The integro-moment method applied to two-dimensional rarefied gas flows

Transcription

1 The tegro-oet ethod appled to two-deoal rarefed ga flow S. Varout (), D. Valougeorg () ad F. Sharpov () () Departet of Mechacal ad Idutral Egeerg Uverty of Thealy Pedo Areo, Volo, 38333, Greece () Departaeto de Fca Uverdade Federal do Paraa Caxa Potal 944, Curtba, , Brazl Abtract. The tegro-oet ethod (IMM) ha bee forulated a ple ad coce aer to olve the twodeoal flow of a gle ga a rectagular duct, decrbed by the learzed BGK equato. Baed o th prototype proble, a detaled coputatoal vetgato of the IMM wth regard to covergece peed, accuracy, torage ad CPU te perfored. Overall, t deotrated that the IMM ay be codered a a relable alteratve coputatoal approach for olvg lear ultdeoal rarefed ga flow. The IMM partcularly utable, whe the dcretzato the olecular velocty pace ut be avoded order to elate the ocllatory behavor of the coputed acrocopc quatte due to the propagato of boudary duced dcotute. Keyword: Vacuu flow, Nao ad croflow, No-equlbru flow, Ketc theory PACS: 5.Dd, , Ab, 5.+y. INTRODUCTION Durg the lat decade there a creaed teret rarefed ga flow due to ther egeerg applcato everal feld cludg vacuu yte. The ot cooly ued coputatoal ethod to olve rarefed flow are probabltc approache baed o the DSMC ethod ad detertc chee baed o the dcrete ordate or velocty ethod, whch fro ow o we wll deote t by DVM. Although, both techque have bee pleeted a ophtcated aer to olve coplex flow cofgurato, there are tll everal ptfall related to ther coputatoal effcecy ad accuracy. For exaple, the DSMC ethod uffer fro tattcal oe whe the flow low, whle the DVM, whch partcular utable for th type of flow, everal occao produce a uphycal ocllatory behavor the acrocopc quatte due to the propagato of boudary duced dcotute de the coputatoal doa (ray effect). A alteratve detertc coputatoal chee for hadlg lear (low) flow the tegro-oet ethod (IMM). Bac forato regardg the forulato of the ethod ad t charactertc are revewed [] ad ore recetly []. The bac advatage of the IMM that the derved equato are dcretzed oly the patal depedet varable. Over the year, the IMM ha bee pleeted a lted uber of proble [3, 4, 5], whle very recetly t ha bee appled to olve the uteady behavor of a ga over a plate haroc ocllatory oto [6]. However, all cae codered o far, the flow oe-deoal. I the preet work, frt the IMM properly forulated to tackle ultdeoal proble. The propoed forulato appled to olve the two-deoal flow of a gle ga a rectagular duct, decrbed by the learzed BGK equato. It tur out that, the exteo of the IMM ult-deoal flow cofgurato, although prcpal ay look traghtforward, t ot trval. Baed o th prototype proble, a detaled 35 Rarefed Ga Dyac: 5-th Iteratoal Sypou, edted by M.S.Ivaov ad A.K.Rebrov. Novobrk 7

2 coputatoal vetgato of the IMM wth regard to covergece peed, accuracy, torage ad CPU te perfored. I addto, certa coputatoal ad prograg ue are reolved a effcet aer order the ethod to be codered a a relable alteratve coputatoal chee for olvg lear ultdeoal flow.. FORMULATION The forulato of the IMM preeted by olvg the two-deoal flow of a gle ga a rectagular duct due to preure gradet. Th proble ha bee olved [7, 8], by pleetg the DVM. Therefore, t ued here a a prototype proble for applyg the IMM two-deoal flow. The cro ecto of the duct H W, wth H ad W deotg the heght ad the wdth of the duct repectvely. By takg the heght H a the charactertc acrocopc legth of the proble, the flow doa Ω W W defed by x, ad y H H, ad t how Fgure. The flow codered a fully developed the z drecto ad ed effect that drecto ay be eglected. Sce the flow codered a otheral, the learzed BGK ketc odel ca be appled, whch for th proble wrtte a ( x, y,ζ, ζ ) δu( x, y) Υ Υ ζ x + ζ y + δυ x y. () x y The ukow dtrbuto fucto Y ha bee reduced fro the learzed dtrbuto fucto, followg the typcal projecto procedure order to elate the z copoet of the olecular velocty vector, whch ow cot of oly two copoet,.e. ζ ( ζ x,ζ y ). The rarefacto paraeter δ proportoal to the vere Kude uber ad t defed by PH δ, () µ v where µ the hear veocty, v the ot probable olecular velocty, whle u ( x, y) the oly ozero copoet of the acrocopc velocty ad t the z drecto. Applyg Maxwell catterg boudary codto wth purely dffue reflecto we fd that the outgog dtrbuto at the boudare becoe + Y b. (3) Equato () wrtte the ore coveet for Y ζ + δy δu, (4) where the ew depedet varable deote the charactertc drecto alog the olecular velocty ζ, ζ (ζ,φ), wth ζ ζcoφ ad ζ ζφ ad x y u ( x, y) Ye ζ ζ dζ. (5) Multplyg Eq. () by exp ( δ/ζ ) ad tegratg the reultg equato alog the charactertc, whle ζ treated a a paraeter, we obta 36 Rarefed Ga Dyac: 5-th Iteratoal Sypou, edted by M.S.Ivaov ad A.K.Rebrov. Novobrk 7

