Numerical Solution of the Taylor-Quette Flow Problem: A Commodious Statistical Approach

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1 Physics Iteratioal (): 8-44, ISSN Sciece Publicatios Numerical Solutio of the Taylor-Quette Flow Problem: A Commodious Statistical Approach Ayoob Karimi, Hassa Zeytooia ad Shahi Navardi Departmet of Civil Egieerig, Islamic Azad Uiversity, Mariva Brach, Ira Departmet of Mechaical Egieerig, Islamic Azad Uiversity, Mariva Brach, Ira Departmet of Mechaical Egieerig, Texas Tech Uiversity, Labae, USA Abstract: Problem statemet: The mai problem i solvig of the Boltzma equatio by statistical methods is its log computatioal time (CPU time). The problem of decreasig of CPU time usage i statistical solutio methods of Boltzma equatio for rarefied vortex flows was studied. Approach: I this study the Boltzma equatio i a rarefied Taylor-Quette flow regime was solved usig the ew Mote Carlo method which is officially amed time Relaxated Mot Carlo method that applied the equilibrium coditios i each time step. Results: The results obtaied from time Relaxated Mot Carlo method for the problem at had were compared with those from the usual Direct Simulatio Mote Carlo method. This compariso showed good agreemet betwee the two sets of results. Coclusio: Whereas, the umber of collisios ad CPU usage time i the suggested method, as compared to the direct simulatio Mote Carlo method, showed a sigificat decrease. Key words: Rarefied gas, Taylor-Quette flow, DSMC method, TRMC method, CPU time usage INTRODUCTION The Taylor-Quette Flow is a flow betwee two coaxial cyliders, i which the outer cylider is fixed ad the ier oe rotates. Taylor (9) while studyig this flow oticed that at certai agular speeds of the ier cylider, vortex flows are created. Karma (94) aalyzed the Taylor-Quette flow i dese flow regimes ad Kuhltau studied the Taylor-Quette flow i rarefied gas flow regimes experimetally (Kuhlthau, 96). Sice the Navier-Stoces equatio ca ot describe rarefied gas flow regimes properly, it is essetial to solve the Boltzma equatio which is valid for ay flow regime (rarefied ad dese). Withi the scope that was developed by Kuhltau, Reichelma ad Nabu umerically solved the Boltzma equatio by the Direct Simulatio Mote Carlo method, ow as the DSMC method ad foud that the Results from the DSMC were i good coformity with the experimetal oes (Reichelma ad Nabu, 99). Stefaov ad Cercigai (99) used the DSMC method to study specially vortex flows betwee two coaxial cyliders. Followig that, Bird, by cosiderig more details, used the DSMC method ad could obtai, i a more exact fashio, the stable solutio for the Taylor-Quette flow (Bird, 994). The mai problem i all previous solutio methods of Taylor-Quette flow (Reichelma ad Nabu, 99; Stefaov ad Cercigai, 99; Bird, 994) is their too log CPU time usage. So usig of other appropriate method which leads to adequate result i less time for problem at had is eeded. O the other had, Carle et al. () solved the Boltzma equatio by usig a method which was compatible with the applicatio of the Maxwellia equilibrium coditios ad Pareschi ad Russo () itroduced the Time Relaxed Mote Carlo method that was proportioal to the method itroduced by Carle et al. (). Pareschi s method is abbreviated as TRMC, i the refereces. Pareschi aalyzed a outer flow problem with his ew method. Sice, Maxwellia equilibrium coditios applied i TRMC method decreases the calculatio time, TRMC method ca lead to appropriate results for the problem at had i smaller time. I (Zeytooia et al., 9), for the first time, the Taylor-Quette flow betwee two coaxial cyliders was solved by the TRMC method ad a compariso betwee some results with those obtaied by the Bird s DSMC method doe. I this study, the authors have completed their before study ad made a complete compariso betwee two methods. The complete compariso of the results from TRMC method with those obtaied by the DSMC Correspodig Author: Hassa Zeytooia, Departmet of Mechaical Egieerig, Islamic Azad Uiversity of Mariva, Mariva, Ira Tel: Fax:

