axally symmetrc bnary gas flow s proposed Dalton s law for partal pressures contrbuted by each speces of a bnary gas mxture (argon and helum) s ncorporated nto numercal smulaton of rarefed axally symmetrc flow nsde a rotatng cylnder by usng the tme relaxed Monte-Carlo (TRMC) scheme and the drect smulaton Monte-Carlo (DSMC) method The results of flow smulatons are compared wth the analytcal soluton and results obtaned by Brd [] The results of the flow smulatons show better agreement than the results obtaned by Brd [] n comparson wth the analytcal solutons However, the results of the flow smulatons usng the TRMC scheme show better agreement than those obtaned usng the DSMC method n comparson wth the analytcal solutons Keywords: Boltzmann equaton; TRMC; DSMC; Rotatng cylnder Introducton Ths paper was recommended for publcaton n revsed form by Assocate Edtor Jun Sang Park * Correspondng author Tel: , Fax: E-mal address: cp@autacr KSME & Sprnger 9 In gas flow problems where the length scale of the system s comparable to the mean free path of molecules n the gas flow the concept of the contnuum s no more vald, Knudsen number greater than [] In ths case the smulaton s done by usng the drect smulaton Monte-Carlo (DSMC) or the collsonal Boltzmann equaton (CBE) methods [, 3] In most cases the drect soluton of the CBE s mpractcable due to the huge number of molecules; however, the mplementaton of the DSMC s often more practcable So far a class of Monte-Carlo method has been used to smulate the rarefed gas dynamc problems The rarefed hypersonc flow s solved by usng the DSMC method by Brd [] The comparson between the Naver-Stokes and the DSMC methods for the smulaton of the crcumnuclear coma was done by Crfo et al [4] The smulaton of the rarefed gas flow through crcular tube of fnte length n the transtonal regme at both low Knudsen number and hgh Knudsen number s done by usng the DSMC method by Shnagawa et al [5] More recently, Botton [6] used the molecular approach based upon a Monte- Carlo smulaton for sodum vapor flow of monoatomc molecules n lqud metal fast breeder reactor bundle Smulaton of flows lke recrculaton flow problems or near contnuum flows s stll expensve due to the partcular nature of the DSMC method [7] However, Paresch et al [8] proposed a modfcaton to the DSMC method to crcumvent ths problem Paresch et al [9] used the tme-relaxed Monte-Carlo (TRMC) method wth the Wld sum expanson [] to approxmate the non-negatve functon descrbng the tme evoluton of the dstrbuton functon of partcles However, the optmal choce of the coeffcents n the Wld sum expanson for the dstrbuton functon of partcles s left as an open problem Paresch et al [, ] smulated numercally the Boltzmann equaton for a two-dmensonal gas dynamc flow around an obstacle usng the TRMC method Ther smulaton showed mprovement over the DSMC method n terms of computatonal effcency Nourazar et al [3] compared the smulaton of the

2 A A Ganae and S S Nourazar / Journal of Mechancal Scence and Technology 3 (9) 848~ Naver-Stokes and the Boltzmann equatons of the axally symmetrc compressble flow past a flat-nosed cylnder at hgh velocty and low pressure wth shock wave usng the Monte-Carlo method The results of the two smulatons are compared n terms of dfferent Knudsen numbers Purpose of the present work In the smulaton of rarefed gas dynamc problems, when the Knudsen number s small, collsons occur at a fast rate; therefore, a knetc treatment (DSMC) of the problem s extremely expensve due to the requred small tme-step Snce the rato of tme scales between macroscopc and mcroscopc effects s large enough, reachng the statonary results of flow characterstcs may almost be mpossble On the other hand, the TRMC schemes allow the use of larger tme-steps than those requred n the DSMC method, therefore allowng one to acheve the statonary results of flow propertes n a comparatvely shorter computatonal tme To our knowledge, none of the researchers so far tred to ncorporate the new dea of usng the Dalton s law for the smulaton of axally symmetrc bnary gas flow mplementng the TRMC scheme In the present work we ntend to smulate the flow propertes of a bnary gas mxture (argon and helum) nsde a rotatng cylnder usng the TRMC scheme and the DSMC method The results of the smulatons usng the two methods are compared wth the analytcal solutons and the results obtaned by Brd, [] for the same case study problem In the smulaton usng the DSMC we follow exactly the same procedures descrbed n Brd, [], and n the smulaton