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1 This tile ppeed in jounl pulished y Elsevie. The tthed opy is funished to the utho fo intenl non-ommeil eseh nd edution use, inluding fo instution t the uthos institution nd shing with ollegues. Othe uses, inluding epodution nd distiution, o selling o liensing opies, o posting to pesonl, institutionl o thid pty wesites e pohiited. In most ses uthos e pemitted to post thei vesion of the tile (e.g. in Wod o Te fom) to thei pesonl wesite o institutionl epositoy. Authos equiing futhe infomtion egding Elsevie s hiving nd mnusipt poliies e enouged to visit:

2 Euopen Jounl of Mehnis B/Fluids 8 () 7 Contents lists ville t SiVese SieneDiet Euopen Jounl of Mehnis B/Fluids jounl homepge: Refied gs flow though ylindil tue due to smll pessue diffeene Sntis Pntzis, Dimitis Vlougeogis Univesity of Thessly, Deptment of Mehnil Engineeing, Volos 8, Geee t i l e i n f o s t t Atile histoy: Reeived June Reeived in evised fom Otoe Aepted 6 Otoe Aville online 6 Noveme Keywods: Lineized flow Kineti theoy Mioflows Vuum flows Knudsen nume Flow of efied gs though ylindil tue onneting two esevois mintined t smll pessue diffeene is onsideed using the isymmeti vesion of the lineised BGK kineti model eqution sujet to Mwell diffuse speul oundy onditions. This is polem of five dimensions in phse spe, solved in fully deteministi mnne using pllelised disete veloity lgoithm. Results inlude flow tes s well s distiutions of density nd veloity petutions, fom the fee moleul up to the slip egime nd fo length-ove-dius (L/R) tios nging fom zeo (oifie flow) up to. The dependeny of the esults on gs eftion, wll ommodtion nd tue length is nlysed nd disussed. It is found tht the Knudsen minimum ppes only t L/R =. Futhemoe, in the se of L/R = it is onfimed tht the esults e ptilly independent of the ommodtion oeffiient. Comping the pesent line esults with oesponding non-line ones, it is seen tht lineised nlysis n ptue the oet ehviou of the flow field not only fo infinitesimlly smll ut lso fo smll ut finite pessue diffeenes nd tht its nge of ppliility is wide thn epeted. Also, the eo intodued y the ssumption of fully developed flow fo hnnels of modete length is estimted though ompison with the pesent oesponding esults. Elsevie Msson SAS. All ights eseved.. Intodution The flow of efied gs though long o shot ylindil tues (inluding oifies) due to smll o lge pessue diffeenes is of mjo impotne in the design nd optimistion of vious types of industil equipment in sevel tehnologil fields. Some of these pplitions inlude mss flow ontolles in gs meteing [], mss spetometi smpling [], miopopulsion in high ltitude nd spe gs dynmis [], pumps nd gs distiution in vuum systems [ 6], memnes nd poous medi in filteing [7,8], gseous devies in mioeletomehnil systems [9,] nd othes [ ]. Refied gs flows though tues lso hve stong theoetil inteest, minly due to the ft tht eltively smll nume of geometi nd flow pmetes is dequte to fully define the polem. The study of suh flows hs llowed the investigtion of mny non-equiliium phenomen in the whole nge of the Knudsen nume. In ddition, they hve een pplied s pototype polems to test the vlidity of poposed kineti equtions, gs-sufe stteing kenels nd intemoleul ollision models, s well s to enhmk the omputtionl effiieny of vious numeil shemes. Fully developed flows in long hnnels, i.e. in hnnels whee the tio of the length ove the hyduli dimete is lge ( sfe estimte is out o moe), hve een onsideed y mny Coespondene to: Physiklish-Tehnishe Bundesnstlt, Belin 87, Gemny. Tel.: E-mil ddess: sntis.pntzis@pt.de (S. Pntzis). esehes [ 6] nd fo vious geometies [7 ], oth numeilly nd epeimentlly. This flow onfigution my e onsideed s the simplest one, sine hnnel end effets e negleted nd the pessue vies only in the flow dietion. The kineti solution is otined only on oss setion of the hnnel fo wide nge of the Knudsen nume nd then the solution fo the whole flow field is otined vi well-known methodology sed on mss onsevtion [,]. Both fo smll nd lge pessue dops etween the inlet nd outlet pessue, the lol pessue gdient is smll nd lineised kineti nlysis my e pplied in omputtionlly effiient mnne, yielding vey ute esults. Epeimentl wok in flows though long hnnels hs lso een pefomed nd vey good geement etween mesuements nd omputtions hs een otined [, 6]. The oesponding flow