Instability of Taylor-Couette Flow between Concentric Rotating Cylinders

Transcription

1 Dou, H.-S., Khoo, B.C., and Yeo, K.S., Instability of Taylo-Couette Flow between Concentic otating Cylindes, Inte. J. of Themal Science, Vol.47, 008, Vol.47, No., Instability of Taylo-Couette Flow between Concentic otating Cylindes Hua-Shu Dou, Boo Cheong Khoo, and Khoon Seng Yeo Temasek Laboatoies, National Univesity of Singapoe, Singapoe 7508 Depatment of Mechanical Engineeing, National Univesity of Singapoe, Singapoe Abstact The enegy gadient theoy is used to study the instability of Taylo-Couette flow between concentic otating cylindes. This theoy has been poposed in ou pevious woks. In ou pevious studies, the enegy gadient theoy was demonstated to be applicable fo wallbounded paallel flows. It was found that the citical value of the enegy gadient paamete K max at tubulent tansition is about fo wall-bounded paallel flows (which include plane Poiseuille flow, pipe Poiseuille flow and plane Couette flow) below which no tubulence occus. In this pape, the detailed deivation fo the calculation of the enegy gadient paamete in the flow between concentic otating cylindes is povided. The calculated esults fo the citical condition of pimay instability (with semi-empiical teatment) ae found to be in vey good ageement with the expeiments in the liteatue. A possible mechanism of spial tubulence geneation obseved fo counte-otation of two cylindes can also be explained using the enegy gadient theoy. The enegy gadient theoy can seve to elate the condition of tansition in Taylo-Couette flow to that in plane Couette flow. The latte easonably becomes the limiting case of the fome when the adii of cylindes tend to infinity. It is ou contention that the enegy gadient theoy is possibly faily univesal fo analysis of flow instability and tubulent tansition, and is found valid fo both pessue and shea diven flows in paallel and otating flow configuations. Keywods: Instability; Tansition; Taylo-Couette flow; otating cylindes; Enegy gadient; Enegy loss; Citical condition.

2 Nomenclatue A, A a, A coefficients s - A amplitude of the distubance distance m B, B a, B coefficients m s - D diamete of the pipe fo pipe flow m E total mechanical enegy of unit volume of fluid J m -3 h, gap width between the inne cylinde and the oute cylinde m H total mechanical enegy loss of unit volume of fluid due to viscosity in steamwise diection J m -3 K function of coodinates (dimensionless). K c citical value of K max fo instability (dimensionless). K max maximum of K in the domain (dimensionless). l half-width of the channel fo plane Poiseuille flow and plane Couette flow m n coodinate in tansvese diection m p static pessue N m - ² adius m 0 aveage adius of inne cylinde and oute cylinde m adius of inne cylinde m adius of oute cylinde m e eynolds numbe (dimensionless). s coodinate in steamwise diection m t time s T Taylo numbe (dimensionless). u velocity component in the main flow diection m s - u 0 velocity at the mid-plane fo plane Poiseuille flow (channel flow) m s - U aveage velocity in the flow passage m s - v velocity component in the tansvese diection m s - v' m Ad, amplitude of the distubance of velocity in tansvese diection m s - W wok done to the unit volumetic fluid by extenal J m -3 x coodinate in the steamwise diection m y coodinate in the tansvese diection m z coodinate in the spanwise diection m

3 adius atio, / angula coodinates ad speed atio, / dynamics viscosity Nm - s kinematic viscosity m²s - density of fluid kg m - ³ shea stess N m - ² angula velocity of the fluid ad s - angula velocity of the inne cylinde ad s - angula velocity of the oute cylinde ad s - a angula velocity of the inne cylinde afte splitting ad s - a angula velocity of the oute cylinde afte splitting ad s - d fequency of the distubance s -. Intoduction Taylo-Couette flow efes to the poblem of flow between two concentic otating cylindes as shown in Fig. [-4]. This teminology was named afte the woks of G. I. Taylo (93) and M. Couette (890). This poblem was fist investigated expeimentally by Couette (890) and Mallock (896). Couette obseved that the toque needed to otate the oute cylinde inceased linealy with the otation speed until a citical otation speed, afte which the toque inceased moe apidly. This change was due to a tansition fom stable to unstable flow at the citical otation speed. Taylo was the fist to successfully apply linea stability theoy to a specific poblem, and succeeded in obtaining an excellent ageement of theoy with expeiments fo the flow instability between two concentic otating cylindes [5]. Taylo s goundbeaking eseach fo this poblem has been consideed as a classical example of flow instability study [6-8]. In the past yeas, the poblem of Taylo-Couette flow has eceived enewed inteests because of its impotance in flow stability and the fact that it is paticulaly amenable to igoous mathematical teatment/analysis due to infinitesimal distubances [-3]. Fo the stability of an inviscid fluid moving in concentic layes, Lod ayleigh [9] used the ciculation vaiation vesus the adius to explain the instability while von Kaman [0] employed the elative oles of centifugal foce and pessue gadient to intepet the instability initiation. Thei goal was to detemine the condition fo which a petubation esulting fom an advese gadient of angula 3

4 momentum can be unstable. In his classic pape, Taylo [5] pesented a mathematical stability analysis fo viscous flow and compaed the esults to laboatoy obsevations. Taylo obseved that, fo small atio of the gap width to the cylinde adii and fo a given otating speed of oute cylinde, when the otation speed of the inne cylinde is low, the flow emains lamina; when the otation speed of the inne cylinde exceeds a citical value, instability sets in and ows of cellula votices ae developed. When the otating speed is inceased to an even highe value, the cell ows beak down and a tubulence patten is poduced. He poposed a paamete, now commonly known as the Taylo numbe, T e h / 0, to chaacteize this citical condition fo instability. Hee, e is the eynolds numbe based on the gap width (h) and the otation speed of the inne cylinde, and 0 is the mean adius of the inne cylinde and the oute cylinde. The citical value of the Taylo numbe fo pimay instability is 708 as obtained fom linea analysis. This value agees well with his expeiments [-3]. Fo Taylo-Couette flow, Snyde has given a semi-empiical equation fo the citical condition fom the collected expeimental data []. Esse and Gossmann have also given an analytical equation fo the citical condition by an simple appoximation, but a constant in the equation have to be fixed using the esult of linea stability analysis []. Howeve, the poblem of Taylo-Couette flow is still fa fom completely esolved despite extensive study [-7]. Fo example, the limiting case of Taylo-Couette flow when the atio of the gap width to the adii tends to zeo should agee with that of plane Couette flow. This includes two possibilities: eithe adius is infinite o gap width is vey small. Thus, the citeion fo instability should eflect this phenomenon. Thee ae some ecent woks tying to addess this issue to some degee of success [8-0]. One may obseves that Taylo s citeion is not appopiate when this limiting case is studied because plane Couette flow is judged to be always stable due to Taylo numbe assuming a null value using Taylo s citeion. This may be attibuted to the fact that Taylo s citeion only consideed the effect of centifugal foce, and does not include the kinematic inetia foce. Theefoe, it is eckoned to be suitable fo low e numbe flows with high cuvatue. Fo otating flow with highe e numbe and low cuvatue, the flow may tansit to tubulence ealie and yet does not violate Taylo s citeion. ecently, Dou [,] poposed a new enegy gadient theoy to analyze flow instability and tubulent tansition poblems. In this theoy, the citical condition fo flow instability depends both on the base flow and the distubance which agees with the expeimental obsevations. Fo a given distubance, the citical condition fo flow instability and tubulent tansition is detemined by the atio (K) of the gadient of total mechanical enegy in the tansvese diection to the loss of total mechanical enegy in the steamwise diection. Fo a given flow geomety and fluid 4

5 popeties, when the maximum of K in the flow field is lage than a citical value, it is expected that instability would occu fo some initial distubances povided that the distubance enegy is sufficiently lage. Fo plane Poiseuille flow (channel flow), Hagen-Poiseuille flow (pipe flow), and plane Couette flow (simple shea flow), the findings based on the theoy ae consistent with the expeimental obsevations; fo the expeimental detemined citical condition, K c = fo all the above mentioned thee types of flows below which thee is no occuence of tubulence. In these compaisons, the distibution of K was calculated fo each flow and the value of K c was obtained using the expeimental data at citical condition [,, 3]. The theoy also suggests the mechanism of instability associated with an inflectional velocity pofile fo viscous flows. The theoy has been extended to cuved flows with simila deivations to paallel flows and thee impotant theoems have been obtained [4]. This theoy has also been employed to study the viscoelastic flows whee the effect of elastic foce is dominating [5]. It should be mentioned that the enegy gadient theoy is a semi-empiical theoy since the citical value of K is obseved and detemined expeimentally and can not be diectly calculated fom the theoy so fa. In this theoy, only the citical condition fo the instability is sought afte and the detailed pocess of instability is not povided. In this study, we apply the enegy gadient theoy to analyze the Taylo-Couette flow between concentic otating cylindes, and aim to demonstate that the mechanism of instability in Taylo-Couette flow can be explained via the enegy gadient concept. Though compaison with expeiments, we show that the enegy gadient function K as a stability citeion is sufficient to descibe and chaacteize the flow instability in Taylo-Couette flow. We also show that plane Couette flow can be consideed as just the limiting case of Taylo-Couette flow when the cuvatue of the walls tends to zeo. Fo flow between concentic otating cylindes, the flow instability may be induced by otation of the inne cylinde o the oute cylinde. If it is induced by the fome, a Taylo votex cell patten will be fomed when the citical condition is violated as in the expeiments; if it is induced by the latte, Taylo votex cell patten will not occu and the flow may diectly tansit to tubulence when the citical condition due to inetia foce is eached as in plane Couette flow [-3, 6]. In this study, only the citical condition fo the fome situation is consideed/teated.. Enegy gadient theoy evisted Dou [] poposed a mechanism with the aim to claify the phenomenon of tansition fom lamina flow to tubulence fo wall-bounded shea flows. In this mechanism, the whole 5

6 flow field is teated as an enegy field. It is poposed that the gadient of total mechanical enegy in the tansvese diection of the main flow and the total mechanical enegy loss fom viscous fiction in the steamwise diection dominate the instability phenomena and hence the flow tansition fo a given distubance. It is suggested that the enegy gadient in the tansvese diection has the potential to amplify a velocity distubance, while the viscous fiction loss in the steamwise diection can esist and absob this distubance. The flow instability o the tansition to tubulence depends on the elative magnitude of these two oles of enegy gadient amplification and viscous fiction damping of the initial distubance. In [], moe detailed deivation has been given to exactly descibe this mechanism, and this theoy is temed as enegy gadient theoy. Hee, we give a shot discussion fo a bette undestanding of the wok pesented in this study. The equation of total mechanical enegy fo incompessible flow by neglecting the gavitational enegy can be witten as [], u ( p u t ) u ( u u). () Fo pessue diven flows, the deivatives of the total mechanical enegy in the tansvese diection and the steamwise diection can be expessed, espectively, as [-4], E n ( p (/ ) u n ) dn ( u ω) dn ( dn u) dn u ( u) n, () E s ( p (/ ) u s ) ( u ω) ds ds ( u) ds ds ( u) s, (3) whee ω u is the voticity. Since thee is no wok input in the pessue diven flows, the magnitude of the total mechanical enegy loss of unit volumetic fluid along the steamwise diection equals to the deivatives of the total mechanical enegy in the steamwise diection, that is H s E. (4) s 6

7 Fo shea diven flows, the deivatives of the total mechanical enegy in the tansvese diection is the same as Eq.(). The enegy loss of unit volumetic fluid along the steamwise diection equals to the deivatives of the total mechanical enegy in the steamwise diection plus the wok done to the fluid by extenal, H s E W s s, (5) whee W is the wok done to the unit volume fluid by extenal. Fo a given base of paallel flow, the fluid paticles may move in an oscillatoy patten in the steamwise diection if they ae subjected to a distubance. With the motion, the fluid paticle may gain enegy ( E ( H ) via the distubance, and simultaneously this paticle may have enegy loss ) due to the fluid viscosity along the steamline diection. The analysis in [, 4] showed that the magnitudes of E and H detemine the stability of the flow of fluid paticles. Fo paallel flows, the elative magnitude of the enegy gained fom the distubance and the enegy loss due to viscous fiction detemines the distubance amplification o decay. Thus, fo a given flow, a stability citeion can be witten as follow fo a half-peiod, and E E A F H n H s d u Ad K u v' K u m Const, (6) E K n. (7) H s Hee, F is a function of coodinates which expesses the atio of the enegy gained in a halfpeiod by the paticle and the enegy loss due to viscosity in the half-peiod. K is a dimensionless field vaiable (function) and expesses the atio of tansvesal enegy gadient and the ate of the enegy loss along the steamline. E V is the kinetic enegy pe unit volumetic fluid, s is along the steamwise diection and n is along the tansvese diection. H is the enegy loss pe unit volumetic fluid along the steamline fo finite length. Futhe, ρ is the fluid density, u is the 7

8 steamwise velocity of main flow, A is the amplitude of the distubance distance, d is the fequency of the distubance, and v' A is the amplitude of the distubance of velocity. m d Futhe, E A H E and H u ae the gadient of total mechanical enegy n s d of unit volumetic fluid in the tansvese diection and the loss of total mechanical enegy of unit volumetic fluid in the steamwise diection, espectively. It can be found fom Eq.(6) that the lage the value of F, the flow is moe unstable. Thee is a constant of F below which the flow emains stable. Fo given distubance, the value of K in the flow field detemines the flow stability. As stated ealie, the atio K is a dimensionless function of the flow field, i.e., a function of coodinates (x,y,z). Since K may vay with the flow paametes and the spatial space, the maximum of K in the domain, i.e., K max, should bound the stability of the flow fo given distubance. As such, a citical value of K max can be used to expesses the citical condition, and is given as K c. The enegy gadient theoy as descibed fo paallel flows in detail in [,], can be extended to the cuved flow [4], if we change the Catesian coodinates (x, y) to cuvilinea coodinates (s, n), to change the kinetic enegy ( m u ) to the total mechanical enegy ( E p u ) (the gavitational enegy is neglected) in the analysis fo the evolution of the distubed fluid paticle, and to make the velocity (u) along the steamline diection [4]. Hee, p is the hydodynamic pessue. Thus, afte these substitutions, the equation (6) and (7) ae also applicable fo cuved flows. These equations can be deived fom the fist pinciple via the same steps as in []. In tem of Eqs.(6) and (7), the distibution of K in the flow field and the popety of distubance may be the pefect means to descibe the distubance amplification o decay in the flow. Accoding to this theoy, it can be found that the flow instability fist occu at the position of K max, fo given distubance, which is constued to be the most dangeous position. Thus, fo a given distubance, the occuence of instability depends on the magnitude of this dimensionless vaiable K and the citical condition is detemined by the maximum value of K in the flow. Fo a given flow geomety and fluid popeties, when the maximum of K in the flow field exceeds a citical value K c, it is expected that instability can occu fo a cetain initial 8

9 distubance [,]. Tubulence tansition is a local phenomenon in the ealie stage, as found in expeiments [4]. Fo a given flow, K is popotional to the global eynolds numbe []. A lage value of K has the big ability to amplify the distubance, and vice vesa. The enegy gadient analysis has suggested that the tansition to tubulence is due to the enegy gadient and the distubance amplification [,], athe than just the linea eigenvalue instability type as expounded and stated in [6, 7]. Both Tefethen et al. [6] and Gossmann [7] commented that the natue of the onset-of-tubulence mechanism in paallel shea flows must be diffeent fom an eigenvalue instability of linea equations of small distubance. In fact, finite distubance is needed fo the tubulence initiation in the ange of finite e as found in expeiments [8]. Dou [,] demonstated that the citeion obtained has a consistent value at the subcitical condition of tansition detemined by the expeimental data fo plane Poiseuille flow, pipe Poiseuille flow as well as plane Couette flow (see Table ). Fo plane Poiseuille flow, both the two definitions of eynolds numbe ae given in Table because diffeent definitions ae found in liteatue. In linea stability analysis, e u 0 l / is geneally used. Hee, u 0 is the velocity at the centeline and l is the half width of the channel. Anothe definition of eynolds numbe, e UL /, is an analogy to that used in pipe Poiseuille flow. Hee, U is the aveage velocity in the channel and L is the width of the channel. Flow type e expession Eigenvalue Expeiments, K max at e c analysis, e c e c (fom expeiments), K c Pipe Poiseuille e UD / Stable fo all e Plane Poiseuille e UL / e u0l / Plane Couette e Ul / Stable fo all e Table Compaison of the citical eynolds numbe and the enegy gadient paamete K max fo plane Poiseuille flow and pipe Poiseuille flow as well as fo plane Couette flow [,]. U is the aveaged velocity, u 0 the velocity at the mid-plane of the channel, D the diamete of the pipe, h the half-width of the channel fo plane Poiseuille flow (L=l) and plane Couette flow. Fo Plane Poiseuille flow and pipe Poiseuille flow, the K max occus at y/l=0.5774, and /=0.5774, espectively. Fo Plane Couette flow, the K max occus at y/l=.0. It can be deduced fom Table that the tubulence tansition takes place at a consistent citical value of K c at about fo both the plane Poiseuille flow and pipe Poiseuille flow, and about 370 fo plane Couette flow. This may suggest that the subcitical tansition in paallel 9

10 flows takes place at a consistent value of K c This finding futhe suggests that the mechanism of flow instability occuing in these basic flows might be the same. In all the paallel flows, obsevation can be made that the tansvese velocity is v=0 and the pessue is constant in the tansvese diection. The vaiation of total mechanical enegy in the tansvese diection is only due to the kinetic enegy u (when the gavitational enegy is neglected). Theefoe, the gadient of the kinetic enegy is the possible souce of amplification of distubance along the tansvese diection. In the steamwise diection, the kinetic enegy is constant, the enegy loss is the pessue dop fo pessue diven flows o the input of the extenal wok fo shea diven flows, which sustains the velocity pofile to keep it constant in the steamwise diection fo lamina flows. Due to zeo tansvesal velocity, the diffusion of enegy in tansvese diection is zeo. Theefoe, fo all the thee paallel flows, including pessue diven flow and shea diven flows, the gadient of kinetic enegy in tansvese diection and the enegy loss along the steamline diection ae the dominating factos fo the flow stability. As such, this can be undestood that the mechanism of flow instability in these paallel flows is the same. It is also noticed that the citical condition fo flow instability as detemined by linea stability analysis diffes lagely fom the expeimental data fo all the thee diffeent types of flows, as shown in Table. Theefoe, linea stability analysis is not a good method to analyze the condition fo tansition to tubulence. Using enegy gadient theoy, it is obseved that the balance of the enegy amplification in tansvese diection and the enegy loss in steamwise diection eally dominate the flow stability. It is also demonstated that the viscous flow with an inflectional velocity pofile is unstable fo both two-dimensional flow and axisymmetic flow [9]. Fom above discussions, fo the plane Poiseuille flow, this said position whee K max > K c should then be the most dangeous location fo flow beakdown, which has been confimed by Nishioka et al s expeiment [30]. Nishioka et al's [30] expeiments fo plane Poiseuille flow showed details of the outline and pocess of the flow beakdown. The measued instantaneous velocity distibutions indicate that the fist oscillation of the velocity occus at y/h=0.50~0.6, as shown by the Fig. 4 in [30]. Nishioka et al. [30] measued the distibution of the instantaneous aveaged velocity in a peiod, and the esults indicate that the oscillation of the velocity always occus in the ange of y/h=0.50~0.6 fo the distubance imposed (the base flow keeps lamina). Fo the pipe flow, in a ecent study, Wedin and Keswell [3] showed the pesence of a "shoulde" in the velocity pofile at about /=0.6 fom thei taveling wave solution. They 0

11 suggested that this position coesponds to whee the fast steaks of taveling waves each the wall. It can be constued that this kind of velocity pofile as obtained by simulation is simila to that found in Nishioka et al's expeiments fo channel flows [30]. The location of the "shoulde" is about the same as that fo K max (at y/h=0.5774). Accoding to the pesent theoy, this "shoulde" may then be inticately elated to the enegy gadient distibution. The solution of taveling waves has been confimed by expeiments ecently [3]. In summay, the mechanism fo instability descibed by the function K is that it epesents the balance between the two oles of distubance amplification by the enegy gadient in the tansvese diection and distubance damping by the enegy loss in the steamwise diection. 3. Enegy Gadient Theoy Applied to Taylo-Couette Flow We shall assume that the distubance to the base flow is peiodic and wave length is elatively small compaed with the scale of the flow geomety. The base flow is assumed to be steady lamina flow. Whethe stability citeia ae witten fo the half-peiod o whole peiod would be the same since the two half-peiod ae skew-symmetical in a peiod. Fo the wall bounded flows consideed hee, the bounday conditions is non-slip. Fo the Taylo-Couette flow, assumptions on the base flow is that expessed by the basic solution (the following Eq.(8-). The distubance is assumed as peiodic along the steamwise diection of the basic flow (ie., along the cicula diection). Unde these assumptions, the same expession given peviously as Eqs.(6) and (7) can be deived fo the cicula flow between concentic cylindes [4]. 3. Velocity distibution fo Taylo-Couette Flow The solution of velocity distibution between concentic otating cylindes can be found in many texts, e.g. [-3]. Fistly, we define the components of the velocity in the tangential and adial diections as u and v, espectively. Assuming v=0 and 0, the Navie-Stokes equations in adial and cicumfeential diections fo steady flows educe to and u dp, (8) d

12 u u 0. (9) Integating Eq.(9) and using the bounday conditions gives the solution of the velocity field as, B u A, (0) whee A and B. () In Eq.(), / and /. is the adius of the inne cylinde and is the adius of the oute cylinde. and ae the angula velocities of the inne and oute cylindes, espectively. 3. Enegy gadient in the tansvese diection The gadient of the total mechanical enegy gadient in the tansvese diection is E ( p / u ) du u u d. () Intoducing Eq.(0) and Eq.() into Eq.(), the enegy gadient in the tansvese diection theefoe is E B B B B A A A A A. (3) 3.3 Enegy Loss Distibution fo Taylo-Couette Flow The following equation fo calculating the adial distibution of ate of enegy loss along the steamline fo Taylo-Couette flow is obtained as [33], dh du, (4) ds u d

13 3 whee is the shea stess. Equation (4) is applicable to flows fo one cylinde otating and the othe at est, and cylindes otating in opposite diections. Fo cylindes otating in the same diection, a diffeent equation must be used [33] d du u ds dh a a a a, (5) whee a u is the velocity in the flow field expessed by u u a assuming that and a is the shea stess in the velocity field expessed by a u. The details of the deivation fo dh/ds can be found in [33] and is not epeated hee. With the velocity gadient obtained fom Eq.(0), the shea stess in (4) is theefoe, B B A B A u u, (6) whee is the dynamic viscosity. Thus, we have 3 B (7) and B A B A B d du u. (8) Intoducing Eqs.(7) and Eq.(8) into Eq.(4), the enegy loss is 3 B B A B A B d du u ds dh 4 4 B A B B A B A B. (9) Fo cylindes otating in same diection, using the same pocedue as that deived fo Eq.(9), the equation can be obtained fom Eq.(5) as,

14 whee dh a dua ds u d a a 4B a A a a and a 4 A a B a, (0) a B a a, () and /, a a /a, a, and a 0. Hee, we have delibeately maintained Eq.(9) and Eq.(0) with simila fom fo the vey pupose that the deivations in subsequent sections below can use essentially the same equation, diffeing only in the coefficients A and B fo Eq.(9) and A a and B a fo Eq.(0). 3.4 Distibution of K Intoducing Eq.(3) and (9 o 0) into Eq.(7), the atio of the gadients of the total mechanical enegy in the tansvese diection and the loss of the total mechanical enegy in the steamwise diection, K, can be witten as, K E / H / s du u u d du u d B A A 4B B 4 A, () whee is the kinematic viscosity. In this equation, the calculations of A and B ae caied out using Eq.(). The evaluations of A and B ae diffeent fo counte otating and co-otating cylindes. Fo cylindes otating in opposite diections, A A and B B (calculated using Eq.()); fo cylindes otating in same diection, A Aa and B Ba (calculated using Eq.()). Intoducing Eqs.(0) and () o () into Eq.(), then simplifying and eaanging, Eq.() becomes, 4

15 5 4 4 K. (3) The evaluations of and ae diffeent fo counte otating and co-otating cylindes. Fo cylindes otating in opposite diections, and ; Fo cylindes otating in same diection, a and a. e-aanging, Eq.(3) can be ewitten as 4 4 K. (4) Using a moe appopiate fom by explicitly showing the eynolds numbe, h e, Eq.(4) can be expessed as e 4 4 h K, (5) whee h is the gap width between the cylindes. If the oute cylinde is at est ( 0 ), and only the inne cylinde is otating ( 0 ), then 0, 0, and. Futhe simplifying Eq.(5), we obtain e h K. (6)

16 Next, by letting y, Eq.(6) is ewitten as K e y e h h ( ) y y y y y y e h ( ) y h h y h h. (7) This equation easily elates to plane Couette flow. Plane Couette flow can have two configuations: two plates move in opposite diections and one plate moves while the othe is at est. Taylo-Couette flow with 0 and 0 coesponds to plane Couette flow fo the latte case. Fom Eq.(7), it can be seen that K is popotional to e in any location in the field. K is an eighth ode function of distance fom the oute cylinde acoss the channel, which is elated to the value of elative channel width h /. The distibution of K along the channel width between cylindes calculated using Eq. (7) is depicted in Fig. fo vaious values of h/ with the inne cylinde otating while the oute cylinde is kept at est ( 0 ). Fo a given e and h/, it is found that K inceases with inceasing y/h and the maximum of K is obtained at y / h fo low values of h/ (h/ <0.43). That is, it eaches its maximum at the suface of the inne cylinde. Fo highe value of h/ (h/ >0.43), the location of K max moves to within the flow located between y/h=0 and y/h=. The cases studied in the liteatue ae usually fo low gap width. We shall focus ou discussion fo the case of h/ <0.43 in this study. The maximum of K fo h/ <0.43 can be expessed as h max e ( ) h K. (8) It is found fom Eq. (8) that K max depends on eynolds numbe and the geomety. As we will see below, the citical stability condition will be detemined by Eq.(8). When the cylinde adii tend to infinity, we have in Eq.(7) 6

17 y h y h, ( ) 4, and 4 h. (9) h Then, Eq.(7) educes to y K e. (30) h This equation at the limit of infinite adii of cylindes is the same as that fo plane Couette flow [3]. The coesponding maximum of K at y=h is K max h e. (3) As discussed in [, 4], the development of the distubance in the flow is subjected to the mean flow condition and the bounday and initial conditions. The mean flow is chaacteized by enegy gadient function K. Theefoe, the flow stability depends on the distibution of K in the flow field and the initial distubance povided to the flow. In the flow egime with high value of K, the flow is moe unstable than that in the flow egime with low value of K. The fist sign of instability should be associated with the maximum of K (K max ) in the flow field fo a given distubance. In othe wods, the position of maximum of K is the most dangeous position. Fo given flow distubance, thee is a citical value of K max ove which the flow becomes unstable. It is not tivial to diectly pedict this citical value K c by theoy as in paallel flows [] since it is obviously a stongly nonlinea pocess and the usual tool fo petubation analysis is not applicable. Nevetheless, it can be obseved in expeiments as those done fo paallel flows. The value of K max when flow instability occus can be taken as a citeion fo instability, and this value is ecoded as K c ; if K max > K c, the flow will become unstable. Thus, the study of distibution of K in the flow field can help to locate the egion whee the flow is inclined to be unstable. In Fig., K inceases with inceasing y/h fo given h/ (at low value of h/ ), and its maximum occus at the inne cylinde. Thus, the flow at the oute cylinde is most stable and the flow at the inne cylinde is most unstable. Theefoe, a small distubance can be amplified at the inne cylinde if the value of K eaches its citical value fo the given 7

18 8 geomety. In othe wods, the inne cylinde is a possible location fo fist occuence of instability, as geneally obseved in the expeiments [5,6]. In Fig., the line fo h/ =0 coesponds to plane Couette flow wheein, one plate moves while the othe is at est, which is a paabola (i.e., Eq.(30)) [3]. It can be found that thee is little diffeence in the distibution of K fo h/ =0.0 and h/ =0. In tems of that view, one may expect that the citical conditions of instability fo these two values of h/ close to one anothe. When h/ inceases, K max deceases. This does not, howeve, imply that the flow becomes moe stable as h/ inceases. This is because the citical value of K max vaies with the vaiation of h/. It will be shown by expeimental data in subsequent sections that K c deceases with the inceasing h/. 4. Compaison with Expeiments at Citical Condition Taylo [5] used a gaph of / vesus / to pesent the esults of the citical condition fo the pimay instability. In ode to use the same chat as Taylo fo ease of efeence, the compaison of theoy with expeiments is also plotted in this way. ewiting Eq.(4), we have 4 4 K. (3) eaanging Eq.(3), the following equation (33) is obtained, 4 4 K. (33) Thus, the citical condition fo a given geomety is given by K c. That is 4 4 K c c. (34)

19 In Equation (34), K c is the citical value of K max at the pimay instability condition, which can be detemined fom expeiments. Fo a given flow geomety, K c is teated as constant fo the initiation of instability as descibed befoe. Afte the value of K c is detemined, the value of ( / ) can be solved by an iteation pocess fo an initial value of and a given value of c /. The calculated esults with the theoy ae compaed with available expeimental data in liteatue [5][][3][6] concening the pimay instability condition of Taylo-Couette flow. Figues 3 to 4 show the compaison of theoy with Taylo s expeiments [5] fo two paametic conditions, while Fig. 5, Fig.6 and Fig.7 show the compaisons of theoy with Coles expeiments [3], Snyde s expeiments [], and Andeeck et al s expeiments [6], espectively. In these figues, the citical value of the enegy gadient paamete K (K c ) is detemined by the expeimental data at 0 and 0 (the oute cylinde is fixed, the inne cylinde is otating). Using the detemined value of K c fo a given set of geometical paametes, the citical value of / vesus / is calculated fo a ange of / as in the expeiments using Eq.(34). In [], the constant in the analytical equation obtained was fixed using the esult of linea stability analysis. It can be seen fom Figs.3-7 then that when the cylindes otate in the same diection, the theoy obtains vey good concuence with all the expeimental data. When cylindes otate in opposite diections, the theoy obtains good ageement with the expeimental data fo small elative gap width (h/ ). Fo lage elative gap width, the theoy has some deviations fom the expeimental data with inceasing negative otation speed of the oute cylinde. The eason can be explained as follows. When the gap is lage and the cylindes ae otating in opposite diections, the flow in the gap is moe distoted compaed to plane Couette flow (linea velocity distibution). This distotion of velocity pofile has an effect on the distibutions of flow enegy loss and enegy gadient. The magnitude of flow enegy loss dh/ds can be calculated by Eq.(9 o 0) and the enegy gadient can be calculated by Eq.(3). On the othe hand, if the otating speed of the oute cylinde is high, the flow laye nea the oute cylinde may ealie tansit diectly to tubulence if the distubance is sufficiently lage [6][3], which has not been the focus of eseaches befoe. This will obviously alte the velocity pofile of the flow and influence the distibution of the enegy gadient function K and the maximum of K (moe discussion will be given in the paagaph below when Fig.8 is intoduced and discussed). Fo example, in Andeeck et al s expeiments [6], when / is -00 and the inne cylinde is at est, the eynolds numbe based on the otation speed of the oute cylinde e ( e h / ) eaches 46. At this value of 46, plane Couette flow has aleady become tubulent (e c = ). Fo counte- 9

20 otating cylindes with cuved steamlines, the tansition must occu ealie than that in plane Couette flow because of the influence of the adial pessue gadient which inceases the adial (tansvese) enegy gadient nea the oute cylinde. The same type of deviation in pediction is also obseved in the compaison of Taylo s mathematical theoy with his expeiments when cylindes otate in opposite diections at lage negative otating speed of oute cylinde; in paticula, if the elative gap is lage [5]. Theefoe, when cylindes otate in opposite diections, futhe study is needed to study the occuence of tubulence as induced by shea flow nea the oute cylinde (caused by convective inetia). This is compaed with the Taylo votex patten as induced by the centifugal foce nea the inne cylinde when only the inne cylinde is otating. It is found that the fist tem on the ight hand side of Eq.(34) is that fo ayleigh s inviscid citeion, and the second tem on the ight hand side of Eq.(34) is due to the effect of viscous fiction. If K c is zeo, Eq.(34) degeneates to the ayleigh s equation. In Figs.3-7, ayleigh s inviscid citeion ( c / ) is also included fo compaison. In Taylo s calculations and expeimental esults [5], it was shown that viscosity has only stabilizing ole to the flow between the concentic cylindes. Fo cylindes otating in the same diection, ou theoy shows vey good ageement with the expeiments and demonstates that viscosity has a stabilizing effect on the flow, as compaed to the inviscid case. Fo inviscid flow at / =0, ( / ) c =0. Fo the viscous flow, the viscosity plays a stabilizing ole and hence gives ise to a cetain theshold quantity of ( / ) not equal zeo, below which the flow is stable. The c mechanism of the stability ole of viscosity is the same as fo the paallel flows [-4]. Viscosity leads to enegy loss along the flow path which leads to the distubance damping when the distubance popagates between the fluid layes. Thus, viscosity inceases the stability of shea flows and enhances the value of ( / ). c In Fig.8, we show the distibution of K along the channel width at the citical condition of K c =77 as shown in Fig.4, which is taken fom Taylo s expeiments fo =3.55cm and =4.035 cm. The value of K c is only dependent on the atio of adius, and is independent of the otating speed of cylindes. It can be seen in Fig.8a that K inceases monotonically fom the oute cylinde to the inne cylinde, when the inne cylinde is otating while the oute cylinde is at est. The maximum of K occus at the inne cylinde, so the stability of the flow is dominated by the K max at the inne cylinde. In Fig.8b, it can be seen that K inceases monotonically fom the oute cylinde to the inne cylinde, when the two cylindes ae otating in same diection and / is lage than /. The maximum of K also occus at the inne cylinde, so the stability of the 0

21 flow is dominated by the K max at the inne cylinde too. In these two pictues, the base flow in the gap is lamina flow. Taylo votex cell patten ae found in these cases as shown in expeiments [5,6]. When the two cylindes otate in opposite diections, the distibution of K geneates two maxima espectively at the inne cylinde and the oute cylinde. In Fig.8c, it can be seen that the maximum at the oute cylinde is not high since the speed of the oute cylinde is small. In this case the base flow in the gap may be still lamina, and the stability of the flow is still completely dominated by the K max at the inne cylinde. If the speed of the oute cylinde becomes high and exceeds cetain citical value, the flow nea the oute cylinde may become tubulence povided that the distubance is sufficiently lage [6][3]. As shown in Fig.8d, the value of K at the oute cylinde (K=367) is about o highe than the citical value fo plane Couette flow to tansit to tubulence (K c =35-370), the flow laye nea the oute cylinde may aleady be tubulent. Thus, although the base flow is lamina nea the inne cylinde but the flow may have tansited to tubulence nea the oute cylinde, at this flow state which is the citical condition of pimay instability deemed dominated by the otation of inne cylinde. Theefoe, as this citical condition is exceeded with inceasing /, but when / is lage and negative, the usual Taylo votex cell patten may not mateialize; instead spial tubulence is geneated [3,6]. This is because the geneation of tubulence nea the oute cylinde has alteed the velocity distibution fom its oiginal lamina behaviou. The ciculation of fluid paticle between the two cylinde sufaces (altenating between the lamina egion and the tubulent egion) foms an intemittent and spial tubulence patten. This may povide a plausible explanation fo the obsevation of spial tubulence patten as found in expeiments [3][6]. As epoduced in Fig.9, Andeeck et al [6] plotted egimes of the flows in tems of o and i as coodinates (shown as Fig. in thei pape). Hee o and i ae the eynolds numbe based on the otating speed of oute and inne cylindes, espectively. The behaviou of the flow may be bette explained using the distibution of K along the gap width, as discussed above. In Figs.0 and, we show the isolines of the K max along the side of inne cylinde in the plane of / vesus / which occus on the suface of the inne cylinde. Because the enegy gadient dominates the flow behaviou and contols the mechanism of the flow instability and tansition, the classification of egimes may be bette undestood using the isolines shown in Figs.0-. By compaing Figs.0- and Fig.9, the egimes of the flow fom expeiments may appea to be aptly chaacteized by the isolines of K max along the inne cylinde in / and / plane. It should be noticed that the isolines of K max fo K max <K c ae exact, while the isolines fo K max >K c ae appoximate because the velocity distibution in the gap can not be

22 accuately expessed by Eq.(0) anymoe due to the fomation of Taylo votices o spial votices/ spial tubulence. It should be made clea that K max is the maximum of the magnitude of K in the flow domain at a given / and / condition and geomety, and K c is citical value of K max at the pimay instability fo a given geomety. It would be (most) inteesting to obtain a unified desciption fo otating flows and paallel flows vis-a-vis the mechanism of instability. As intoduced befoe, although the citical eynolds numbe diffes geatly in magnitude fo plane Couette flow, plane Poiseuille flow and pipe Poiseuille flow, the citical value of the K max is about the same fo all the thee mentioned kinds of flows (35-389). Plane Couette flow is the limiting case of Taylo-Couette flow when the cuvatue of walls is zeo. The limiting value of citical condition of Taylo-Couette flow should be the same as that fo plane Couette flows. Lundbladh and Johansson s diect numeical simulation poduced a citical condition of ec=375 fo plane Couette flow [34]. Anothe thee (independent) eseach goups also obtained ec=370 0 in expeiments via flow visualization technique duing the peiod [35-37]. Some subsequent expeiments showed a lowe citical eynolds numbe of 35 [38-39]. In ode to include all possible esults, the data can be classified as in the ange of fo plane Couette flow. Ou deivation has shown that K max =e fo plane Couette flow as indicated by Eq.(30). Using these data fo ec, the citical value of K max fo plane Couette flow is taken to be K c =35-370, below which no tubulence occus egadless of the distubance. Authos h h/ / c e c K c (cm) (cm) (cm) (cm - ) Taylo (93) Coles (965) Snyde (968) Gollub & Swinney (975) Andeeck et al (986) Hinko (003) Pigent & Dauchot (004)

23 Table Collected data fo the detailed geometical paametes fo the expeiments and the citical condition detemined fo the case of the oute cylinde at est ( 0 ) and the inne cylinde otating ( 0 ). In Table, expeimental data ae collated fo the citical condition of the pimay instability in the Taylo-Couette flows. A most inteesting esult fo small gap flow was obtained by Hinko [8] ecently. This esult is useful to claify how the Taylo-Coutte flow is elated to plane Couette flow. Hinko obtained e c =350 fo the flow in small gap of concentic otating cylindes with h/ =0.0. Unde this citical condition, the Taylo numbe is T=350 Χ0.0=5. This value is quite diffeent fom the geneally acceptable theoetical value of 708. Fo this expeiment, K c =338 is obtained using Eq.(8). This value appoaches the citical value fo plane Couette flow of All the expeimental data fo the pimay instability in Taylo-Couette flows ae depicted in Fig. by plotting K c vesus the elative gap width h/. The citical value K c of K max fo plane Couette flow, plane Poiseuille flow and pipe Poiseuille flow ae also included with h/ =0. Fo all the wall-bounded paallel flows, K c =35 389, which ae calculated fom the expeimental data [,, 3]. It is noted fom Fig. that K c deceases with inceasing h/, which depends on e and h/ as also shown by Eq.(8). When h/ tends to zeo, the value of K c tends to the value of plane Couette flow. It may be obseved fom Fig. that thee seems a coelation/elationship fo the K cuve fo all types of wall-bounded flows (including Taylo-Couette flows, plane Couette flow, plane Poiseuille flow, and pipe Poiseuille flow). Howeve, it should be mentioned that the citical condition fom expeiments fo Taylo-Couette flow is the pimay instability fo cell patten fomation which coesponds to the infinitesimal distubance (which is consistent with the pediction by linea stability analysis [5]), while the citical conditions fo wall bounded paallel flows wee those to sustain the tubulence (spot) below which no tubulence can be geneated which coesponds to finite amplitude distubance. This diffeence petaining to the initial distubance needs to be futhe investigated in futue. Taylo numbe has been used to descibe the stability of Taylo-Couette flow as discussed in the intoduction [-5]. The Taylo numbe fo the case of the oute cylinde fixed ( 0 ) is, T e h / 0, with e h. The citical value fo instability is Tc=708 fom linea stability calculation [-]. When h/ 0 tends to zeo, the flow educes to plane Couette flow. In tems of the Taylo numbe, when h/ 0 tends to zeo ( 0 tends to infinite), T=0 and definitely T<Tc; this means that the flow is always stable. In othe wods, by stating Tc=708, the 3

24 citical e is infinite if h/ 0 tends to zeo. This contadicts the expeimental esults of plane Couette flow. Obviously, if the Taylo-Couette flow is elated to plane Couette flow, then the Taylo numbe may not be sufficient o appopiate to descibe the tansition. It is only applicable fo concentic otating cylindes with the magnitude of h/ 0 not vey lage o vey small. Taylo [5] used mathematical theoy and linea stability analysis and showed that linea stability theoy agees well with expeiments. Howeve, the linea stability theoy pesupposes an infinitesmall distubance as intoduced fo the (otating) Taylo-Couette flow just as fo the empeiments. Fo the paallel flows, the stability conditions wee obtained at finite amplitude of distubance as fo the associated expeiments. A unified pictue linking the stability citeion/citeia established between the Taylo-Couette flow and paallel flows may be inconguous. On the othe hand, as shown in this pape, the pesent theoy is valid fo all of these concened flows. Theefoe, it is postulated that the enegy gadient theoy is at the vey least a moe feasible univesal theoy fo flow instability and tubulent tansition, and which is valid fo both pessue and shea diven flows in both paallel flow and otating flow configuations. 5. Conclusion In this pape, the enegy gadient theoy is applied to Taylo-Couette flow between concentic otating cylindes. The deivation fo the enegy gadient function K is given fo Taylo-Couette flow, which is also elated to plane Couette flow. The limit of infinite cylinde adii of Taylo-Couette flow coesponds to plane Couette flows. The theoetical esults fo the citical condition found have vey good concuence with the expeiments in the liteatue. The conclusions dawn ae: () The enegy gadient method as a semi-empiical theoy is valid fo otating flows. The citical value of K max fo pimay instability in Taylo-Couette flows is a constant fo a given geomety as confimed by the expeimental data. Theefoe, this may suggest that the enegy gadient function K is a vey easonable paamete to descibe the instability in Taylo-Couette flow. () The isoline chat on the plane of / vesus / may povide a basic physical explanation of the egimes of flow pattens found in the expeiments of Andeeck et al.[6] (3) All wall-bounded shea flows shae the same mechanism fo the instability initiation based on the elative dominance between the gadient of total mechanical enegy and the loss of total mechanical enegy in the flow. The limit of Taylo-Couette flow at vey small gap width becomes that of plane Couette flow. 4

25 (4) The K function is useful fo elating the plane Couette flow to the Taylo-Couette flow. It has a clea physical concept and meaning. On the othe hand, Taylo numbe is not valid o appopiate in the limiting case of Taylo-Couette flow at vey small gap width when the adii of cylindes tend towads infinity. (5) The enegy gadient theoy can function as a plausible univesal theoy fo flow instability and tubulent tansition and which is valid fo both pessue and shea diven flows in both paallel flow and otating flow configuations. It is shown in the pesent study that thee ae some similaities between Taylo-Couette flow and plana Couette flow. Acknowledgements The authos thank D. HM Tsai fo helpful discussions. efeences. S. Chandasekha, Hydodynamics and Hydomagnetic Stability, Dove, New Yok, 96, P. G. Dazin and W. H. eid, Hydodynamic stability, Cambidge Univesity Pess, nd Ed., Cambidge, England, 004, P. Chossat and G. Iooss, The Couette-Taylo Poblem, Spinge-Velag, H. Schlichting, Bounday Laye Theoy, (Spinge, 7th Ed., Belin, 979), 83-; G. I. Taylo, Stability of a viscous liquid contained between two otating cylindes, Philosophical Tansactions of the oyal Society of London. Seies A, 3 (93), Donnelly,.J., Taylo-Couette flow: the ealy days, Physics Today, No., 44 (99), Tagg, The Couette-Taylo poblem, Nonlinea Science Today, 4 (994) M. Benne, H. Stone, Moden classical physics though the wok of G. I. Taylo, Physics Today, No.5, 000, L. ayleigh, On the dynamics of evolving fluids, Poceedings of the oyal Society of London. Seies A, Vol. 93, No (Ma., 97), T. von Kaman,934, Some aspects of the tubulence poblem. Poc. 4 th Inte. Cong. Fo applied Mech., Cambidge, England, Also Collected woks (956), Vol.3, Buttewoths Scientific Publications, London, H. A. Snyde, Stability of otating Couette flow. II. Compaison with numeical esults, Phys. Fluids, (968) A. Esse and S.Gossmann, Analytic expession fo Taylo Couette stability bounday, Phys. Fluids, 8(7), 996, D. Coles, Tansition in cicula Couette flow, J. Fluid Mech., (965), E.. Kuege, A. Goss &.C. DiPima, On the elative impotance of Taylo-votex and non-axisymmetic modes in flow between otating cylindes, J. Fluid Mech. 4 (966), J.P. Gollub, and H.L.Swinney, Onset of tubulence in a otating fluid, Phys. ev. Lett., 35, 975, C.D. Andeeck, S.S. Liu, and H.L. Swinney, Flow egimes in a cicula Couette system with independently otating cylindes, J. Fluid Mech., 64 (986), pp W.F. Langfod,. Tagg, E.J. Kostelich, H.L. Swinney & M. Golubitsky, Pimay 5