usng the TRMC we follow exactly the same procedures descrbed n Paresch et al [] Descrpton of the case study problem A cylnder rotates at a tangental velocty of meter/second, and the radus of the cylnder s meter The gas mxture nsde the cylnder contans 5% argon and 5% helum, the ntal temperature of the gas mxture nsde the rotatng cylnder s Kelvn, and the ntal pressure of the gas mxture nsde the cylnder s 76 Pascal absolute These data are the same as those depcted by Brd [] Mathematcal formulatons The Boltzmann Equaton The Boltzmann equaton for the temporal evoluton of partcles velocty dstrbuton functon for speces p s wrtten as []: t ( npfp) Vp ( npfp) s q = r = Q n f ( p p, nqfq) () In Eq (), n s the number densty of molecules of speces, f s the probablty dstrbuton functon of the molecules of speces havng velocty V, f s the probablty dstrbuton functon of the molecules of speces havng velocty V, s Knudsen number whch s equal to the rato of λ, the mean free path between collsons, to the characterstc length L The subscrpts p and q represent the partcular speces The blnear collsonal operator Qn ( p fp, nf q q) that descrbes the bnary collsons of the molecules s gven by: ( p p, nqfq) Q n f s 4π q= = (, ) * * nn p q fp fq fpfq σ Vrpq Ω dωdvq () Where, V r s the relatve velocty between the pq molecules of speces p and speces q and Ω s a vector of the untary sphere The kernel σ s a non negatve functon whch s descrbed as []: ( r, pq ) α ( θ) σ V Ω = b V α (3) rpq Where, θ s the scatterng angle between V r pq and Vr Ω The varable hard sphere (VHS) [] pq model s often used n numercal smulaton of rarefed gases, where, bα ( θ ) = C wth C a postve constant and α = The value of C s equal to [], C = σ pq Substtutng Eq (3) nto Eq () one gets the followng: Q( npfp, nqfq) = s 4π * * nn p qσ pqvrpq fp fq fpfq dωdvq q= (4)

3 85 A A Ganae and S S Nourazar / Journal of Mechancal Scence and Technology 3 (9) 848~868 Substtutng the blnear collsonal operator (Eq (4)) nto Eq (), the Boltzmann equaton s wrtten as: ( npfp) Vp ( npfp) t r = Q( npfp, nqfq) s 4π * * = nn p qσ pqv rpq fp fq fpf q dωdvq q= (5) In Eq (5), n s the number densty of molecules of speces, f s the probablty dstrbuton functon of the molecules of speces havng velocty V, f s the dstrbuton functon of the molecules of speces * havng velocty V, f s the post-collson probablty dstrbuton functon of the molecules of speces * havng velocty V and f s the post-collson probablty dstrbuton functon of the molecules of speces havng velocty V The subscrpts p and q represent the partcular speces The TRMC Scheme In a bnary gas mxture flow wth two dfferent speces p and q, the Boltzmann equaton can be wrtten for the speces separately ( npfp) Vp ( npfp) t r = Q( npfp, npfp) Q( npfp, nqfq) (6) In the present work the Dalton law s used to calculate the pressure; the pressure of the mxture s calculated as P = Pp Pq, where P p s the partal pressure of the speces p, and P q s the partal pressure of the speces q The densty of the mxture s calculated as ρm = yarρar yheρhe, where ρ m s the mxture densty, y Ar s the molar fracton of argon, ρ Ar s the densty of argon, y He s the molar fracton of helum and ρ He s the densty of helum Frst, we assume that only the argon gas exsts nsde the rotatng cylnder and the flow s smulated by usng the TRMC and the DSMC methods Second, we assume that only the helum gas exsts nsde the rotatng cylnder and the flow s smulated by usng the TRMC and the DSMC methods In both cases the effect of collsons of molecules of argon and the effect of collsons of molecules of helum are consd- Q n f, n f = ered separately, ( p p q q) t Q n f = r ( npfp) Vp ( npfp) ( p p, npfp) (7) We splt Eq (7) [4] nto an equaton for the effect of collson, ( n f ) r, and an equaton for p p the effect of convecton, Qnf ( p p, nf p p) = The equaton for the effect of collson s wrtten as []: t = ( npfp) Q( npfp, npfp) (8) For smplcty we omt the ndex and replace np fp wth f, where f s the mass densty functon of the molecules of speces havng velocty V Then Eq (8) s wrtten as: t ( f ) = Q( f f ),, and the collson term s wrtten as: 4π * * Q( f, f ) = σv r f f ff dωdv 4π 4π * * σvf r fd dv f σvfd r dv = Ω Ω = P( f, f ) µ ( ) f V Where, 4π 4π VfdΩ r Vd r fd µ ( V ρ ) = σ V = σ Ω V = κ m (9) () s the mean collson frequency for the molecules havng velocty V, ρ s the densty of the gas, m s the mass of a molecule of the gas, κ s a molecular 4π constant κ = σvd r Ω and ρ = fd m V (Wld, 95) In a specal case n whch σ Vr s ndependent of V (Maxwellan molecules), we have: r κρ µ ( V ) = µ = () m Substtutng Eqs () and () nto Eq (9): f = Q( f, f ) = P( f, f ) µ f t ()