2 Phy. Itl. (): 8-44, method, shows that obtaied results are i good agreemet with those of the DSMC ad that the calculatio time required to get stable results for the TRMC method, as compared to the same for the DSMC method, is sigificatly less. MATERIALS AND METHODS The appropriate mathematical model of the aalysis of rarefied gas dyamics is the Boltzma equatio: f + v. x f = Q f, f t x Ω R v R With iitial coditios: ( = ) = f x,v,t f x,v () I relatio (), f is the desity distributio fuctio (a o-egative fuctio) ad depeds o the positio x, the velocity v ad the time t. I Eq., is the Kudso umber which is proportioal to the mea free path betwee the collisios. Dual collisio operator, which appears as Q (f, f) i Eq. defies the collisio betwee two molecules i a mooatomic gas as: = ( * ) Q f, f v σ v v, ω R S ( ( ) ( ) ) f v f v f v f v dω. d v * * * () I the relatio above, ω is a vector o the uit sphere S R which shows the directio of the molecule s collisio. σ Kerel is a o-egative fuctio which specifies the details of iteractio betwee two molecules. Molecule s velocities after the collisio ( v,v * ) deped o the velocities before the collisio ( v, v * ) ad are defied as: v = v+ v + q ω v * = ( v+ v q ω) ( * ) () The relative velocity of two molecules with respect to each other is defied as q = v v* If f is the desity i the phase space, the; the macroscopic desity is defied as: 9 ρ = f d v (4) R Similarly, the macroscopic velocity is defied as: u = f v d v ρ (5) R ad the temperature ad eergy of the fluid ca be obtaied from: T = v u f d v ρ (6) R E v f d v = (7) R I the equilibrium state, where Q( f, f ) =, every local distributio fuctio has a Maxwellia distributio form as: ρ u v M ( ρ, u, T ) ( v ) = exp / ( π T) T (8) I relatio (8), T,u,ρ are the mea temperature, velocity ad desity of the gas, respectively. To solve the Boltzma equatio by Mot Carlo method, trasiet step (cosiderig Q = ) ad collisio step (cosiderig x f = ) will be solved separately. To begi the umerical solutio of each steps simplest discritizatio, the first order time discretizatio of the Boltzma equatio, is discussed. After discretizatio, the mai problem seems to appear i the way of solvig the collisio step. Therefore, it is discussed this issue i more details. So with cosiderig oly collisio step: f = Q f,f t (9) By usig the simple Forward Euler method for discretizatio of relatio (9) (which is used mostly for discretizatio) (Bird, 994), we will have: + t f = f + Q f,f () It ca be see that the essetial coditio for the umeric scheme stability is that t should be i the same order of amplitude as the mea free path. So i small value of, this discretizatio method is ot a

3 Phy. Itl. (): 8-44, proper oe. I other hads, i the implicit discretizatio method, stability coditio becomes less importat, but other problems associated with solvig the implicit collisio operator (that is, isufficiet data for the iitial coditios required for the solvig), maes this method, ot a proper oe, also. So, a appropriate solutio for a oliear equatio lie (9), which, i additio to havig ucoditioal stability, ca still remai valid ear the dese rage, ca be obtaied as follows. Cosiderig a variable chage such as P( f, f ) = Q( f, f ) + µ f Eq. 9 becomes (Carle et al., ): f = P ( f,f ) µ f t () Iitial coditio i () is similar to (), µ is a costat parameter ad P is a liear two-directioal operator. By chagig the variables ad defiig ew variables (Carle et al., ), we get: τ = µ t / ( ) e, ( τ ) = F v, f v, t e µ t / () I which τ is the Relaxed Time. Also Eq. chages to the simple form as follow: F = P F, F τ µ () With boudary coditios F( v, τ = ) = f ( v). Cosiderig Eq. as a Qushi equatio, it ca be solved by the followig power series: ( τ ) = τ F v, f v = = f v f v = (4) The (6) power series is ow as Wild extesio. Cosiderig the relatio P ( f, f ) = Q ( f, f ) + µ f i the Boltzma equatio, we have: = = (7) lim f v limf v, t M v t where, M( v ) is the local Maxwellia distributio fuctio. Cosiderig this case, the followig solutio, which is based o the Maxwellia equilibrium for m, ca be obtaied (Pareschi ad Russo, ): + µ t / µ t / f v e e f v = = ( ) µ t / ( e ) M( v) + (8) I (8), f = f ( t) ad t is a small time step. Cosiderig µ = 4 π σ ρ, the followig result will be derived (Pareschi ad Russo, ) (to have a o egative result, such assumptio is essetial): P( f, f ) = Q( f, f ) + 4 π σ ρ f (9) Here, σ is the maximum collisio sectio i the flow field ad is calculated accordig to (). I relatio (), σ is the collisio sectio betwee two specified molecules: σ = max σ ( vi- v j ) () v i,v j Cosiderig the result from (8), It should be oted that, this method ca be derived through differet weight fuctios which mae the best estimate for high order coefficiet i Wild extesio (Pareschi ad Russo, ); so, i the overall form: I which, f fuctios ca be calculated from the relatio (5): + h h + h= µ = = (5) f v P f, f,,... By substitutig the mai variables (the oes before the variable chage), we get the followig equatio which is the solutio of the Qushi Eq. 9 as: (6) = µ t / µ t / = ( ) f v, t e e f v 4 m + f = A f + A M = () f is calculated from (5) ad A = A ( t) is a oegative weight fuctio that should satisfy the followig coditios: Compatibility coditios: τ τ lim A τ / τ = lim A τ / τ = =,...,m + ()

