VISUALIZED DEVELOPMENT OF ONSET FLOW BETWEEN TWO ROTATING CYLINDERS

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1 ISFV14-14 h Inernaional Symposim on Flo Visalizaion Jne 1-4, 1, EXCO Daeg, Korea VISUALIZED DEVELOPMENT OF ONSET FLOW BETWEEN TWO ROTATING CYLINDERS Takashi Waanabe.*, Yorinob Toya**, Shohei Fjisaa* *Gradae School of Informaion Science, Nagoya Universiy, Nagoya Japan ** Deparmen of Mechanical Engineering, Nagano Naional College of Technology, Nagano Japan KEYWORDS: Main sbjec(s): roaing flo, flo developmen, Taylor-Coee flo Flid: incompressible flid Visalizaion mehod(s): compaional flid dynamics, pos processing Oher keyords: Ginzbrg-Landa eqaion, finie lengh cylinders ABSTRACT : The onse flo developing beeen a roaing inner cylinder and a saionary oer cylinder is invesigaed by he nmerical approach and he Ginzbrg-Landa (GL) eqaion model. This flo is ell knon as Taylor-Coee flo. The lenghs of cylinders are finie and he boh end of he cylinders are saionary alls. When he lenghs of he cylinder are finie, he pichfork bifrcaion changes he flo from Coee flo o Taylor vorex flo a a criical Reynolds nmber. In case ha he cylinder lenghs are finie, he effec of he end alls are no negligible even he cylinder is long, and he Ekman layer develops a loer Reynolds nmbers and i indces vorex flo near he end all. The iniial perrbaion of flo developing in an infinie space is modeled by he GL eqaion. In his paper, e invesigae he applicabiliy of he nseady one-dimensional GL eqaion model o Taylor vorex flo developing from res and esimae he effec of end alls. The modeling parameers are he iniial and final amplides and he ime consan. The ime variaion of he velociy componens are fied by he solion of he model eqaion. The resl says ha he ime consan drasically changes in he axial direcion of he cylinders. Insead, he change is relaively gradal in he radial direcion. 1 Inrodcion The flos beeen coaxial roaing and/or saionary cylinders have been exensively invesigaed by he heoreical, experimenal and nmerical approaches [1]. This flo is knon as Taylor-Coee flo, and he infinie cylinder lenghs are assmed in he ideal condiions. This ype of flos can be fond in many flid machinery and chemical reacors, and he invesigaion of he flo behavior is imporan for no only he scienific ineress b engineering areas. In he realisic flo, he lenghs of he cylinders are finie and he effec of he end alls of he cylinders are no negligible []. The developmen of he perrbaions developing in a finie space is sally modeled by Ginzbrg- Landa (GL) eqaion of flo moion []. The early experimenal sdy by Gollb and Feilich [4] measred he radial velociy componen and esimaed groh of he perrbaion amplide ih he deviaion of he Reynolds nmber from is criical vales. They conclded ha he hird-order GL model is enogh o explain he flo behavior and he sloing don is effec is ell prediced, hile he higher model fails o explain he phenomena ell. Abshagen e al [5] invesigae he ime variaion ISFV14 Daeg / Korea 1 1

2 T. WATANABE, Y. TOYA, S. FUJISAWA of he velociy profile along he line consising of he midpoins among o cylinders. The nmerical mehod hey inrodced reflec he effec of he end alls by a homoopy parameer τ [,1], here τ = means he sress-free bondary condiion and τ = 1 corresponds o a solid saionary end alls. Their resl shos ha, hile he flo ih τ = follos Landa amplide eqaion, he nmerical and experimenal onse and decay of he radial velociy componen across he criical Reynolds nmber have differen ime consans given by he GL eqaion even he end all effec τ is small. Czamy and Lepo [6] and Manneville and Czarny [7] examined he flo for he larger aspec raio ha is he fracion of he lengh of he cylinder o he idh beeen o cylinders. The sdy on he GL model in Taylor vorex flo menioned above is limied o he flos a he midpoin of he cylinders. In his paper, e sdy enire region of he azimhal secions. In he follos, e explain or research mehod in sec. and give he resl in sec.. Finally, e conclde in Sec. 4 Research Mehod and Parameers Idenificaion of he Ginzbrg-Landa Eqaion.1 Flo Field and Nmerical Mehod We invesigae he Taylor flo ih a roaing cylinder and a saionary oer cylinder and he saionary end alls. The coordinae sysem sed here is he cylindrical coordinae shon in Fig. 1, hose origin is he cener of he boom end all. The geomerical parameers are he radis raio η of o cylinders and he aspec raio Γ defined by he raio of he cylinder lenghs o he cylinder gap idh. In his sdy e assme ha he radis raio η is The governing eqaion is he ime-dependen o-dimensional incompressible Navier-Sokes eqaion and he eqaion of coniniy: The reference lengh is he gap idh of he cylinder and he reference velociy is he circmferenial velociy of he inner cylinder. Then, e ge he non-dimensional form eqaions: 1 + ( ) = p + (1) Re z = () R o T Where = (, v, ) is he velociy vecor ih is R in componens in he radial, azimhal and axial direcion, is he ime, p is he pressre, Re is he Reynolds nmber based on he reference vales. The bondary Z p condiion is he non-slip condiion and i is formlaed as follos: L θ T = ( i,,) on he inner cylinder () Z lo = on he oer cylinder and end alls (4) These eqaions ogeher ih he bondary condiions are solved by he finie difference mehod ih hird order accracy for he convecion erm and he second Fig. 1 Coordinae Sysem erm accracy for he oher spaial erms. r ISFV14 Daeg / Korea 1

3 VISUALIZED DEVELOPMENT OF ONSET FLOW BETWEEN TWO ROTATING CYLINDERS. Ginzbrg-Landa Eqaion and Fiing Mehod The GL eqaion sed in his sdy is he hird order eqaion given by da A τ = ε A = d (5) A A( ) = e / T here A is he amplide, ε represen he relaive variaion from he criical poin, and τ and A are scaling facors[5]. The general solion of he GL eqaion is A f e + ( A / T f / A ) i. (6) 1 In his eqaion A i and A f are he iniial and final amplide, respecively, and T is he ime consan defining he groh rae of he amplide. We esimae he Ai, A f and T by fiing he eqaion o he ime series of velociy componens obained in he mehod in sec..1. Le f i ( i =,1, N 1) be he N-series of velociy componen. Then e inrodce he evalaion fncion N = 1 1 J ( fi A( )} (7) i= Where he A() is given by eq. (6). We find minimm poin of J ih respec o A i, A f and T, ha is, J J J =, =, = (8) T A i A f These eqaions are solved by he Neon s mehod. The ime range of he fiing is careflly seleced. When he vales are almos near he iniial vale or he final vale, hese vales may inclde some erroneos vales. Therefore, e firs fix he iniial and final ime poin and se he daa f i in he range of xmax / e f i xmax / e here x max is he maximm in he profile and e is he Napier s nmber. Some ypes of velociy profile ih ime are fond, and hey are shon in Fig.. Type 1 shos a monoonos decrease of increases. Type has exremm and hen he vale change exceeded is iniial vale. Type also have exremm b is vale does no exceed is iniial vale. Type 4 has more han one exrema. We change he fiing range of ime in each profile ype. Becase he GL eqaion is a model o capre he iniial developmen, e c off he ime series of daa a he peak of he fies exremm. This mehod gives niqe profiles and a fiing resl. (ⅰ) Type1 (ⅲ) Type (ⅱ) Type (ⅳ) Type4 Fig. Types of velociy variaion ih ime ISFV14 Daeg / Korea 1

4 T. WATANABE, Y. TOYA, S. FUJISAWA Resls.1 Onseing Flo from Res In his paper, e sho he resl a Γ = 4., ha is very shor cylinder. The flo sddenly sars a res, and he Reynolds nmber is ha is he criical Reynolds nmber for he onse of vorex flo in infinie lenghs of cylinders [8]. Figre represens he posiions a hich he profiles of he fied resl are shon. In his figre, he roaing inner cylinder is o he lef and he saionary oside is o he righ. Posiions (a) and (b) are near he inner all, posiion (c) is a he mid-locaion in he radial direcion, and posiion (d) is near he oer end of he flo region. For he convenience, e inrodce a o-dimensional coordinae (i, k) ha has is origin (, ) a he loer lef and he coordinae a he pper righ of (8, ) in Fig.. The fiing resls of he radial velociy componen are shon in Fig. 4. In hese resls, he red line shos he calclaed resl, green line shos he fied resl by he hird order solion given by eq. (6), and ble line represens he fied resl of he firs order exponenial solion. Every resl shos monoonos ime variaion and he hird-order fied crve gives good resl hile he firs-order fiing crve diverge a he earlier sage. This means ha he sloing don effec is favorable in he resls. The fied crve a (a) gives a shores ime consan han ohers. This is becase he Ekman layer begins o develop ih he roaion of he inner cylinder. As ill be noed laer, he posiion (c) is near he cener of a second Taylor vorex from he boom. The profile a his posiion shos a monoonic increase of he velociy componen hile anoher profile gives decreases. Hoever, all of hese flo profiles are ype (1) shon in Fig : real.4 : hird order : firs order (b) 1 (a) (i,k) = (1,1) A i = A f = , T = (b) (i,k) = (1,16) A i = A f = , T = 6.48 (a) (c) (d) Fig. Locaions here he fied resls are shon in his paper and he coordinae sysem in he azimhal secion (c) (i,k) = (4,8) A i = A f =. 1 -, T = (d) (i,k) = (7,1) A i = A f = , T =.8 Fig. 4 Time variaion of he radial velociy componen and heir fied crves ISFV14 Daeg / Korea 1 4

5 VISUALIZED DEVELOPMENT OF ONSET FLOW BETWEEN TWO ROTATING CYLINDERS The resl for he velociy componens are shon in Fig. 5. This case also ell modeled by he hird-order fiing crves. Please be sre ha he vale of he velociy componen a posiion (b) is an order of magnide smaller han hose a oher posiions. This is becase he posiion (b) is on he midline in he axial direcion and no axial velociy appears hen he flo is symmery ih respec o he axial direcion. The conor of he ime consan T in he azimhal secion is shon in Fig. 6. The pper lo and loer lo represen he resl of he radial velociy componen and he axial velociy componen, and from lef o righ, he T conor profile, ypes of velociy variaion (Fig. ) and he final resl of he flo vecors are shon. In he figre of he variaion ype, ble, cyan, yello and red color represen o Type 1,, and 4, respecively. The Ekman layer firs appears on he end alls. Therefore he ime consan of he radial velociy componen change is small near he corners beeen he inner cylinder and end alls. The ime consan near he end alls gros ih he radial posiions. This means ha he flo developmen propagae from he corners o he oer region. Apar from he end all, he profile of he vale of he ime consan in he radial direcion is small and he vorex flo gros ih an almos fixed speed. Alhogh he radial variaion is small, some drasic changes are fond in he axial direcion. The locaions of hese changes are shon ih he black arros on he lef side of he conor. These locaions almos correspond o he ceners of vorices fond in he final flo field. The drasic change in he axial direcion is de o he change of he ypes of he flo variaions ih ime. In Fig., he ype 1 ends o give larger ime consan T han ha of he ypes, and 4. When he variaion changes from ype 1 o ype, or 4 as he measring posiion changes in he axial direcion, he ime consan becomes small, and vice versa. The conor of he ime consan T of he axial velociy componen has o sors of lines corresponding o drasic changes. One is he line ha originaes from he corners beeen inner cylinder and end alls and penerae ino he almos cener region of he flo field. The oher appears a he je region here he radial flo formlaed by he vorices is an oer flo. Similarly ih he radial velociy componen, he ime consan depends on he ype changes of he velociy variaions.. Developing and Decaying Flo : real : hird order : firs order 1 (a) (i,k) = (1,1) A i = A f = , T = (c) (i,k) = (4,8) A i = A f = , T = (d) (i,k) = (7,1) A i = A f = , T =.4 Fig. 5 Time variaion of he axial velociy componen and heir fied crves (b) (i,k) = (1,16) A i = A f = , T = Abshagen e al [5] invesigaed he difference of he ime consans for he developing flo ih a sbcriical sae and for he decaying flo ih a criical sae. In his paper, e see he flo a he aspec ISFV14 Daeg / Korea 1 5

6 T. WATANABE, Y. TOYA, S. FUJISAWA raio Γ = 4. and he radis raio η = We have fond ha he criical Reynolds nmber for he onse of he inner vorices lies beeen Re = 4. and 44.. Figre 7 shos he ime consan obained from he radial and axial velociy componens of he developing and decaying flo beeen Re = 4. and 44.. The developing flo and he decaying flo have similar profiles of he ime consans. In deail, he ime consans of he developing flo are larger han hose of he decaying flo. This resl conradics ih ha obained by Abshagen e al. [5]. One reason for his conradicion is he difference is he aspec raio. In or case, he lenghs of he cylinders are shor and he effec of he end all is dominan o even o he inner vorices. (C) (C) (A) (B) (A) (B) 7.5 (a) Time consan of radial velociy componen. (b) Type of ime variaion of radial flo componen. (c) Finally formed for-vorex normal mode flo.. (d) Time consan of axial velociy componen. (e) Type of ime variaion of axial flo componen. Fig. 6 Time variaion of he axial velociy componen and heir fied crves. (e) Finally formed for-vorex normal mode flo (same figre as (c)). (a) Time consan of radial velociy componen for he developing flo. (b) Time consan of axial velociy componen for developing flo. (c) Time consan of radial velociy componen for he decaying flo. (d) Time consan of axial velociy componen for decaying flo. Fig. 7 Time consans in he developing and decaying flo beeen Re = 4. and 44.. ISFV14 Daeg / Korea 1 6

7 VISUALIZED DEVELOPMENT OF ONSET FLOW BETWEEN TWO ROTATING CYLINDERS 4 Conclsions The developmen of he flo beeen he roaing inner cylinder and he saionary oer cylinder is invesigaed by he nmerical approach. The lengh of he cylinder is very shor (Γ = 4.) and he gap idh is no small (η =.6667), and he effec of he end alls of he cylinders is large. When he onseing flo gros from res, he radial velociy componen firs begins o emerge near he corners of he inner cylinder all and he end alls. The conor of he ime consan T for he radial velociy componen has lines here he vales T changes drasically. These lines are almos locaed sraigh lines hrogh he ceners of he finally formed vorices. The ime consan of he axial velociy componens has o sors of lines: one begins a he corners beeen he inner cylinder and he end alls and he oher is locaed near he je of he flo field. The criical Reynolds nmber in or condiion lies beeen 4. and 44.. The developing and decaying flos beeen hese Reynolds nmbers are examined. The conor of he ime consan T have similar profiles for he developing and decaying flos, hile he vale of T is larger for he developing flo. This resl conradics ih ha obained in he previos ork by Abshagen e al. [5]. References 1. Tagg R, he Coee-Taylor problem, Nonlinear Science Today, Vol. 4, No., pp. -5, Dcher C and Mller SJ, Spaio-emporal mode dynamics and higher order ransiions in high aspec raio Neonian Taylor-Coee flos, Jornal of Flid Mechanics, Vol. 641, pp , 9.. Hirshfeld D and Rappor, DC., Groh of Taylor vorices: a moleclar dynamics sdy, Physical Revie E, Vol. 61, No. 1, pp. R1-R4,. 4. Gollb JP and Feilich MH, Opical heerodyne es of perrbaion expansions for he Taylor insabiliy, Physics of Flids, Vol. 1, No. 5, pp , Abshagen J, Meincke, P, Pfiser G., Cliffe KA and Mllin T, Transien dynamics a he onse of Taylor vorices, Jornal of Flid Mechanics, Vo. 476, pp. 5-4,. 6. Czamy O and Lepo RM, Time scales for ransiion in Taylor-Coee flo, Physics of Flids, Vol. 19 (7), pp , Manneville, P and Czarny O, Aspc-raio dependence of ransien Taylor vorices close o hreshold, Theoreical and Compaional Flid Dynamics, Vol., pp. 15-6, Koschmieder, EL, Bénard Cells and Taylor Vorices, Cambridge Universiy Press, 199. Copyrigh Saemen The ahors confirm ha hey, and/or heir company or insiion, hold copyrigh on all of he original maerial inclded in heir paper. They also confirm hey have obained permission, from he copyrigh holder of any hird pary maerial inclded in heir paper, o pblish i as par of heir paper. The ahors gran fll permission for he pblicaion and disribion of heir paper as par of he ISFV14 proceedings or as individal off-prins from he proceedings. ISFV14 Daeg / Korea 1 7