4 A A Ganae and S S Nourazar / Journal of Mechancal Scence and Technology 3 (9) 848~ The frst-order tme dscretzaton of Eq () s wrtten as: f f n n n f n n n = P( f, f ) µ f t n n µ t P( f, f n µ t ) = f µ (3) The probablstc nterpretaton of Eq (3) s the followng In order a partcle s sampled from f n, a partcle s sampled from f n, wth probablty of ( µ t ) and a partcle s sampled from ( n n P f, f ) µ, wth probablty of µ t However, the above probablstc nterpretaton fals f the rato of µ t s too large because the coeffcent of n f on the rght hand sde may become negatve Ths means that n the stff regon (where the Knudsen number s small << ) the tme step becomes extremely small; therefore, the method becomes almost unstable near the flud regme [5, 6] To crcumvent the problem, a new ndependent varable and a new dependent varable F ( V, ) are defned as: t ( e µ ) (, ) (, ) = (4) t F V = f V t e µ (5) Therefore, Eq () s rewrtten as: F = P( F, F) µ F = = ( V, ) f ( V,) (6) Eq (6) s the Cauchy problem, and t has a power seres soluton as follows [4]: Expandng the summaton terms: F 3 = f f 3 f3 4 f4 P F F P f f f f f f (, ) = (, ) Substtutng Eq (9) nto Eq (6): 3 f f 3 f3 4 f4 = P f f f f f f µ P( f, f) P( f, f) P( f, f) = P( f, f) P( f, f) µ P( f, f) P( f, f) P( f, f) = µ P( f, f) P( f, f) P( f, f) P( f, f) = µ { P( f, f) P( f, f) } (, ) (9) () Equatng the coeffcents of correspondng powers of, we can fnd for f as: f = P( f, f) µ f = P( f, f) µ f3 = P( f, f) P( f, f) µ 3 3 () Therefore, the functon f s found by the recurrence formula as: F = P F F µ F (, ) ( V, ) f ( V,) = = (7) k f = P f f k µ (, ) k h k h h= k =,, () Substtutng Eq (7) nto Eq (6): F = = k k k fk ( k ) fk k= k= k k P( F, F) = P fk, fk k= k= (8) Convertng Eq (7) nto orgnal varables we obtan the followng formula representng the soluton to the Cauchy problem 9: k (3) k= µ t µ t k (, ) ( ) ( ) f vt = e e f v A class of numercal schemes based on a sutable

5 85 A A Ganae and S S Nourazar / Journal of Mechancal Scence and Technology 3 (9) 848~868 truncaton for m of Eq (3) s derved [4]: m µ t µ t k f ( vt, ) e ( e ) fk ( v) k= µ t m ( e ) M( v) n Where f f ( n t) = (4) = and t s tme step The quantty M n Eq (4) s the asymptotc soluton of the equaton and called Maxwellan Eq (4) can be generalzed by usng dfferent weght functons ncludng the nfluence of the hgher order coeffcents In general the TRMC scheme s wrtten as: m n f Af k k Am M k= = (5) the weght functons ( ) k Where, the functons f k are gven by Eq () and A are non-negatve functons that satsfy the consstency, conservatve and the asymptotc preservaton condton (Gabetta et al, [4]) In ths smulaton the frst order TRMC of Eq n n (5) as f = Af Af AM s used 3 Analytcal solutons The energy E of a partcle n an axally symmetrc gas mxture flow nsde a rotatng cylnder s gven as [7], E( r) = Iω = mr ω (6) The rotatonal effect s the same as addtonal external feld actng on the system and may be wrtten as: U( r) = mr ω (7) Usng the Boltzmann dstrbuton for the partcle number densty and substtutng for U( r ) from Eq (7): ( ) mr ω U r n( r) = Aexp = Aexp (8) kt kt Where the normalzaton factor A can be determned by N = n( r) dv gvng: ( ) R π U r N = Aexp rdrdθ kt R π mr ω = Aexp rdrdθ, kt R mr ω N = π A exp rdr = kt R mr ω π A exp d( r ) kt R kt mr ω = π A exp m ω kt π AkT mr ω = exp mω kt Then for A can be wrtten: (9) (3) mr A Nmω ω = π kt exp (3) kt Therefore: mω r exp Nmω kt n( r) = π ktl mω R exp kt (3) Where, N s the total number of molecules, m s the mass of a molecule of gas, ω s the angular velocty, k s the Boltzmann constant, T s the absolute temperature, L s the length of the cylnder, R s the cylnder radus and r s the radal dstance 4 Boundary and ntal condtons 4 Boundary condtons At the boundary where the axs of symmetry exsts the rule of specular reflecton s consdered The specular reflecton rule s mplemented for molecules at the sold surface boundary wth normal velocty to the sold boundary beng reversed and those wth parallel velocty to the sold surface boundary remanng unchanged At the top and the bottom boundary of the cylnder the rule of specular reflecton and at the wall of the cylnder the rule of dffuson reflecton are consdered The dffuse reflecton rule s mplemented for molecules at the sold surface boundary wth a velocty component equal to the tangental

6 A A Ganae and S S Nourazar / Journal of Mechancal Scence and Technology 3 (9) 848~ velocty at the cylnder wall equal to meter/second and the other component of the velocty s equal to the most probable velocty accordng to the equlbrum Maxwellan dstrbuton Therefore, the resultant velocty s equal to the velocty that was obtaned n accordance wth the knetc theory of gases In the dffuse reflecton rule the temperature of the reflected molecule and the temperature of the sold wall boundary are requred to be equal, and the veloctes of the reflected molecules are dstrbuted accordng to equlbrum Maxwellan dstrbuton 4 Intal condtons The ntal values of the gas mxture pressure and the temperature are 76 Pascal absolute and Kelvn, respectvely The molar concentratons of the gas mxture nsde the cylnder consst of 5% argon and 5% helum The ntal veloctes of the molecules are obtaned based on the knetc theory 5 The Numercal procedures In the smulaton of the problem usng the DSMC method, two grd systems are chosen The frst grd system (Fg ) s used to calculate the averagng of flow propertes Ths grd system s chosen to be fne enough n order to ncrease our computatonal accuracy The grd system s refned up to where the varatons of the flow propertes are not substantal (the varatons of the flow propertes less than %) The second grd system s chosen to be very fne (the mesh sze s equal to tmes the mean free path of the molecules); therefore, the collsons of the molecules are controlled wthn each mesh accurately Our grd system conssts of 5 tmes meshes and the total number of model molecules s 357 The number of real molecules s obtaned based on the gas densty and Avogadro s number where each model molecule conssts of * 5 real molecules In the TRMC scheme the same grd system (5 tmes ) and the same number of model molecules, 357, are used The number of real molecules s obtaned based on the densty of gas and Avogadro s number For the axally symmetrc flows, the dstrbuton of the modeled molecules s lnearly proportonal to the radal dstance The volume of the mesh far from the axs of symmetry s much larger than the volume of the mesh close to the axs of symmetry Therefore, the number of the modeled molecules far from the axs of symmetry s much greater than the number of modeled molecules close to the axs of symmetry Ths leads to the unformty of densty n the radal drecton The sze of the mesh s on the order of the mean free path, and the tme step n our smulaton s chosen to be tmes the mean collson tme Brd [] The molecules are dstrbuted n the mesh system accordng to the normal dstrbuton The ntal velocty of the molecules s chosen based on the knetc theory of gases The drecton of velocty of the molecules s chosen randomly based on equlbrum Maxwellan dstrbuton We then start to advance wth tme and the new poston of the molecules s desgnated The collsons of the model molecules are done based on the varable hard sphere (VHS) model proposed by Brd [] We then contnue advancng n tme untl the statstcal fluctuatons of the flow propertes are mnmum In the smulaton of gas mxture (argon and helum) the flow nsde the hgh speed rotatng cylnder the gas mxture pressure nsde the cylnder s ntally 76 Pascal absolute, the radus of the cylnder s meter, the tangental velocty at the cylnder wall s meter/second, the gas mxture nsde the cylnder conssts of 5% argon and 5% helum, and the temperature of the gas mxture nsde the cylnder s Kelvn The total number of 357 model molecules s consdered to be dstrbuted lnearly along the radal coordnates The tmestep of the smulaton s chosen to be -6 second and the smulaton s done untl 57 seconds of real tme (8 teratons) n DSMC method In the TRMC scheme the tme-step of the smulaton s chosen to be 5* -6 second and the smulaton s done untl 57 seconds of real tme (56 teratons) The calculatons are performed by an IBM compatble personal computer wth 8 GHz CPU and 5 MB RAM 5 Algorthm of the Drect Smulaton Monte Carlo Method (DSMC) Fg The structure of mesh system for the cylnder of radus meter and heght of meter DSMC Algorthm (for the VHS collson model molecules) T = 73K ref

7 854 A A Ganae and S S Nourazar / Journal of Mechancal Scence and Technology 3 (9) 848~868 d p = 33, for He ref dq = 47, for Ar ref m r = m p m q m p m q 3 k = J K 6 t = Sec Dstrbute the ntal locatons of the partcles accordng to the unform dstrbuton for n t = to n tot Gven{ v n, =,, N} Defne the local Knudsen number ( ) Calculate dpq = ( dref ) pq Γ Compute an upper bound { kt ( ) ( ref m pq r v v )} (( π 4 ) d ( )) v v pq ( 5 γpq ) γ pq σ = max for the cross secton, σ s updated n each collson Set µ = 4πσ Set Nc = Iround( µ N t/ ( )) Select N c dummy collson pars (, ) unformly among all possble pars, and for those Compute the relatve cross-secton σ = ( π 4) d pq v v Generate unform random numbers (Rand) f Rand < σ σ - Perform the collson between and, and * compute the post-collson veloctes v and * v accordng to the collsonal law - Generate two unform random numbers ξ, ξ - Set α = cos ( ξ ), β = πξ - Set φ = ( cosβsnα snβ snα cos α) T * v = ( v v) v v φ, - Set * v = ( v v) v v φ n * - Set n * v, v else n n - Set n n v, v n n - Set v for the N Nc partcles that have not been selected End for Calculate macroscopc propertes: macroscopc flow velocty: v = ( mnv mnv p p p q q q), densty: ρ ρ = nm, thermal Velocty: v v, temperature: T = ( n m v n m v p p p q q q ), Tr 3kn pressure : P nkttr ( n m v n m v p p p q q q ) = = 3 Durng each step, all the other N Nc partcle veloctes reman unchanged Here, by Iround(x), we denote a sutable nteger roundng of a postve real number x In our algorthm, we choose: [ x] wth probablty [x] - x Iround(x) = [ x] wth probablty x - [x] 5 Algorthm of the Tme Relaxed Monte Carlo Method (TRMC) TRMC Algorthm(frst order TRMC scheme for the VHS collson model molecules) T = 73K ref dp = 33, for He ref dq = 47, for Ar ref m r = m p m q m p m q 3 k = J K 6 t = Sec Dstrbute the ntal locatons of the partcles accordng to the unform dstrbuton for n t = to n tot Gven{ v, n =,, N} Defne the local Knudsen number ( ) Calculate dpq = ( dref ) pq Γ Compute an upper bound { kt ( ) ( ref m pq r v v )} ( 5 γpq ) (( π 4 ) d ( )) v v pq γ pq σ = max for the cross secton, σ s updated n each collson Set = exp ( ρσ t / ) Compute A( ) =, A( ) = Set Nc = Iround( NA /) Select Nc dummy collson pars (, ) unformly among all possble pars Compute the relatve cross-secton σ = ( π 4) d pq v v Generate unform random numbers (Rand) f Rand < σ σ - Perform the collson between and, and

8 A A Ganae and S S Nourazar / Journal of Mechancal Scence and Technology 3 (9) 848~ (a) (b) (c) (d) Fg Number of molecules dependency test: (a) Densty, (b) Number densty, (c) Temperature, (d) Swrl velocty compute the post-collson veloctes * v and * v accordng to the collsonal law - Generate two unform random numbers ξ, ξ α = cos ξ, β = πξ - Set ( ) - Set φ = ( cosβ snα snβsnα cos α) T * v = ( v v ) v v - Set * v = ( v v ) v v n * - Set n * v, v else n n - Set n n v, v φ, φ n n - Set v for the N Nc partcles that have not been selected - Set NM = Iround( NA ) - Select N M partcles among those that have not collded, and compute ther mean momentum and energy - Sample N M partcles from the Maxwellan wth the above momentum and energy, and replace the N M selected partcles wth the sampled ones n n Set v for all the N N c N M partcles that have not been selected End for

9 856 A A Ganae and S S Nourazar / Journal of Mechancal Scence and Technology 3 (9) 848~868 Calculate macroscopc propertes: macroscopc flow velocty: v = ( mnv mnv p p p q q q), ρ densty: ρ = nm, thermal Velocty: v v, temperature: T = ( n m v n m v p p p q q q ), Tr 3kn pressure : P = nkt = Tr ( nmv nmv p p p q q q ) 3 6 The Number of model molecules dependency test The number of model molecules dependency test s done by usng more model molecules n each mesh The smulaton s performed by usng 357 model molecules n the grd system; however, n the number of model molecules dependency test the number of model molecules s ncreased to 57 n our grd system The results of smulaton of flow characterstcs for the two cases (357 model molecules and 57 model molecules n the grd system) at 4 seconds of real tme of flow smulaton are obtaned and compared wth each other Fgs (a), (b), (c) and (d) show the number of molecules dependency test for densty, number densty, temperature and swrl velocty, respectvely Fg (a) shows the number of model molecules dependency test for the mxture densty The maxmum dscrepancy between the results of the smulaton of the number of model molecules dependency test for the mxture densty for the two cases (357 model molecules and 57 model molecules n the grd system) s less than 4% at 65 meter of radal dstance nsde the cylnder Fg (b) shows the number of model molecules dependency test for number denstes of argon and helum The maxmum dscrepancy between the results of smulaton of number of model molecules dependency test for the number densty of argon and helum for the two cases (357 model molecules and 57 model molecules n the grd system) s less than 73% at 65 meter of radal dstance nsde the cylnder and less than 35% at meter of the radal dstance nsde the cylnder, respectvely Fg (c) shows the number of model molecules dependency test for the temperature The maxmum dscrepancy between the results of smulaton of the number of model molecules dependency test for the temperature for the two cases (357 model molecules and 57 model molecules n the grd system) s less than 66% at 75 meter of radal dstance nsde the cylnder Fg d shows the number of model molecules dependency test for the swrl velocty The maxmum dscrepancy between the results of smulaton of number of model molecules dependency test for the swrl velocty for the two cases (357 model molecules and 57 model molecules n the grd system) s less than 437% at 65 meter of radal dstance nsde the cylnder 7 Dscusson of results Fg 3(a) shows the comparson of the results of smulatons usng the DSMC method and the TRMC scheme wth the results of Brd, [] for the varatons of the swrl velocty along the radal coordnates Comparson of our results of smulaton usng the DSMC method for the swrl velocty wth the results of Brd [] shows good agreement at second of real tme ( teratons); however, comparsons of our results of smulatons usng the DSMC method at 57 seconds of real tme (8 teratons) and the results of smulaton usng the TRMC scheme at 57 seconds of real tme (56 teratons) wth the results of Brd, [] show hgh dscrepances The dscrepances are due to the lack of suffcent real tme (number of teratons) n the calculaton of the Brd [] smulaton In that smulaton the DSMC method s used; however, larger tme-steps are not allowed []; therefore, one s not able to reach the statonary flow smulaton Comparsons of our results of smulaton usng the DSMC method at 57 seconds of real tme (8 teratons) for the swrl velocty wth the results of smulaton usng the TRMC scheme at 57 seconds of real tme (56 teratons) show good agreement Fg 3(b) shows the tme evoluton of the varatons of swrl velocty along the radal coordnates for dfferent teratons ( to 8 teratons) Fg 4(a) shows the comparson of the results of smulatons usng the DSMC method and the TRMC scheme wth the results of Brd [] for the varatons of the gas mxture (argon and helum) temperature along the radal coordnates Comparson of the results of smulaton usng the DSMC method for the gas mxture (argon and helum) temperature wth the results of Brd, [] shows good agreement at second of real tme ( teratons) However, comparsons of our results of smulatons usng the

10 A A Ganae and S S Nourazar / Journal of Mechancal Scence and Technology 3 (9) 848~ (a) (b) Fg 3 Swrl velocty: (a) Comparson between the results of smulaton usng the DSMC method, the TRMC scheme and the results of Brd, 994 for the varatons of the swrl velocty along the radal coordnates, (b) Tme evoluton of the varatons of swrl velocty along the radal coordnates from the 5 second of real tme (5 teratons) up to 57 seconds of real tme (8 teratons) (a) (b) Fg 4 Temperature: (a) Comparson between the results of smulaton usng the DSMC method, the TRMC scheme and the results of Brd, 994 for the varatons of the temperature along the radal coordnates, (b) Tme evoluton of the varatons of temperature along the radal coordnates from the 5 second of real tme (5 teratons) up to 57 seconds of real tme (8 teratons) DSMC method at 57 seconds of real tme (8 teratons) and the results of smulaton usng the TRMC scheme at 57 seconds of real tme (56 teratons) wth the results of Brd, [] show hgh dscrepances The dscrepances are due to the lack of suffcent real tme of computng (number of teratons) n the Brd, [] calculatons Ths s because n the Brd calculatons usng the DSMC method one s unable to choose larger tme steps; however, n the smulaton usng the TRMC scheme one s allowed to choose larger tme-steps (see []), therefore reachng larger real tme for calculatons Comparsons of our results of smulaton usng the DSMC method at 57 seconds of real tme (8 teratons) for the gas mxture (argon and helum) temperature wth the results of smulaton usng the TRMC scheme at 57 seconds of real tme (56 teratons) show good agreement Fg 4(b) shows the tme evoluton of the varatons of the gas mxture (argon and helum) temperature along the radal coordnates for dfferent