3 Y δ ϕ ζ ). (6) δ δ/ζ δ /ζ ( x, y, ζ, ) u( x', y' ) e d + ( e To fd Eq. (6) we have alo ued the boudary codto (3). Subttutg Eq. (6) to Eq. (5) ad after oe route apulato we fd the followg tegral equato for the acrocopc velocty: u δ δ ( x, y) T ( δ) u( x', y' ) d + T ( δ ) δ (7) A t how Fgure, the dtace fro a pot x, y up to the boudary alog the charactertc le the oppote drecto of the olecular velocty ζ, the par ( x ', y' ) deote pot alog the tegrato path, ], wth x' x coφ ad y' y φ, whle [ ( ) T ω exp ζ (8) ( ω) ζ ζ dζ are the well kow Abraowtz fucto of order [9]. FIGURE. The flow doa Equato (7) olved uercally a teratve aer to yeld the ukow acrocopc velocty. Aother acrocopc quatty of practcal teret, whch ca be readly obtaed oce the velocty feld kow, the hear tre teor defed by the vector equato Π Π xz yz ( x, y) ( x, y) δ T co φ φ δ ( δ) u( x', y' ) d + T ( δ ) Alo, the o-deoal flow rate gve by the double tegral W H W H co φ. (9) φ H G u( x, y) dxdy. () W 37 Rarefed Ga Dyac: 5-th Iteratoal Sypou, edted by M.S.Ivaov ad A.K.Rebrov. Novobrk 7

4 Before we cloe th ecto t teretg to ote that at the free olecular lt,.e. δ, the frt ter at the rght had de of Eq. () vahe ad the, by expadg the Τ ( ω) for ω we deduce the plfed expreo l u ( x, y) δ 4, () whch after t ubttuted Eq. (6) ca be tegrated aalytcally to obta a cloed for expreo for the flow rate [7]. 3. THE NUMERICAL SCHEME The two-deoal flow doa dvded rectagular eleet deoted by ( j), wth,..., I ad j,..., J. The geoetrcal ceter of t eleet detered by x ( /) x ad y j ( j /) y, where x /( A I) ad y / J, whle A H / W the apect rato. The uercal grd how Fgure. Equato (7) wrtte the for FIGURE. The coputatoal grd u W H ( x, y) K( x, y x', y' ) u( x', y' ) dx' dy' + Q( x, y W H : ) () ad the by aug that the ukow velocte rea cotat at each grd eleet, t ay be approxated a I J j j + j u K u Q,,..., I ad j,..., J, (3) where 38 Rarefed Ga Dyac: 5-th Iteratoal Sypou, edted by M.S.Ivaov ad A.K.Rebrov. Novobrk 7

5 Κ j y + y x + x K y y x x ( x, y ) ' j : x', y' dx' dy. (4) ad Q j ( ) Τ δ dφ. (5) δ The double tegral (4) over a rectagular eleet ca be reduced to gle tegral f the varable ad φ, tead of x ' ad y', are ued. A t ee Fgure, the tegrato varable deote the dtace betwee the ode (, ) x ad the de of the cell x, y ), whle φ the correpodg agle wth repect to the x ax. y j ( Followg th procedure the tegrato wth repect to becoe aalytcally ad we have to tegrate uercally oly wth repect to φ. I partcular, whe ad j we fd ϕ ϕ3 ϕ4 ϕ ( ) + ( ) + ( ) + Κ j T δ T δ T δ3 T ( δ4 ), (6) ϕ ϕ ϕ3 ϕ4 where the dtace ad the agle are etated by ug typcal trgooetrc arguet ad ther value are: x x x, coϕ y y y, ϕ 3 x x + x, coϕ 4 y y + y (7) ϕ ad y j y + y ϕ ta, x x x y j y y ϕ 3 ta, x x + x y j y y ϕ ta, x x x y j y + y ϕ 4 ta (8) x x + x The four agle (8) are how the detal of Fgure. For the pecfc cae of reduced to ad j the above reult / 4 j 8 x Κ j T δ. (9) coϕ The etato of all reag tegral cludg the oe the ource ter (5) ay be perfored uercally by ug Newto Cote forula. We have foud that the applcato of the trapezodal rule adequate to provde reult of good accuracy. Oce all tegral are coputed Eq. (8) reduced to a algebrac yte, whch olved by pleetg typcal terato chee ad the the ukow u, wth,..., I ad j,..., J are etated. j 4. REMARKS ON THE COMPUTATIONAL EFFICIENCY OF THE IMM Followg the IMM procedure that we have decrbed detal Secto ad 3, the flow proble of a gle ga through a rectagular duct ha bee olved for a apect rato A,.5 ad.. I all flow cofgurato we have ued x y, wth J, 5 ad, whle I J / A. The reult, addto to the velocty feld, clude 39 Rarefed Ga Dyac: 5-th Iteratoal Sypou, edted by M.S.Ivaov ad A.K.Rebrov. Novobrk 7

6 the hear tre teor ad the overall quatte of the flow rate ad the drag coeffcet at the wall. The accuracy of the reult ha bee teted everal way. For δ ad δ, aalytcal reult are avalable, whle for the teredate value of δ the reult have bee copared wth the extg correpodg reult baed o the DVM [7, 8]. Fally, aother way to judge the accuracy to expect fro the ethod by etatg the drag coeffcet, whch cotat ad depedet of δ. Sce, th flow proble ha bee olved before ad all flow quatte of practcal teret have bee publhed we chooe ot to tabulate the reult of the preet work, whch are baed o the IMM. Itead, we prefer to proceed wth oe qualtatve ad quattatve reark regardg the coputatoal charactertc of the IMM a well a t plu ad u copared to the DVM. It ha bee foud that the IMM coverge for all value of δ. The peed of covergece fat for δ., t low dow a δ creaed ad becoe very low for δ. The uber of terato requred for covergece of the ae order wth the correpodg terato requred the DVM. A theoretcal tablty aaly ay be appled future work, at leat for ple ketc odel uch a the BGK, to cofr thee experetal fdg. The requred CPU te per terato the IMM proportoal to N, where N I J the total uber of ode, whle the correpodg CPU te the DVM proportoal to N K, where K the uber of dcrete olecular velocte. Therefore, for the preet proble, a log a the uber of ode the phycal pace le tha the uber of dcrete velocte the olecular velocty pace ( N < K ), the IMM ru fater tha the DVM ad the ve vera whe N > K. I geeral, the requred CPU te per terato the IMM proportoal to D N, where N the total uber of ode the coputatoal doa ad D the uber of phycal deo the proble. Regardg accuracy ad torage dead we ay ay that at th tage the DVM perfor better. However, the IMM approach ay be further proved, whle the DVM approach, over the year, ha bee coputatoal optzed. Probably, a ore ophtcated uercal chee, keepg the plcty of the oe propoed Secto 3 ad upgradg at the ae te t accuracy ay be approprate. The ot portat advatage of the IMM copared to the DVM that dcretzato perfored oly the phycal pace. Th of partcular portace proble wth dcotute the boudary or the tal codto. The, the DVM thee dcotute are propagatg the coputatoal doa ad yeld reult wth a uphycal ocllatory behavor. Th ptfall copletely elated the IMM. Overall, we have foud that the IMM a relable coputatoal tool for olvg th prototype two-deoal proble. Baed o th experece we expect a lar behavor other ore coplex ultdeoal flow a well. ACKNOWLEDGMENTS The work of two of the author (S.V. ad D.V) ha bee partally fuded by the Aocato EURATOM Hellec Republc. Th upport gratefully ackowledged. Alo, the thrd author (F. Sh.) thak CNPq (Brazl) for the upport of h reearch. REFERENCES. M. N. Koga, Rarefed ga Dyac, Pergao Pre, Oxford, F. Sharpov ad V. Selezev, Data o teral ga flow, J. Phy. Che. Ref. Data, 7, , C. Cercga ad C. D. Paga, Phy. Flud, 9, 67, V. G. Cheryak, B. T. Porodov ad P. E. Suet, Zh. Tekh. Fz., 43, 4, P. E. Suet ad V. G. Cheryak, Ivz. Akad. Nauk SSSR. Mekh. Zhdkot Gaza N6, 7, F. Sharpov ad D. Kalepa, Ga flow ear a plate ocllatg logtudally wth a arbtrary frequecy, ubtted. 7. F. Sharpov, Rarefed ga flow through a log rectagular duct, J. Vac. Sc. Techol. A, , D. Valougeorg ad S. Nar, Accelerato chee of the dcrete velocty ethod: Gaeou flow rectagular crochael, SIAM J. Sc. Cop., 5, , M. Abraowtz ad I. A. Stegu, Hadbook of Matheatcal Fucto wth Forula, Graph, ad Matheatcal Table. 9th ed., New York: Dover, page 89, Rarefed Ga Dyac: 5-th Iteratoal Sypou, edted by M.S.Ivaov ad A.K.Rebrov. Novobrk 7