though shot hnnels poses muh lge omputtionl diffiulties due to the inesed dimensionlity of the polem nd moe impotntly due to the ft tht end effets must e onsideed in the simultion y inluding dequtely lge pts of the upstem nd downstem ontines in the omputtionl domin. As esult, the equied simultion time is signifintly inesed. Ely woks investigte slit nd oifie flows ne the fee moleul egime [7 9], with ptiul emphsis on flow into vuum, ut thei nge of ppliility is smll. In ode to otin the ehviou of the flow fo ny eftion egime, the most ommonly implemented nd suessful ppohes ely on the Diet Simultion Monte Clo (DSMC) method []. At lge pessue diffeenes, DSMC hs een used, due to its simpliity nd high uy, in the whole nge of eftion fo the /$ see font mtte Elsevie Msson SAS. All ights eseved. doi:.6/j.euomehflu...6

3 S. Pntzis, D. Vlougeogis / Euopen Jounl of Mehnis B/Fluids 8 () 7 Fig.. Geomety nd oodinte systems. solution of high speed flows though slits [ ], oifies [,,] nd shot hnnels [,6,7]. These woks hve limittions in the smll pessue diffeene nge nd the investigtions losest to the pesent wok wee onduted y Shipov [] fo oifie flow with vlues of pessue tio up to.9. It is well known tht DSMC, t lest in its oiginl vesion, is not suitle fo the simultion of low speed flows. Non-line kineti model equtions tkled y the Disete Veloity Method (DVM) hve lso een pplied in [8 ] fo polems of plne nd isymmeti geomety (slits, pltes, oifies nd tues) nd this ppoh ould ltentively e used fo ny vlue of pessue tio. In the se of smll pessue diffeenes, the litetue is the limited. Akin shin et l. [] hve solved the lineised nonisotheml slit polem y the integl moment method employing the Shkhov kineti model [,6]. Flow though slit hs lso een emined y Hsegw nd Sone [7] fo smll pessue diffeenes with the BGK model. Shipov hs pplied the DVM to solve the lineised isotheml [8] nd non-isotheml [9] slit polem, whee the stisftion of the Onsge theoem is veified. Shkhov solved the polem of lineised isotheml flow in hnnels [,] nd lso omped with the fully developed flow in slightly diffeent mnne []. This poedue signifintly edued the diffeene etween the two ppohes ut pio knowledge of the pessue gdient is equied, whih is diffiult to otin without the omplete numeil simultion. Thee e sevel epeimentl investigtions egding flow though slits nd oifies [,]. Results inlude mss flow tes, dishge oeffiients nd intepolting fomuls. Tues of smll length-to-dius tios hve lso een studied epeimentlly y Seeknth [], Fujimoto nd Usmi [], Mino [] nd Voutis et l. [6] fo wide nge of pessue tios in the tnsition egime. Howeve, vey few woks del with low pessue diffeenes [7,8]. In this wok, we pply pllelised DVM lgoithm to investigte flows though iul tues diven y smll pessue diffeenes in wide nge of the Knudsen nume. The tue geomety nges fom oifie up to length-ove-dius tio equl to nd the hnnel end effets e onsideed y inluding pt of the upstem nd downstem ontines. Low pessue diffeenes hve not een emined etensively in the pst nd detiled study of this polem is impotnt in ode to otin elile solutions fo onditions whee the omputtionl ost of DSMC is vey high. Results e povided in dimensionless fom fo the flow tes nd the mosopi distiutions nd thei dependeny on gs eftion, wll ommodtion nd tue length is nlysed nd disussed. The nge of ppliility of lineised theoy is emined vi ompison with non-line BGK esults. Also, the eo intodued y the ssumption of fully developed flow fo hnnels of modete length is estimted though ompison with the oesponding pesent esults.. Fomultion.. Flow onfigution Conside two esevois filled with montomi, efied gs t slightly diffeent pessues nd onneted y ylindil tue though whih flow is indued. The pessue of the upstem nd downstem esevois is P = P + P nd P = P espetively, s seen in Fig.. The fields of ll ulk quntities hnge only long the dil nd il dietions, ˆ nd ˆ, while emining onstnt in the zimuthl ngle ϑ due to the isymmety. Theefoe, in ode to desie the geomety, only the tue length L nd dius R e equied. Even though ll wlls nd esevois e mintined t onstnt tempetue T, smll vitions of tempetue e epeted in the flow field. Due to the eltively smll length of the tue, the flow is not fully developed nd theefoe potion of the two ontines is inluded in the simultions in ode to popely impose the oundy onditions nd tke into ount the hnnel end effets.

4 6 S. Pntzis, D. Vlougeogis / Euopen Jounl of Mehnis B/Fluids 8 () 7 The si flow pmete is the efeene eftion pmete δ, defined hee s δ = RP µ υ () whee µ is the gs visosity t efeene tempetue T nd υ = Rg T is the most pole moleul veloity with R g eing the gs onstnt. It is noted tht δ is invesely popotionl to the Knudsen nume. The flow onfigution is defined y the geometil tio L/R nd the efeene eftion pmete δ nd the ojetive is to otin the solution of this flow in tems of these two pmetes... Govening kineti equtions Fo suffiiently long times nd due to the nely isotheml ntue of the pessue diven flow, the stedy stte BGK kineti model eqution my e used f ξ ˆ ξ ϑ f ˆ θ + ξ f ˆ = ν f M f () whee ξ = (ξ, ξ ϑ, ξ ) is the moleul veloity veto nd θ [, π] is the oesponding ngle in the ˆ ϑ plne, f is the unknown distiution funtion nd ν = P/µ is the ollision fequeny. The Mwellin distiution is defined y f M ˆ, ˆ = n ˆ, ˆ πrg T ˆ, ˆ / ep ξ û ˆ, ˆ R g T ˆ, ˆ. () Mosopi quntities, suh s the nume density n ˆ, ˆ, ulk veloity û ˆ, ˆ nd tempetue T ˆ, ˆ, e otined y tking ppopite moments of f. The oodintes in the physil ˆ, ˆ nd moleul veloity spes (ξ, ξ ϑ, ξ ) e shown in Fig. nd onstitute five-dimensionl phse spe fo the distiution funtion. Sine the pessue diffeene etween the upstem nd the downstem vessel is smll (P/P ), the distiution funtion n e lineised s f = f ( + hp/p ) () with f eing Mwellin t the efeene onditions. All quntities e epessed in dimensionless fom s follows: = ˆ R, = ˆ R, = ξ υ, ρ (, ) = n (, ) n n P P, τ (, ) = T (, ) T T P P, p (, ) = P (, ) P P û (, ) P, u (, ) = P P υ P whee ρ, τ, u, p e the petutions of density, tempetue, veloity nd pessue. The ight ontine onditions e tken s efeene quntities. As finl step, the moleul veloity veto = (, ϑ, ) is tnsfomed to ylindil oodintes = p, θ,. Thus, we otin p os θ h p sin θ h θ + h + δh = δ ρ + τ + u. (6) () Similly, the mosopi quntity petutions e epessed in tems of the petution h ρ = π h π / p ep d p dθd (7) u = π / u = π / π π h p ep d p dθd (8) h p os θ p ep d p dθd (9) τ = π h π / p ep d p dθd. () The petution of pessue is found y p = ρ + τ. The mosopi veloity veto hs only two omponents u = (u, u ) due to the isymmety of the flow. It is lso noted tht the pessue diffeene is not inluded in these epessions nd is tken into ount only duing the dimensionlistion of esults oding to Eq. (). This is typil in line solutions. The most impotnt quntity fo ptil pplitions is the mss flow te though the hnnel, defined y Ṁ = πm R n ˆ, ˆ û ˆ, ˆ ˆdˆ () with m eing the moleul mss. The mss flow te is lineised to yield Ṁ LIN = πmn υ P P u (, ) d () nd then the lineised mss flow te is non-dimensionlised y the fee moleul (δ = ) solution fo flow though n oifie ( tue of zeo length) given y Ṁ FM = R πp/υ [], whih n e esily etted y the method of hteistis. Results e pesented in dimensionless fom oding to W LIN = ṀLIN Ṁ FM = πg () whee G = u (, ) d () is the edued flow te. Even though the epession fo G ontins the il veloity whih depends on, G (nd W LIN ) is ptilly onstnt in ny oss-setion. It is lso noted tht oding to [8] the lineised nlysis is vlid when P P () fo δ < nd P P δ (6) fo δ >... Boundy onditions The fomultion is ompleted y poviding the oundy onditions. Moleules enteing fom the fee sufes (A), (B), (F), (G)

5 S. Pntzis, D. Vlougeogis / Euopen Jounl of Mehnis B/Fluids 8 () 7 7 (s shown in Fig. ) onfom to Mwellin distiution oding to the onditions of the oesponding vessel. Thus, fo the left vessel, we hve n = n + n, T = T nd û = nd theefoe the petution fom the equiliium distiution is h + A,B = ρ in = (n + n) n n (P/P ) = (P + P) P P (P/P ) =. (7) Similly, it is found tht in downstem fee sufes (F), (G) the petution of the inoming distiution is h + F,G =. (8) Fo the wlls (C), (D), (E), diffuse speul oundy onditions e imposed, i.e. h + = α M ρ w + ( α M ) h (9) whee α M is the Mwell ommodtion oeffiient nd h is the distiution of impinging ptiles. The ρ w onstnts e found y imposing the impemeility ondition (u n = ) nd the veloity integls (8) (9). The finl epession is ρ w = I impinging + ( α M ) I speul α M I depting. () The nottion impinging efes to the distiution of moleules hitting the wll, while the wods speul nd depting denote the distiution of ptiles stteed speully nd nonspeully, espetively. The integls of Eq. () e π θ I depting = p φ θ, p, π θ ep p dp dθd () θ I impinging = h p φ θ, p, I speul = θ ep p dp dθd () π θ h speul π θ p φ θ, p, ep p dp dθd () whee = C, D, E (see Fig. ) nd C : θ C =, θ C = π, =, =, φ C θ, p, = D : θ D =, θ D = π/, =, =, φ D θ, p, = p os θ E : θ E =, θ E = π, =, =, φ E θ, p, =. () All of the ove integls e lulted numeilly fo onsisteny esons. Finlly, t the is of symmety ( = ), denoted in Fig. s (H), moleules e efleted speully, i.e. h + H p, θ, = h H p, π θ, () whee θ [, π/].. Numeil sheme.. Itetive lgoithm nd disetistion The numeil sheme is sed on the Disete Veloity Method fo the tetment of the thee-dimensionl moleul veloity spe. The ontinuum spetum of the ylindil omponents p nd is disetised y the Legende polynomil oots mpped in, p,m nd,,m espetively, while the moleul veloity ngles e unifomly distiuted in [, π] due to the isymmetil popeties of the flow. The solution is otined y n itetive poedue, whee the min unknown is the distiution funtion. Initilly, the petution of density is set equl to unity upstem, zeo downstem nd vies linely long the tue, while the petutions of veloity nd tempetue e zeo eveywhee. This estimtion is hosen in ode to elete onvegene nd is used in omintion with the govening eqution (6) to lulte the vlue of the distiution funtion h. The distiution funtion is futhe used to genete new vlues fo the ulk quntities vi the oesponding moments (7) (). These quntities e e-used in the govening eqution to otin new estimtes fo h nd this poedue is epeted until pope onvegene iteion, imposed on the ulk quntities, is stisfied. A seond ode disetistion sheme hs een pplied in the two-dimensionl physil spe, deived in the sme wy s in [9] y integting the govening eqution in, θ, in n ity disetistion intevl, ting in oth pts of (6) with the opeto A = k + k / k k / θj +θ j / θ j θ j / i + i / i i / ( ) ddθd. (6) Then, ll integtions n eithe e ied out nlytilly, y eliminting the deivtives, o numeilly y the tpezoidl ule, using the vlues of the distiution t the limits of the disetistion intevl, e.g. t k k /, k + k /. The tpezoidl ule is thus using the seond-ode eo. The finl disetised epession is l p os θ j h l,m i+,j+,k+ + hl,m i+,j,k+ hl,m i,j+,k+ hl,m i,j,k+ i + h l,m i+,j+,k + hl,m i+,j,k hl,m i,j+,k hl,m i,j,k l p sin θ j h l,m i+,j+,k+ θ j hl,m i+,j,k+ + hl,m i+,j+,k i+ h l,m i+,j,k + h l,m i,j+,k hl,m i,j,k + h l,m i,j+,k+ hl,m i,j,k+ i + m h l,m i+,j+,k+ + hl,m i+,j,k+ + hl,m i,j+,k+ + hl,m i,j,k+ k h l,m i+,j+,k hl,m i+,j,k hl,m i,j+,k hl,m i,j,k h l,m i+,j+,k+ + hl,m i+,j,k+ + hl,m i,j+,k+ + hl,m + δ i,j,k+ 8 + h l,m i+,j+,k + hl,m i+,j,k + hl,m i,j+,k + hl,m i,j,k = δ ρi+,k+ + ρ 8 i,k+ + ρ i+,k + ρ i,k l + p + m τ i+,k+ + τ i,k+ + τ i+,k + τ i,k + l p os θ j u,i+,k+ + u,i,k+ + u,i+,k + u,i,k + m u,i+,k+ + u,i,k+ + u,i+,k + u,i,k. (7) In this disetised fom, the indies i, k efe to the physil gid, l, m efe to the disete veloity mgnitudes p nd espetively, while j efes to the disete veloity ngle. The sign

6 8 S. Pntzis, D. Vlougeogis / Euopen Jounl of Mehnis B/Fluids 8 () 7 Fig.. Shemtil epesenttion of the mhing sheme. (Fo intepettion of the efeenes to olou in this figue legend, the ede is efeed to the we vesion of this tile.) Fig.. Stemlines fo L/R = with () δ =. nd () δ =. Fig.. Stemlines fo L/R = with () δ =. nd () δ =. following some indies indites the position of the gid point in ompison to the disetistion point, fo emple h l,m i+,j+,k+ = h i + i, k + k, l, θ p j + θ j, m. It is lso noted tht the vlues of the distiution funtion e stoed only on the limits of the disetistion intevl, (i±, j±, k±) nd not on the entl point (i, j, k). This epession is pplied fo ny intevl, egdless of the gid distnes i, k nd the ngle disetistion θ j. It is lso usle fo intevls ontining gid points with = fte the pplition of the l Hospitl ule on the indeteminte ftions... Pllelistion nd memoy hndling The omputtionl effot n e distiuted in sevel poessos y notiing tht the distiution funtions of diffeent veloity mgnitudes n e lulted independently fom one nothe [6]. As esult, the ode n e esily pllelised in the moleul veloity spe. Eh poesso solves the kineti eqution fo goup of veloities nd infomtion on mosopi quntities nd impemeility onstnts is ehnged etween the poessos t the end of eh itetion. In this mnne, the tnsmission of the distiution is iumvented, getly eduing the ost of pllel ommunition. The pllelistion lgoithm hs een tested in othe polems [] with sevel poessos, displying vey good sling hteistis (e.g. 9% effiieny fo 6 oes). Fo pllelistion of even lge sle, the pllelistion my e etended in the physil spe. Memoy hndling tehniques hve lso een used to edue stoge equiements euse of the five-dimensionl ntue of the distiution funtion fo this polem. Due to the veloity mgnitude independeny, tempoy y n e lloted nd ovewitten fte teting eh mgnitude. Futhemoe, the dimensionlity of this y n e edued even moe y stoing the distiution only in the pts of the domin equied y the mhing sheme of the disetised govening eqution. Fo emple, s seen in Fig., the distiution is stoed only t positions

7 S. Pntzis, D. Vlougeogis / Euopen Jounl of Mehnis B/Fluids 8 () 7 9 Fig.. Pessue petution distiutions t the symmety is with () L/R =, () L/R = nd () L/R = (the esults fo δ =. nd δ = oinide). (ed ows) nd (geen ows), fo motion towds the positive dietion. Blue ows indite the oundy onditions, while dshed ows denote the symmetil pts of the distiution whih e lso negleted. These tehniques pemit hving two-dimensionl y fo the distiution funtion nd getly edue memoy limittions. In this mnne, tues of lge L/R tios n e onsideed, sine the size of the distiution y is only detemined y the height of the entne/eit egions nd the nume of the moleul veloity ngles... Computtionl gid, numeil pmetes nd enhmking The omputtionl gid ws non-unifom, with ptiul emphsis on the uy ne the wll ones. The disetistion intevls vy oding to i = ( + η) i (8) nd similly fo the -dietion. The smllest intevls, e lose to the ones. It is well known [7] tht the upstem/downstem egions must e quite lge, sine the mss onsevtion lw indites tht the mosopi quntities onvege to the unifom ontine vlues t vey high distne fom the tue. Theefoe, ontine egions of size L on = R on = R wee used, fte veifying tht esults do not hnge fo lge vlues y pmeteistion uns. The vege esidul pe node hs een hosen s onvegene iteion N totl esidul = N totl i= ρ i ρ p i + τ i τ p i + u,i u p + u,i u p (9),i whee the p supesipt denotes the oesponding quntities in the pevious itetion nd N totl is the totl nume of nodes. The disetistion pmetes used e displyed in Tle. Fo the esults shown hee, intevls hve een used in the fist unit length ound the ones, with the nume of totl physil nodes vying etween 6 nd 6. The nume of disete veloities nged etween 8, nd,. A typil omputtionl itetion tkes ppoimtely s with CPU oes. The totl nume of itetions needed to eh onvegene depends stongly on δ. Inditively, fo the stisftion of the iteion shown in Tle, it is ound, nd, fo δ =., nd, espetively. The dependene on the tue length is weke, due to the ft tht the initil ondition of line distiution within the hnnel ppoimtes ette the finl solution fo longe tues. The fulfilment of the mss onsevtion piniple (u ),i + u = () whih hs een otined y tking the ppopite moment of Eq. (6) in the moleul veloity spe, ws emined y lulting the left hnd side of Eq. () in the whole field. It ws shown

8 S. Pntzis, D. Vlougeogis / Euopen Jounl of Mehnis B/Fluids 8 () 7 Fig. 6. Ail veloity distiutions t the symmety is with () L/R =, () L/R = nd () L/R = (the esults fo δ =. nd δ = oinide). Tle Computtionl pmetes. Minimum physil spe intevl = 6.7 Physil disetistion inement fto η Disete veloity ngles N θ in (, π) Disete veloity omponents p nd 6 6 Mimum vlue of veloity omponents p,m nd,m Convegene iteion (esidul) 8 9 tht ll vlues e vey lose to zeo. Futhemoe, the flow te G given y Eq. () ws lulted fo ll ses in sevel positions long the tue nd it ws found to e onstnt in t lest thee signifint figues. Finlly, the fee moleul solution is etieved with vey good uy. Ovell the numeil esults pesented in the net setion e onsideed ute up to signifint figues within (±) in the lst figue.. Results nd disussion.. Flow tes Results fo the dimensionless flow te W LIN e pesented in Tle fo vious vlues of the efeene eftion pmete δ nd the length of the tue L/R, epesenting flows fom the fee moleul up to the slip egime though oifies nd tues Tle Dimensionless flow te W LIN fo vious vlues of δ nd L/R, with puely diffuse efletion (α M = ). δ L/R of smll nd modete length. In ptiul, δ = [,.,.,.,,,, ] nd L/R = [,.,,,,, ], while puely diffuse efletion is ssumed on the wll nd α M =. It is seen tht, t eh vlue of δ, the flow te W LIN is deesed s L/R is inesed. This is well epeted sine s the hnnel length is inesed the lol pessue gdient is edued while the wll fition is inesed, leding to lowe ulk veloities. Howeve, W LIN is not dietly popotionl to the invese of L/R. With egd to δ, it is seen tht fo eh L/R the flow te W LIN is monotonilly inesed long with the eftion pmete δ, while fo L/R = the Knudsen minimum phenomenon, well known in the se of long tues, is oseved. Thus, fo L/R = nd stting fom the fee moleul limit (δ = ), the flow te is slightly deesed

9 S. Pntzis, D. Vlougeogis / Euopen Jounl of Mehnis B/Fluids 8 () 7 d Fig. 7. Ail veloity distiutions t vious positions fo δ =. with () L/R =. nd () L/R = nd fo δ = with () L/R =. nd (d) L/R =. up to δ =. nd then it is inesed long with δ. It is noted tht the Knudsen minimum is vey shllow omped to the one oseved in tues of infinite length nd it ppes to smlle vlue of the eftion pmete (somewhee in the nge δ =..) thn epeted. Inditive esults fo inomplete ommodtion e pesented in Tle, whee the dimensionless flow te W LIN is tulted fo δ = [,.,, ] nd L/R = [,, ], with α M = [.,.8]. It is seen tht in the se of L/R = the flow te emins onstnt fo ll vlues of the ommodtion oeffiient (see lso the fully ommodted esults of Tle ) nd theefoe is ptilly independent of the wll ommodtion popeties. This is one of the fvoule popeties of oifie flow, mking it n idel onfigution fo the evlution of numeil shemes, kineti models nd intemoleul potentils, s well s fo ompison with epeimentl dt, sine the fto of gs-sufe intetion n e negleted in this se. Simil popeties hve een found fo slit flow [8]. Fo tue of finite length, W LIN is inesed s α M is deesed nd the intetion eomes moe speul. This tend is moe dominnt s δ is deesed nd L/R is inesed, whih is epeted sine sufe ommodtion popeties ply moe impotnt ole fo highly efied tmosphees nd longe hnnels. In ode to estimte the intodued eo when the ssumption of fully developed flow (i.e. L/R ) is pplied to hnnels of modete length, ompison with oesponding esults of the pesent wok is pefomed. In Tle the flow te W FD, otined Tle Dimensionless flow te W LIN fo vious vlues of δ nd L/R, with diffuse-speul efletion (α M ). L/R α M δ y the lineised fomultion fo fully developed flow is shown fo the two longest tues emined hee (L/R =, ) nd fo the sme nge of the eftion pmete ( δ ). The quntities in penthesis efe to the eltive devition, whih hs een otined y omping the fully developed esults of Tle with the oesponding ones in Tle. It is seen tht the geement is impoved s L/R nd δ e inesed. This is esonle sine, s the length of the tue is inesed, the hnnel ends ffet eltively smlle pt of the geomety, while high vlues of δ esult in smlle flow development lengths nd theefoe fste dpttion to the fully developed flow pofile. The mimum devition, whih is 9.8%, ous fo L/R = nd δ = s it ould e epeted, while the minimum one is 6.7% nd is found fo L/R = nd δ =. These esults povide n estimte of the uy to epet when fully developed solutions e implemented in efied flows

10 S. Pntzis, D. Vlougeogis / Euopen Jounl of Mehnis B/Fluids 8 () 7 d Fig. 8. Rdil veloity distiutions t vious positions fo δ =. with () L/R =. nd () L/R = nd fo δ = with () L/R =. nd (d) L/R =. Tle Dimensionless flow te W FD fo vious vlues of δ nd L/R, otined y the fully developed flow ppoimtion, with puely diffuse efletion (α M = ); the numes in penthesis denote the oesponding eltive eo. δ L/R.67 (9.8%). (.9%)..6 (7.%). (.%)..9 (9.7%). (.8%)..6 (.%). (.8%).8 (8.9%).9 (9.%).9 (7.%).7 (8.%).6 (.9%).8 (6.7%).6 (.%).6 (6.7%) though iul hnnels of modete length due to smll pessue diffeenes... Mosopi distiutions Stemline plots e shown in Figs. nd fo L/R =, nd δ =.,. These stemlines e inditive fo the othe vlues of L/R nd δ used in this wok. It is onfimed tht the impemeility ondition is lwys stisfied, poduing ompletely hoizontl lines inside the hnnel, even fo the demnding se of the eltively long hnnels. The stemlines e symmetil ound = L/ (R) nd stutues ppeing in non-line flows, suh s voties, e sent. The distiutions of petued pessue nd il veloity long = e shown in Figs. nd 6 espetively fo sevel vlues of L/R nd δ. The petued pessue f upstem is equl to unity, then it is edued though the tue nd finlly eomes zeo f downstem. It is seen tht the pessue pofiles fo eh L/R e quite simil fo ll vlues of δ. Also, they vy nely line t the ente of the hnnel. This is moe evident in the se of the long tue L/R = nd it is esonle sine in this se the hypothesis of fully developed flow is ptilly fulfilled. The diffeenes e mostly loted in the gdient of pessue, detemining the flow te. The il veloity f upstem is zeo, then it is inesed up to = L/ (R) whee its mimum vlue is ehed nd finlly it is deesed to zeo f downstem. Fo L/R = the il veloity ehes plteu fte distne of ound one tue dius fom the inlet nd then mintins this vlue fo the est of the tue, showing lose esemlne to fully developed pofile. The petued pessue nd edued il veloity e ntisymmeti nd symmeti espetively out = L/ (R). The il nd dil mosopi veloity omponent distiutions e shown in Figs. 7 nd 8 espetively t the entne ( = ), middle ( = L/ (R)) nd eit ( = L/R) of the tues with L/R =., nd δ =.,. Due to the popeties of the

11 S. Pntzis, D. Vlougeogis / Euopen Jounl of Mehnis B/Fluids 8 () Fig. 9. Pessue petution fo δ = with () L/R =, () L/R =, () L/R =. P(,) p(,) p(,) flow, the il veloity pofile is identil t the entne nd eit of the tue (see Fig. 7), while the dil omponent is ntisymmeti ound the tue middle (see Fig. 8). Fo δ =., the il veloity tkes smll vlues distiuted in the now nge nd hteised y signifint jump on the wlls, while fo δ = moe ponouned poli pofile is otined. The dil veloity distiution ppohes line fom fo the most line se (δ =., L/R = ), while it is lwys zeo in the middle. In ll ses, lge δ nd smlle L/R led to lge veloity mgnitudes. A moe omplete view of the flow field is pesented in Figs. 9 fo vious vlues of δ nd L/R. The effet of hnging the tue length is emined fo onstnt eftion pmete δ =, with L/R =, L/R = nd L/R = in Figs. 9 nd, whee the oesponding pessue nd il veloity fields e shown. The pessue distiution ound the hnnel ends gdully eomes lose to the ontine vlues s the length ineses, i.e. the pessue ontou olouing t eh ontine is moe unifom fo L/R = thn fo L/R =,. This hppens euse the e of the esevois ffeted y the hnnel flow is smlle fo longe hnnels, due to the smlle indued gs veloities. Also, smll egion of slightly highe nd lowe pessue thn the ontine vlues is lso oseved just ove the hnnel openings. The il veloity is signifintly edued fo longe tues nd seems to develop nely onstnt pofile inside the hnnel nd in eltively shot distne fom the hnnel ends. The end influene on the veloity pofile seems to fde wy ound one unit of dimensionless length inside the hnnel fo L/R = nd δ =. The effet of hnging the eftion pmete δ is shown in Figs. nd fo tue of L/R =, with δ =., nd. No signifint hnges ou fo the pessue petutions, esides slightly lge devition of pessue in the ontines fom the equiliium vlues s δ is inesed. The il veloity vlues e inesed long with δ. It is noted tht the non-smooth distiutions of the mosopi quntities in Figs. e used y the so-lled y effets. These e pesent minly t δ =., used y the oundy indued disontinuities of the distiution funtion whih popgte though the flow field nd eome moe dominnt t smll vlues

12 S. Pntzis, D. Vlougeogis / Euopen Jounl of Mehnis B/Fluids 8 () Fig.. Ail veloity fo δ = with () L/R =, () L/R =, () L/R =. u (,) u (,) u (,) of δ. This is disdvntge of the disete veloity method, whih is not esily iumvented. Howeve, it is epeted to hve no signifint effet on ovell quntities, suh s the flow te... Compison with non-line esults In this susetion, ompison of the pesent line flow tes with oesponding ones sed on the non-line BGK eqution, denoted y W NL, is pesented. Fo this pupose the flow tes W NL fo P /P =.9 (o P/P =.) nd L/R =,, e tulted in Tle. The esults hve een otined y the non-line BGK model [6] nd hve een found to e in vey good geement with oesponding DSMC esults []. It is seen tht the geement etween the line flow tes W LIN nd the oesponding non-line ones W NL is vey good in genel. The diffeenes etween e edued s δ is deesed nd s L/R is inesed. This tend is eplined sine s the gs eomes moe efied nd the tue longe the ulk veloity is deesed leding to smll Mh nd Reynolds numes nd to line symmeti flow field. In sevel ses the geement is good to ll thee signifint figues shown, while the wost se is fo δ = nd Tle Dimensionless flow te W NL fo vious vlues of δ nd L/R with P /P =.9 nd α M =, otined y the non-line BGK model [6]. δ L/R W LIN W NL W LIN W NL W LIN W NL L/R =, whee the eltive eo is out.%. Oviously, s the pessue diffeene is deesed the geement will e lso signifintly impoved. Bsed on the ove it my e stted tht fo pessue tio vlues up to P/P =. the pesent BGK line fomultion gives stisftoy esults in good geement with the non-line BGK fomultions in the fee moleul nd tnsition egimes. This emk hs two implitions: Fist, lineised equtions povide oet esults t wide nge thn epeted fom the

13 S. Pntzis, D. Vlougeogis / Euopen Jounl of Mehnis B/Fluids 8 () 7 p(,) Fig.. Pessue petution with L/R = nd () δ =., () δ =, () δ =. p(,) p(,) mthemtil deivtion, i.e., the pessue diffeene must e smll ut finite nd not infinitesimlly smll s speified y theoy. Seondly, lineised theoy n e used s omplimenty ppoh fo onsidele nge of pessue tios, fo whih othe omputtionl methods, suh s the DSMC method, eome omputtionlly ineffiient. It is noted tht the omputtionl effiieny of the pesent DVM lgoithm is ptilly independent of the pessue diffeene.. Conluding emks Refied gs flow though ylindil tues due to smll pessue diffeene hs een investigted y the lineised BGK model sujet to Mwell diffuse-speul oundy onditions. The kineti equtions e solved y implementing n effiiently pllelised nd memoy stoge hndling disete veloity lgoithm. The investigtion oves flow in the fee moleul up to the slip egime though tues of vey shot (inluding oifies) up to modete lengths (L/R = ). The quntittive ehviou of the flow te nd the mosopi distiutions of pessue (o density) nd veloity is emined in detil in tems of the eftion pmete δ nd the tue length tio L/R. The flow te is monotonilly inesed fo L/R elow, whee the Knudsen minimum shows up in the nge δ =... In the se of oifie flow (L/R = ), it is onfimed tht the esults e ptilly independent of the ommodtion oeffiient, mking this flow onfigution vey dvntgeous fo enhmking nd ompison with epeimentl dt. Comping the pesent line esults with oesponding non-line ones, it is seen tht line nlysis n ptue the oet ehviou of the flow field in the fee moleul nd tnsition egimes not only fo infinitesimlly smll ut lso fo smll ut finite pessue diffeenes nd it is gued tht its nge of ppliility is wide thn epeted. This is vey impotnt euse line nlysis hs solid nd well known theoetil kgound, whih is vey helpful in the oust nd elile numeil solution of the govening kineti equtions. Finlly, the intodued eo when the ssumption of fully developed flow is pplied to hnnels of modete length is estimted y ompison with oesponding esults is pefomed. It is hoped tht the pesent wok my e useful in engineeing pplitions s well s in ompisons with epeimentl esults whih e vey limited in efied flows in the tnsition egime due to smll pessue diffeenes.

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