11 858 A A Ganae and S S Nourazar / Journal of Mechancal Scence and Technology 3 (9) 848~868 (a) Fg 5 Comparson between the results of smula-tons usng the DSMC method, the TRMC scheme for the varatons of the gas mxture (argon and helum) pressure along the radal coordnates teratons ( to 8 teratons) Fg 5 shows the comparson of the results of smulatons usng the DSMC method wth the TRMC scheme for the varatons of the gas mxture (argon and helum) pressure along the radal coordnates Fg 5 shows that at 57 seconds of real tme (8 teratons) the values of pressure around the wall of the cylnder are approxmately 8 tmes bgger than the values of pressure around the center of the cylnder (b) 7 Comparsons of number densty results wth the analytcal soluton The analytcal soluton of Kuo [7] s only used here as a reference to make the comparson of the results of the present smulaton, snce the soluton of Kuo [7] s based on the assumpton that the flow s n equlbrum Here n the present smulaton one assumes that the results of our smulaton reached the equlbrum condton as well However, ths assumpton s not completely vald Therefore, the analytcal soluton of Kuo [7] can only be used as a reference and not as an absolute source to udge the accuracy of the TRMC scheme Fgs 6(a), 6(b) and 6(c) show the comparsons between the DSMC method, the TRMC scheme and the results of Brd [] wth the analytcal soluton for the varatons of the densty of gas mxture, the number densty of helum and the number densty of argon along the radal coordnates, respectvely Comparson of our results of smulaton for the densty of the gas mxture [Fg 6(a)] wth the analytcal soluton (c) Fg 6 He number densty, Ar number densty and Densty: (a) Comparson between the DSMC scheme, the TRMC scheme and the results of Brd 994 wth the analytcal soluton for the varatons of the number densty of helum along the radal coordnates, (b) Comparson between the DSMC scheme, the TRMC scheme and the results of Brd 994 wth the analytcal soluton for the varatons of the number densty of Argon along the radal coordnates, (c) Comparson between the DSMC scheme, the TRMC scheme and the results of Brd 994 wth the analytcal soluton for the varatons of the densty along the radal coordnates

12 A A Ganae and S S Nourazar / Journal of Mechancal Scence and Technology 3 (9) 848~ shows reasonable agreement, and the agreement s pronounced for the radal dstance of r > 5m ; however, the comparson of our results of smulaton usng the DSMC and the TRMC wth the analytcal soluton shows better agreement than the results of Brd, [] [Fg 6(a)] Comparson of our results of smulaton for the number densty of helum [Fg 6(b)] wth the analytcal soluton shows reasonable agreement, and the agreement s pronounced for a radal dstance of r > 5m However, comparson of the results of smulaton of Brd, [] for the number densty of helum [Fg 6(b)] wth the analytcal soluton shows hgh dscrepances The dscrepances of the Brd results are due to the lack of suffcent real tme n the smulaton usng the DSMC method In the smulaton usng the DSMC method, one s not allowed to choose larger tme-steps; therefore, reachng the larger real tme s very dffcult However, n the smulaton usng the TRMC scheme, one s allowed to choose larger tme-steps n order to reach larger real tme Comparson of our results of smulaton for the number densty of argon [Fg 6(c)] wth the analytcal soluton shows better agreement than the comparson of the results of smulaton of Brd [] for the number densty of argon [Fg 6(c)] wth the analytcal soluton 8 Conclusons The comparson of the results of smulaton usng the DSMC method wth the results of smulaton usng the TRMC scheme for the swrl velocty and the temperature show good agreement However, the comparsons of the results for the swrl velocty and the temperature of Brd [] wth the analytcal soluton show hgh dscrepances The comparsons of the results of smulatons usng the DSMC method and the TRMC scheme wth the analytcal soluton for the densty, number densty of helum and number densty of argon show good agreement However, the comparsons of the results for the densty, number densty of Helum and number densty of argon of Brd [] show hgh dscrepances The conclusons are summarzed as follows: Due to the requred small tme-steps n the DSMC smulatons, the dscrepances of the results usng the DSMC method are pronounced n comparson wth the results of smulatons usng the TRMC scheme The comparsons of the results of smulatons usng the TRMC scheme for the densty and the number densty wth the analytcal soluton show better agreement than those obtaned by DSMC method Havng larger tme-steps n the smulaton usng the TRMC scheme allows one to reach statonary results for the flow characterstcs n shorter tme; therefore, the results of smulatons usng the TRMC scheme show mprovement over the results of smulatons usng the DSMC method In the present smulaton, there are two sources of approxmaton errors: the approxmaton errors nherent n the selecton of larger tme-step used n the TRMC scheme, and the approxmaton errors due to modelng the molecular collson, the VHS model, used n the present work These two sources of approxmaton errors nteract wth each other The nature of the nteracton of the two sources of approxmaton errors s very complcated However, the results of our smulaton n the present work show that the approxmaton errors nherent n the selecton of larger tme-steps n the TRMC scheme counteract the other sources of approxmaton errors nherent n the smulaton modelng, such as modelng of molecular collsons, the VHS model, used n the present work Therefore the results of smulaton, usng the TRMC scheme, show mprovement over the results of smulaton usng the DSMC method Acknowledgment The authors would lke to thank Mr S Navard and Mr S M Hossen for ther helpful scentfc dscussons durng preparaton of ths work References [] G A Brd, Molecular Gas Dynamcs and the Drect Smulaton of Gas Flows, Oxford Unv Press, London, (994) [] R W Hockney and J W Eastwood, Computer smulaton usng partcles, McGraw Hll Internatonal Book Co (98) [3] K Nanbu, Theoretcal bass of the drect smulaton Monte Carlo method, In Boff, V and Cercgnan, C edtors, Proceedng of the 5 th Internatonal Symposum on Rarefed Gas Dynamcs, (986), Pages [4] J E Crfo, G A Lukanov, A V Rodonov, G O Khanlarov and V V Zakharov, Comparson between Naver-Stokes and Drect Mote-Carlo smulaton of the crcumnuclear coma, Icarus 56, (),

13 86 A A Ganae and S S Nourazar / Journal of Mechancal Scence and Technology 3 (9) 848~ [5] H Shnagawa, H Setyawan, T Asa, Y Sugyama and K Okuyama, An expermental and theoretcal nvestgaton of rarfed gas flow through crcular tube of fnte length, Pergamon, Chemcal Engneerng Scence, 56, (), [6] M Botton, Molecular Approach to Sodum Vapor Condensaton, Rewettng and Vaporzaton n LMFBR Bundle under Hypothetcal Accdent Condtons, J Nuclear Scence and Technology, 4 (5), (4), pp [7] R S Myong, A generalzed hydrodynamc computatonal model for rarfed and mcro scale gas flows, Journal of Computatonal Physcs 95, (4), [8] L Paresch and R E Calfsch, An Implct Monte- Carlo Method for Rarefed Gas Dynamcs, J Comput Phys 54, (999), 9 [9] L Paresch and G Russo, Tme Relaxed Monte- Carlo Methods for the Boltzmann Equaton, SIAM J Sc Comput 3, (), [] E Wld, On Boltzmann s Equaton n the Knetc Theory of Gases, Proc Camb Phl Soc, 47, (95), 6-69 [] L Paresch, Trazz, S, Asymptotc Preservng Monte Carlo Methods for the Boltzmann Equaton, Transport Theory Statst Phys, 9, (5), pp45-43 [] L Paresch, S Trazz, Numercal Soluton of the Boltzmann Equaton by Tme Relaxed Monte-Carlo (TRMC) Method, Internatonal Journal of Numercal Method n Fluds 48, (5), [3] S S Nourazar, S M Hossen, A Ramezan and H R Dehghanpour, Comparson between the Naver-Stokes and the Boltzmann equatons for the smulaton of an axally symmetrc compressble flow wth shock wave usng the Monte-Carlo method, Computatonal Methods and Expermental Measurements XII, WIT Transacton on Modelng and Smulaton, 4, 6-69, (5), WIT Press [4] E Gabetta, L Paresch and G Toscan, Relaxaton Schemes for Nonlnear Knetc Equatons, SIAM J Number Anal, 34, (997), pp [5] K Nanbu, Drect Smulaton Scheme Derved From the Boltzmann Equaton, Journal of the Physcal Socety of Japan, 49, (98), pp 4-49 [6] H Babovsky, On a Smulaton Scheme for the Boltzmann Equaton, Mathematcal Method n the Appled Scences, 8, (986), pp 3-33 [7] L Y Kuo, Problems and solutons on Thermodynamcs and Statstcal Mechancs, World Scentfc Publcaton (99) Seyed Salman Nourazar receved a BS degree n Mechancal Engneerng from Amrkabr Unversty of Technology n 974 He then went on to receve hs MS and PhD degrees from Concorda and Ottawa n 978 and 993, respectvely Dr Nourazar s currently an Assocate Professor at the Department of Mechancal Engneerng n Amrkabr Unversty of Technology, Tehran, Iran Dr Nourazar s research nterests are n the area of Computatonal Flud Dynamcs, Advanced Mathematcs, Flud Mechancs and Rarefed Gas Dynamcs Amr Abbas Ganae receved a BS degree n Mechancal Engneerng from Scence & Technology of Iran n 99 He then went on to receve hs MS degrees from Amrkabr n 995 Ganae s research nterests are n the area of Computatonal Flud Dynamcs, Flud Mechancs and Rarefed Gas Dynamcs He started PhD courses at 3 Now he s gong to receve hs PhD degree tll July 9