4 Phy. Itl. (): 8-44, Coservatio coditio: = τ A =, [,] Threshold coditio: τ lim A τ = =,...,m () (4) Oe fuctio which satisfies the above coditios ad also is based o relatio (8) cocept is: A A = τ τ, =,...,m = τ (5) It should be oted that aother weight fuctio could be selected, but it must be checed to mae sure that it is a optimal choice (Pareschi ad Russo, ). Now with cosiderig the TRMC ad DSMC solutio methods (both of them are based o the Mote Carlo method), for solvig the previous equatios, At first, it is preseted a Direct Simulatio of the Mote Carlo method which correspods to Eq. ad is compatible with the Nabu-Babovsy algorithm. It should be poited out that the molecule here meas a model molecule which, by itself, represets may more molecules. Cosiderig f as a probability distributio of the particles, we have: f ( v,t ) d v (6) R ρ = = through the sum total of partial probabilities ( µ t / ) of f plus t / µ of P f, f / µ. It should be metioed that the iterpretatio of Eq. 7, whe the value of t / is too large, is ot true (because the f coefficiet o the right had side of the equatio may come out egative). So, i a semi-dese flow, due to the mea free path of the flow beig small, the time step of the solutio should be reduced to satisfy the restrictio coditio. Cosequetly, this method of solutio taes much time ad practically is ot usable. Velocities of the molecules after collisio are obtaied as: v + v v v i j i j v i = + ω v + v v v i j i j v j = ω (8) Compoets of ω vector o the uit sphere are accordig to: cos φ siθ ω = si φ siθ cos θ θ = arccos ξ = φ = π ξ (9) ξ ad ξ are radom umbers which will be geerated i a radom maer o [,] iterval. Further iformatio o that is available i (Bird, 994). O the other had, the TRMC method ca also be used to determie the f distributio fuctio. The TRMC method is based o cosiderig three terms of the Eq. right had accordig to the followig: Havig used the variable chage, the Boltzma s f = A f + A f + A M () ew form was the relatio (). With discritizatio ad usig the Forward Euler method ad usig f ( v ) as Sice relatio () satisfies the coditio of (), the probabilistic expressio of this relatio is that, the f ( v, t) we should have: positio of a molecule at f + could be determied through the summatio of partial probabilities A of f P t t ( f, f + µ µ ) f = f + (7) plus A of f = P( f, f ) / µ plus A of the Maxwellia µ fuctio M. for calculate of A, A, A amouts, it is essetial to evaluate τ from Eq. ad also use of It should be oted that if µ = 4 π σ, the Eq. 5. Relatio is valid o whole rage of µ t / P( f, f ) / µ. It should also be observed that betwee zero ad ifiite which is Ispire with DSMC P( f, f ) / µ = f is the first term of the Wild expasio. method (which based o Nabu-Babosi) that Thus, the probabilistic Iterpretatio of Eq. 7 is that, µ t / <. If time step size selected as a big amout as the positio of a molecule at f +, could be determied µ t /, calculate of distributio fuctio o + 4

5 step is based o Maxwellia distributio sample taig. I other words it ca be said that f + covert to distributio fuctio o equilibrium coditio (Pareschi ad Russo, ). RESULT AND DISCUSSION Figure shows the problem s geometries. The coditios presumed for solvig the problem are similar to the coditios used by Stefaov ad Cercigai (99) ad Bird (994) all of whom solved the Taylor- Quette problem by the DSMC method. The radius of the outer cylider, r, is twice the radius of the ier cylider ad the gas betwee the two cyliders is a mooatomic gas with hard spheres model (i the simulatio, the molecular properties of Argo has bee cosidered). The gas, at the iitial coditios, is static ad uiform. Desity is so selected that the mea free path becomes ( r r ) / 5. The selected legth alog the rotatio axis is 4 times the distace betwee the two cyliders. The two ed plates alog the rotatio axis are assumed with the axial symmetry boudary coditio ad the cylider surfaces are assumed with the reflectio boudary coditio i equilibrium. At t =, the ier cylider rotates with a speed that is three times the most probable speed. Hece, i these coditios, The Kudso umber is. ad the velocity ratio (sec) is.. I both methods, 4 calculatio cells are cosidered alog the axial directio ad cells alog the radial directio with cells havig similar dimesios to each other, the fluid field is divided ito 4, cells. 7 The time step value is selected as t = for the two solutio models ad each molecule of the model stads for 4 real molecules. Because of the existig high temperature differeces ad high desity gradiets created by the supersoic rotatioal speed of the cylider, the local ad effective values of may o dimesioal parameters, such as the Kudse umber, are differet from their omial values at stable flow coditios (begiig ad ed coditios). Therefore, at the begiig of the problem s solutio, may small vortices are geerated which together, gradually, form larger vortices ad after about rotatios of the ier cylider, three large vortices form betwee the two coaxial cyliders. It ca be see from Fig. that the system s stable flow aswer obtaied from the TRMC method, taes the form of three vortices whose size, geometry ad physical properties are similar to each other ad the oly differece is their directio of rotatio which chages for every other vortex. The Phy. Itl. (): 8-44, 4 result, thus obtaied, totally agrees with the results from (Bird, 994). Also, i Fig., desity cotours are preseted. Sice the three vortices i Fig. have similar flow properties, for a more precise aalysis, oly the middle vortex is cosidered. I Fig. ad 4, desity results obtaied from the TRMC ad DSMC methods are show. As it was predicted, the desity icreases i the radial directio, but, i the axial directio, because of existig vortex flows, it decreases ad icreases alterately. Due to the high speed of the ier cylider, gas temperature icreases sigificatly. The field temperature icrease cotiues util the heat trasferred to the surface equals the study doe by the surface of the rotatig ier cylider. As it ca be observed i Fig. 5 ad 6 which represet the results obtaied from the TRMC ad DSMC methods, the maximum temperature occurs ear the ier cylider. As the flow moves away from the ier cylider s surface, its temperature decreases. Also, it ca be see that the temperature gradiet iside the vortex flow, i the axial directio is greater tha the temperature gradiet o the outside of the vortex. Fig. : Axial cuttig of two coaxial cyliders Fig. : Strai lies ad desity cotours o Taylor- Quette flow (by TRMC method) Fig. : Nodimetioal desity (desity ratio to iitial desity) cotours i oe vortex o radial ad axial directio (by TRMC method)

6 Phy. Itl. (): 8-44, Fig. 4: Nodimetioal desity (desity ratio to iitial desity) cotours i oe vortex o radial ad axial directio (by DSMC method) Fig. 7: Nodimetioal rotatioal speed cotours (rotatioal speed ratio to the most probable velocity) i oe of Taylor-Quette flow vortexes o radial ad axial directio (by TRMC method) Fig. 5: Nodimetioal Temperature (temperature ratio to iitial temperature) cotours i oe of Taylor-Quette flow vortexes o radial ad axial directio (by TRMC method) Fig. 8: Nodimetioal rotatioal speed cotours (rotatioal speed ratio to the most probable velocity) i oe of Taylor-Quette flow vortexes o radial ad axial directio (by DSMC method) Fig. 6: Nodimetioal Temperature (temperature ratio to iitial temperature) cotours i oe of Taylor-Quette flow vortexes o radial ad axial directio (by DSMC method) I Fig. 7 ad 8, o dimesioal rotatioal speed cotours ( w / v ) alog the radial ad axial directios m of the cyliders ad i the distace betwee the two cyliders are show. As ca be expected, the maximum rotatioal speed of the flow occurs ear the rotatig ier cylider. I Fig. 9, the umbers of collisios for te succeedig time steps are show. 4 Fig. 9: Number of collisios i two methods (TRMC ad DSMC methods) o te (cosecutive time step i stable coditio CPU time required for the DSMC is 6 h ad for the TRMC is 4. h. Calculatios are performed o a Petium 4,.8 GHz processor. CPU time for the calculatios based o the TRMC method is 6% of the CPU time required for the DSMC. This shows that the TRMC is faster tha the DSMC. The mai reaso for the icrease i calculatio speed is that the umber of collisios i each time step is reduced.

7 Phy. Itl. (): 8-44, CONCLUSION By comparig Fig., 5 ad 7 with Fig. 4, 6 ad 8, respectively ad i the same order, it is observed that the results obtaied from the TRMC ad the DSMC methods have may similarities. I geeral, due to the applicatio of the Maxwellia equilibrium coditios i each time step, the results obtaied from the TRMC are more uiform ad show less fluctuatios. Therefore, the TRMC method, by elimiatig these fluctuatios, provides more fittig results, whe compared to the DSMC scheme. However, the mai advatage of the TRMC method over the DSMC, is i the reductio of the time required to process the program algorithm. The mai reaso for this time reductio i the TRMC method is that the umber of itermolecular collisios (which egage the bul of the computatios), especially i more dese areas of the flow, decreases that is because of exertig of Maxwellia equilibrium coditio i each time step. ACKNOWLEDGMENT The Computig Cetre of Mechaical Departmet of Islamic Azad Uiversity of Mariva is acowledged for allowig us to use their advace computers. The authors also gratefully acowledge the software support of Mechaical Departmet of Islamic Azad Uiversity of Mariva. REFERENCES Bird, G.A., 994. Molecular Gas Dyamics ad the Direct Simulatio of Gas flows. d Ed., Oxford Uiversity Press, Lodo, ISBN: , pp: 78. Carle, E., A.M.C. Carvalho ad E. Gabetta,. Cetral limit theorem for Maxwellia molecules ad trucatio of the wild expasio. J. Comm. Pure Applied Math, 5: DOI:./(SICI)97-() Karma, V., 94. Some aspects of the turbulece problem. Proceedig of the 4th Iteratioal Cogress for applied Mech., Cambridge, Eglad, pp: Kuhlthau, A.R., 96. Recet low desity experimets usig rotatig cylider techiques. Proceedig of the st Iteratioal Symposium o Rarefied Gas Dyamics, Sept. -, Pergamo, Lodo, pp: 9-. Pareschi, L. ad G. Russo,. Asymptotic preservig Mote Carlo methods for the Boltzma equatio. Tras. Theory Stat. Phys., 9: DOI:.8/ Pareschi, L. ad G. Russo,. Time relaxed Moet Carlo methods for the Boltzma equatio. SIAM. J. Sci. Comput., : 5-7. DOI:.7/S Reichelma, J. ad K. Nabu, 99. Mote Carlo direct simulatio of the Taylor istability i rarefied gas. Phys. Fluids, 5: DOI:.6/ Stefaov, S. ad C. Cercigai, 99. Mote Carlo simulatio of the Taylor-Couette flow of a rarefied gas. J. Fluid Mech., 56: 99-. DOI:.7/S9769 Taylor, G.I., 9. Stability of a viscous liquid cotaied betwee two rotatig cyliders. J. Philosoph. Tras. R. Soc. Lod., : DOI:.98/rsta.9.8 Zeytooia, H., S. Sulaimay., S. Navardi ad S. NourAzar, 9. The solvig of the Taylor Quette flow with time relaxed Mot Carlo method: A axisymmetrical model. Proceedig of the 8th Iteratioal symposium i Computer Method i Mechaic, May 8-, Zieloa Gora, Polad, pp: