Mathematical model and finite-volume solution of a three-dimensional fluid flow between an eccentric cylinder and a cone

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1 INTERNATIONAL JOURNAL OF MATHEMATICAL MODEL AND METHOD IN ALIED CIENCE olum 6 Matmatcal modl and fnt-volum soluton of a t-dmnsonal flud flow btwn an ccntc cylnd and a con E. Konava L. avn A. Konav. Akpov Abstact T goal of ts pap s to dtct and to justfy t basc paamts and opatonal condtons n ydodynamc sals wt tn lay flud flow wn t dmnsonalty of spac nflunc of nta and vscous focs n consstnt cannot b nglctd. T matmatcal modl of t-dmnsonal nfocd and sa flud flow n an ccntc cannl btwn a statonay out con and a otatng nn cylnd s basd on t Nav-toks quaton and t contnuty quaton. uc flows occu n t ydodynamc sals and flud flm bangs. On t bass of smlaty toy and dmnsonal analyss t sgnfcanc of t tms n t Nav toks quaton and t contnuty quaton w consdd wt t valus of smlaty cta suc as ynolds numb Eul numb and ots and gomty paamts suc as t ccntcty and concty paamts. T numcal soluton and smulaton pogam a basd on t fnt-volum mtod wt t calculaton scms psntd n ts pap. T sults psnt vlocty and pssu flds togt wt som ntgal caactstcs.g. lakag as a functon of concty and ccntcty. T sults of numcal solutons w compad wt sults of wdly known analytcal solutons and wt t sults calculatd by ot numcal mtods. Kywods Con-cylnd gap fnt volum ncompssbl flud Nav-toks quaton. I. INTRODUCTION ATHEMATICAL modlng of nfocd and sa flows of Mvscous fluds n gaps of vaous gomty s a topcal quston n ydodynamcs. Among xampls of suc flows a flows n noncontact ydodynamc sals and flud-flm Ts wok was suppotd n pat by t Mnsty of Educaton and cnc of t Russan Fdaton und t pojcts No. 9..4/K and No. 363 of a pat of t stat task fo t "tat Unvsty - EC" n and und gant of t sdnt of t Russan Fdaton No 4.Z6.4.6-МК. E.. Konava s wt t Educaton-cnc-sac nsttut of nfomaton tcnologs tat Unvsty Educaton-cnc-oducton Complx Oyol 3 Russa cospondng auto to povd pon: ; -mal: lnoks_box@mal.u. L. A. avn s wt t Faculty of Nw Tcnologs and oducton Automaton tat Unvsty Educaton-cnc-oducton Complx Oyol 3 Russa -mal: savn@ostu.u. A.. Konav s wt t Faculty of Nw Tcnologs and oducton Automaton tat Unvsty Educaton-cnc-oducton Complx Oyol 3 Russa -mal: usako@nbox.u... Akpov s wt t Faculty of Nw Tcnologs and oducton Automaton tat Unvsty Educaton-cnc-oducton Complx Oyol 3 Russa -mal: vapov@nbox.u bangs wc a wdly usd n mcancal ngnng mtallugcal and ockt ndusty. It s wdly known tat t a many lf-tm and lablty qumnts fo sals and bangs [] []. Matmatcal and smulaton modls of t t dmnsonal nfocd and sa flow of vscous ncompssbl flud n t gap btwn t stady stat con and t otatng ccntc cylnd a studd n ts atcl. T man quatons n ts matmatcal modl a t Nav- toks quaton and t contnuty quaton [3]. T tdmnsonal flow n t cannl und study t tcknss of wc s vaabl n all dctons dos not allow to us tadtonal appoacs to fom a poblm of t tn lays of flud flow n t flud-flm bangs and t ydodynamc sals. Fo xampl as t wll b sown blow t s mpossbl to omt nta tms and som dsspatv tms of t Nav- toks.. t s mpossbl to us t ynolds quaton [3] o vaaton appoac dscbd n [4] [7]. In ts cas t psnt sac focuss on t numcal soluton of t Nav-toks quaton systm consdng all t nta and dsspatv tms. II. MODELING A. Matmatcal modl T flow of t vscous ncompssbl Nwtonan flud n t confuso s und nvstgaton. T flow gon s fomd by t statonay tuncatd con stato and t otatng cylnd oto wc a sown n Fg.. T Con as ad R and R spctvly. T Cylnd wt adus s offcntd n con and s otatng at a constant angula vloctyω. Und pssu t flud flows fom on nd towads t cannl snkag and scaps fom t ot nd und pssu. IN:

2 Fg. gomty of t cannl T flud s assumd to fll up t wol cannl t flow s lamna. T tmpatu s assumd as constant. T Nav-toks quaton and t contnuty quaton a t fundamntal quatons wc dscb t flow pocss [3] [] and n tnso fom look lk: D σ w s a dnsty of flud s a vlocty vcto s a gadnt of t pssu σ D s a stss dvato s a vlocty dvgnc s a tnso poduct. T stss dvato s dtmnd by t Nwton's gnald ypotss: σ µd D w µ s a coffcnt of t dynamc vscosty D s a stan-at dvato. Fo t ncompssbl mdum: D T. 3 Du to t nondmnsonalaton by mans of caactstc quantts systm n cylndcal coodnats can b aangd as follows: Eu Eu Eu 4 w L a non-dmnsonal adal and axs coodnats a nondmnsonal adal tangntal and axs componnts of t vlocty vcto s a non-dmnsonal pssu functon ηγ ηγ γ L R R L γ η a non-dmnsonal coffcnts and gomtc paamts µ Eu a ynolds and Eul numbs spctvly s a caactstc vlocty R j j s a adal claanc s a ccntcty. T quatons of t cannl boundas may b sown as follows: ]. [; sn cos π α ηγ γ α ηγ R INTERNATIONAL JOURNAL OF MATHEMATICAL MODEL AND METHOD IN ALIED CIENCE olum 6 IN:

3 INTERNATIONAL JOURNAL OF MATHEMATICAL MODEL AND METHOD IN ALIED CIENCE olum 6 ystm 4 can b solvd wt bounday condtons smultanously. T bounday condtons fo t vlocty componnts can b psntd as follows: ω R R R. Fo pssu functon on t nds of canal:. 6 Bcaus t flud flow canal s closd fom t tangntal coodnat dcton t podcal condtons can b dscbd as follows: F F π B. Analyss of modl F F π. 7 As was sad bfo t flows n noncontact sals and fludflm bangs a nvstgatd so t flow tcknss s vy small. T st of man paamts and ts od of magntuds a psntd n Tabl I. I T man paamts od of magntuds aamt Low lvl Upp lvl m - - m L m - - n pm Δ a 7 μ a s - kg/m 3 3 Usng od-of-magntud analyss t s asy to dtmnng wc tms n t quatons a vy small latv to t ot tms [] [3] [6] [7]. T valus of t tms of quatons 4 a psntd n Tabl II t gomty paamt coffcnt doman s 4 to. In od to coffcnt two cass avalabl fstly f t concty paamt as t sam magntud wt t latv gap η and scondly f t concty paamt xcds t latv gap η by on o mo ods of magntud. Also t Eul numb Eu and t ynolds numb ods of magntud a consdd n follow conclusons: - f t concty paamt s lss tan 3 and t ynolds numb s lss tan tn t vlocty adal componnt t ntal tm and t vlocty componnts dvatvs n t tangntal and axs dctons a nglgbl; - f t ynolds numb s mo tan tn t ntal tms and t vscosty tms as t sam magntud and f t concty paamt s mo o qual to 3 all t vlocty componnts as sgnfcant valus. II Equaton 4 tms od of magntuds T Nav-toks quaton Intal tms Dsspatv tms D σ Eu Eu - Eu - - T contnuty quaton Accodng to t od-of-magntud analyss wdly usd assumptons of ydodynamc toy of lubcaton [3] [6] a accptabl f t concty paamt s lss tan 3 and tat s t flow btwn two cylnds actually. In ts study t s ncssay to consd t Nav-toks quaton n ts complt fom. Tus t matmatcal modl of t sacd pocss as a look 4-7 and conssts of fou nonlna patal dffntal quatons wt fou unknown functons. III. NUMERICAL CALCULATION Numcal calculatons of quatons 4-7 a basd on t fnt o contol volum mtod F..M.. By mans of t F..M. t s possbl to gt an adquat soluton vn fo a cud ms bcaus of guaantd fulfllng of t fundamntal laws of consvaton [8] [9]. Equatons 4 n tnso fom look as follows: Eu w a non-dmnsonal Hamltonan opato. Accodng to t flow gon gomty t lmnt s by O dcton s vaabl and dpnds on t O coodnat. T lmnt ss masud wt t O and O coodnats a constant. Fg. as t dsctaton pncpl vsualaton n cas of t coaxal flow gon IN:

4 INTERNATIONAL JOURNAL OF MATHEMATICAL MODEL AND METHOD IN ALIED CIENCE olum 6 a a a E a a n a s ω N E a k a m EuΔΔ K M Δ Δ n s a n an n a n nω an a ω n q N Δ Δ Δ Δ N ank anm kn Eu mn a k k akk akωωk ank Nk ask k k K a Δ K kk a m Eu kω b b ω b ω n b n s b s k b k m m b Fg. Flow gon dsctaton: a -adal and axal scton and b 3-d F fo Nav-toks quatons axs O Accodng to appoac [8] - [] t followng opaton s t volum ntgaton of quatons 8 n ac fnt volum: dω d dd sown n t Fg. n a cuvlna coodnat systm. Usng t Ostogadsk fomula t s possbl to dcas t dgt of t dvatvs of t vlocty vcto: n d Eu dω Ω n d n d. w n s a unt nomal vcto on t spctvly sufac of F. Wn w calculat sufac ntgals on ac F sufac Fg. and us t man-valu tom vlocty componnts n t Nav-toks quatons can b appoxmatd by t xponntal functons.g. componnt of t fst quaton n ac F [ ; ] looks as follows: xp. xp Δ Aft tat systm 9 tuns to: 9 w E FE a E xp E F a n n xp n n a F xp a s F s xp s s F xp Δ a k m E k k F m k m a xp m m Δ a a E a a n a s a k a m F E E ΔΔ F ns ΔΔ ns F k m Δ Δ E Δ km E n s Δ Δ tc. n s k m km ystm s a dsct analogu of 8 and som of ts coffcnts nclud unknown functons dsct soluton. Du to t Nav-toks quaton consstng non-lna tms t soluton sac pocdu s tatv. Usng t fst t quatons of t systm wt an appoxmat pssu dstbuton an appoxmat vlocty dstbuton s calculatd.. t quaton systm s solvd of t followng fom: A f Ω IN:

5 INTERNATIONAL JOURNAL OF MATHEMATICAL MODEL AND METHOD IN ALIED CIENCE olum 6 w A a j a coffcnts bfo ncmnts of vlocty componnts n t Nav-toks quaton n all nods of dsct flow gon - appoxmat pssu dstbuton - t valus of t vlocty componnts on t Ω bod of t aa. T soluton of ts quaton systm s mplmntd usng t Gauss dl mtod t convgnc s povdd by t suffcnt attbut [8]: a - fo all quaton a j and a > - at last on quaton a j T fulfllmnt of ts condton s povdd by t fatu of t matx A coffcnts fo wc t dagonal lmnts qual t sum of sd lmnts wt all t cospondng unknown componnts of vlocty fo nstanc fo t fst quaton: a. a j T fulfllmnt of t scond condton s povdd by t known componnts of vlocty on t bounday of t aa.. by t bounday condtons sttng so t cospondng sd componnts mov to t gt-and pat tus dcasng t sum of t manng â j. Nxt t systm can b wttn fo ac unknown functon psntd n a fom of a sum of ts valu on t pvous taton and som ncmnt F : F F. F T sults of t o taton can b takn as t soluton of som asymptotc poblm. Tn t quaton systm fo t ncmnts F calculaton on vy taton can b obtand by mans of subtacton of t cospondng quatons on t pvous stp fom t fst t quatons of t systm on t stp addng t contnuty quaton to clos t systm on t stp: a ae a ω an N as a k K am M be E be a n a n n anω an a n ωn q s ank anm bn N bn kn mn ak k ak k akω ωk ank Nk ask k ak kk a m bk K bk c ω n n s s k k m c ω c c c c c ω n s k c ω c c c k cm m n s w b E EuΔΔ b N EuΔΔ b. Δ K Eu k kω m c ω Δ Δ ω c n s ΔΔ c.δδ k m ω. It s mo convnnt to wt ts systm down n a matx fom: A C B f w B a coffcnts bfo ncmnts of pssu n t Nav-toks quaton C a coffcnts bfo ncmnts of vlocty componnts n t contnuty quaton f s a gt-and pat of t dsct analogu of t contnuty quaton wc ncluds t valus of t vlocty componnts on t pvous taton. Matx ncluds os block bcaus of t contnuty quaton so ts matx dtmnant appoacs o and ts nvson s dffcult to ac. As t sult t systm of ts quatons may b solvd as follows: xpss n tms of t vcto of vlocty ncmnts n t fst quaton of and st t n t scond quaton. Hby w can n t fst plac fnd pssu ncmnts and tn - vlocty componnts ncmnt. T coffcnts of t systm of quatons dsct analogu nclud xponntal functons of t vlocty componnts appoxmaton Fg.3. Fut tos xponntal functons can b appoxmatd by mans of t pow functons as polynomals of fft od [8]. H t coffcnts wc a fst on t lst a sown as follows: E f E AE < E f E. E f E E [ ;] a E. E f E E ;] E > <. f [ ;] a f. f ;] f > Fg. 3 t coffcnt appoxmaton IN:

6 INTERNATIONAL JOURNAL OF MATHEMATICAL MODEL AND METHOD IN ALIED CIENCE olum 6 T appoxmaton accuacy n t ang und factos study s lss tan.% and suc substtuton sgnfcantly dcass t calculaton tm. I. DICUION Blow som smulatd sults fo t vscous ncompssbl flud nfocd and sa flow n t ccntc gap btwn t out con and t nn cylnd wt nput data a psntd s Tabl III. III T nput data dfnton ssu dop Δ Fquncy n a pm Radus m Gap m Lngt L m Eccntcty m -4.. Concty Dnsty scosty μ kg/m 3 a s T vlocty dstbutons axal componnt wt spct to lngt and tcknss bot n t gon of t maxmal gap s psntd n Fg. 4. As t can b sn n Fg. 4 t maxmum vlocty valu s acd on t lp of t cannl. Consdng t smulaton sults t s stablsd tat t ccntcty ncas lads to t nonlna lakag ncas. T concty paamts ncas wt a fxd gap at t nlt of t cannl lads to t dop n lakag Fg.6. Also fo t cas of t nfocd flow n t small nono concty and t o ccntcty gon smulaton t sults w compad wt t appoxmat soluton of G. Nktn [6] [7]. T sult of ts compason s about % o n dg gon concty owv t o ncass as t apx angl of t con ncass. Fg. 6 Lakag as a functon of concty and ccntcty Fo t cas of coaxal and o concty flow gon t sults w compad wt a wll-known analytcal soluton and wt ot sults smulatd by t fnt lmnt mtod FEM and t fnt dffnc mtod FDM. T sults of suc compason of vaous mtods smulaton a psntd n tabl I. Fg. 4 axal vlocty componnt: a - ov a nomal and tangntal coodnat at t nlt and b - ov a nomal and axal coodnat I T smulaton o Numb of lays toug tcknss 7 Und constans of t axal pssu dffnc and t nn cylnd otaton pssu n t axal dcton s nonlna wt xtmum pont as sown n Fg. a. Also n Fg. b t pssu appaanc n t tangntal dcton wt t maxmum pont n t tnnst gap gon wc dfns t bang capacty of t lubcant lay. Fg. ssu functon: a - n t axal and b - tangntal dcton Max. o FDM % Max. o FEM % Max. o FM % Obvously t F mtod as smallst pcntag of o and adquat sults vn on a cud ms.. CONCLUION o t pap psnts t matmatcal modl of a vscous ncompssbl flud flow n a cannl btwn a statonay con and an ccntc otatng cylnd. Ts modl s dffnt fom t known modls of t flud flow n t flud-flm bangs [3] [6] [8]-[] n t followng ways: fstly t allows to consd a smultanous acton of t pssu and t sa flow scondly t taks a vaabl cannl gomty IN:

7 INTERNATIONAL JOURNAL OF MATHEMATICAL MODEL AND METHOD IN ALIED CIENCE olum 6 nto account and t fnally allows to consd t nflunc of t nta focs on t basc pyscal valus flds. T matmatcal modl was ntally tansfomd nto a nondmnsonal fom wc allowd to mplmnt an analyss of t nflunc of vy tm of t quatons. It was dtmnd tat fo t flows n a cylnd-con cannls w t consty paamt s not g tan < 3 t ynolds assumptons [3] a vald gadng t smallnss of t nomal vlocty componnt and t vlocts dvatvs ov t tangntal and axal coodnats and gadng t nsgnfcant cang n pssu ov a lays tcknss. Fo t cass wn consty 3 and t numb > t s ncssay to consd t nta tms and t nflunc of t nomal vlocty componnt; t s also mpotant to consd t cang n pssu ov t lays tcknss and t cannls lngt. As a sult t solutons basd on t psntd matmatcal modl n lgt of ts complxty w obtand numcally. T dsct analogu of t modl was obtand basd n t mtod of fnt volums ffctvnss of wc s causd by mplmntaton of t consvaton laws n vy lmntay volum wc maks t dffnt fom all ot ms mtods. Usng spcfc xampls t was sown tat calculaton scms basd on t FM allow to obtan adquat sults vn on a cud ms. [] A. Z. and A. Al-af Flow Btwn Fnt tady Rotatng Eccntc Cylnds. Totcal and Computatonal Flud Dynamcs vol pp. 8. [3] C. u L. Wang Y. T. Cw and N. Zao Numcal study of ccntc Coutt-Taylo flows and ffct of ccntcty on flow pattns. Totcal and Computatonal Flud Dynamcs vol. 8 4 pp [4].. Baanan Adjustmnt of Dsspatv Tms to Impov Two and T- Dmnsonal Eul Flow olutons. WEA Tansactons on Flud Mcancs vol. pp. -4. [] H. Abbass. Tuk and.b. Nasalla Intpolaton functons n contol volum fnt lmnt mtod Computatonal Mcancs vol.3 3 pp [6].A. Matsnkovsky Roto vbaton of cntfugal macn: Hydodynamcs of stcto canals Ukan: umdu. [7] G.A. Nktn Goov and labynt sals of ydaulc agggats Moskow: Masnoston 98 n Russan. [8] L. avn A. Konav and E. Konava Totcal aspcts of modlng flud flm flow n jounal bangs and sals. XIIt Intnatonal cntfc-tcncal confnc als and salng tcnology of macns and dvcs. Woclaw Kudowa Zdoj oland pp [9] Ca-Wan Cang-Jan and Cao-Kuang Cn Nonlna dynamc analyss of a flxbl oto suppotd by mcopola flud flm jounal bangs. Intnatonal jounal of ngnng cnc vol pp. -7. [] L. avn E. Konava and A. Konav Modlng of flud flow n t gap of con-cylnd sals. XIIt Intnatonal cntfc-tcncal confnc als and salng tcnology of macns and dvcs. Woclaw Kudowa Zdoj oland pp ACKNOWLEDGMENT T autos would lk to tank to oganng commtt of t 9t Intnatonal Confnc on Ccuts ystms Communcatons and Computs CCC fo t possblty fo us to publs ts atcl n t NAUN jounal. REFERENCE [] A.. Ivanov.A. Koobcnko and A.. ostak Dsgn of t sals of pumps wt nd and tubo pump assmbls n lqud ockt ngn oon tat Unvsty n Russan. [] W. Todd Lndsy and Daa W. Clds T ffct of convgn and dvgn axal tap on t otodynamc coffcnts of lqud annula pssu sals. Toy vsus xpmnt AME pp [3] H. Yuko Hydodynamc Lubcaton. Tokyo: png 6 p. [4] L. avn A. Konav and E. Konava aatonal pncpl n t ydodynamc lubcaton toy. Intnatonal Jounal of Matmatcal Modls and Mtods n Appld cncs vol. 9 pp [] G. A. Kon and T. M. Kon Matmatcal Handbook fo cntsts and Engns. Dov ublcatons. [6] L.M. Mln-Tomson Totcal ydodynamcs. Fous dton London: Macmlan and Co LTD 96. [7] M.G. Kanovc Flm lubcaton bangs. Moskow 963 n Russan. [8].. atanka Numcal Hat Tansf and Flud Flow. McGаw- Hll Hmsp ublsng Copoaton 98. [9] J. L and Z. Cn A nw stabld contol volum mtod fo t statonay toks quatons. Adv. Comput. Mat. vol. 3 9 pp. 4. [].H. Cou Analyss and convgnc of a covolum mtod fo t gnald toks poblm. Mat. Comput. vol pp.8 4. [].R. abbag-yad M.T. Alkams M. Esmal and N.E. Mastoaks Fnt volum analyss of two-dmnsonal stan n a tck pp wt ntnal flud pssu. Intnatonal Jounal of Matmatcal Modls and Mtods n Appld cncs vol. 8 pp IN: