Instability of Taylor-Couette Flow between Concentric Rotating Cylinders
Hua-Shu Dou, Boo Cheong Khoo, Nhan Phan-Thien, and Koon Seng Yeo Fluid Mechanics Division, Department of Mechanical Engineering National University of Singapore, Singapore 119260, SINGAPORE Email: [email protected]
Abstract The energy gradient theory is used to study the instability of Taylor-Couette flow between concentric rotating cylinders. In our previous studies, the energy gradient theory was demonstrated to be applicable for wall bounded parallel flows. It was found that the critical value of the energy gradient parameter K at subcritical transition is about 370-389 for wall bounded parallel flows (which include plane Poiseuille flow, pipe Poiseuille flow and plane Couette flow) below which no turbulence occurs. In this paper, the detailed derivation for the calculation of the energy gradient parameter in the flow between concentric rotating cylinders is provided. The theoretical results for the critical condition of primary instability obtained are in very good agreement with the experiments found in literature. The mechanism of spiral vortices generation for counter-rotating of two cylinders is also explained using the energy gradient theory. The energy gradient theory can also serve to relate the condition of flow transition in the Taylor- Couette flow to that in the plane Couette flow. The latter reasonably becomes the limiting case of the former when the radii of cylinders tend to be infinite. It is our contention that the energy gradient theory is possibly universal for analysis of flow instability and turbulent transition, and which is valid for both pressure and shear driven flows in both parallel and rotating flow configurations.
Keywords: Flow instability; Transition; Taylor-Couette flow; Rotating cylinders; Energy gradient; Energy loss; Critical condition; Taylor vortices
Contents: 1. Introduction 2. Energy Gradient Theory 3. Energy Gradient Theory Applied to Taylor-Couette Flows 3.1 Velocity distribution for Taylor-Couette Flow 3.2 Energy gradient in the transverse direction 3.3 Energy Loss Distribution for Taylor-Couette Flow 3.4 The K parameter 4. Comparison with Experiments at Critical Condition 5. Further Discussion on Rayleigh and Taylor Criteria 6. Conclusions
1 1. Introduction Taylor-Couette flow refers to the problem of flow between two concentric rotating cylinders as shown in Fig.1 [1-4]. This terminology was named after the works of G. I. Taylor (1923) and M. Couette (1890). This problem was first investigated experimentally by Couette (1890) and Mallock (1896). Couette observed that the torque needed to rotate the outer cylinder increased linearly with the rotation speed until a critical rotation speed, after which the torque increased more rapidly. This change was due to a transition from stable to unstable flow at the critical rotation speed. Taylor was the first to successfully apply the linear stability theory to a specific problem, and succeeded in obtaining excellent agreement of theory with experiments for the flow instability between two concentric rotating cylinders [5]. Taylor’s groundbreaking research for this problem has been considered as a classical example of flow instability study [6- 8]. In the past years, the problem of Taylor-Couette flow has received renewed interests because of its importance in flow stability and particularly amenable to rigorous mathematical treatment/analysis due to infinitesimal disturbances [1-3]. For the stability of an inviscid fluid moving in concentric layers, Lord Rayleigh [9] used the circulation variation versus the radius to explain the instability while von Karman [10] employed the relative roles of centrifugal force and pressure gradient to interpret the instability initiation. Their goal was to determine the condition for which a perturbation resulting from an adverse gradient of angular momentum can be unstable. In his classic paper, Taylor [5] presented a mathematical stability analysis for viscous flow and compared the results to laboratory observations. Taylor observed that, for small ratio of the gap width to the cylinder radii and for a given rotating speed of outer cylinder, when the rotating speed of the inner cylinder is low, the flow keeps to being laminar; when the rotating speed of the inner cylinder exceeds a critical value, instability sets in and rows of cellular vortices are developed. When the rotating speed is increased to an even higher value, the cell rows breakdown and turbulence pattern is produced. He proposed a parameter, now commonly known as the 2 Taylor number, T = Re ()h / R1 , to characterize this critical condition for instability. Here, Re is the Reynolds number based on the gap width (h) and the rotating speed of the inner cylinder, and R1 is the radius of the inner cylinder. The critical value of Taylor number for primary instability is 1708 as obtained from linear analysis. This value agrees well with his experiments [1-3]. However, the problem on Taylor-Couette flow is still far from completely resolved despite the extensive works [11-17]. For example, the limiting case of Taylor-Couette flow when the ratio of the gap width to the radii tends to zero should be in accord with the status of plane Couette flow. Thus, the criterion for instability should reflect this phenomenon. There are two recent works trying to address this issue to some degree of success [18-19]. One may observe that Taylor’s criterion is not appropriate when this limiting case is studied because plane Couette flow is judged to be always stable due to Taylor number assuming a null value using Taylor’s criterion. This may be attributed to the fact that Taylor’s criterion only considered the effect of centrifugal force, and does not include the kinematic inertia force. Therefore, it is reckoned to be suitable for low Re number flows with high curvature. For rotating flow with higher Re number and low curvature, it may transit to turbulence earlier and yet does not violate Taylor’s criterion. Recently, Dou [20] proposed a new energy gradient theory to analyze flow instability and turbulent transition problems. In this theory, the critical condition for flow instability and turbulent transition is determined by the ratio (K) of the energy gradient in the transverse direction to the energy loss in the streamwise direction for the given disturbance. For a given flow geometry and fluid properties, when the maximum of K in the flow field is larger than a critical value, it is expected that instability would occur for some initial disturbances provided that the disturbance energy is sufficiently large. For plane channel flow, Hagen-Poiseuille flow, and plane Couette flow, the findings based on the theory are consistent with the experimental observations; for the experimental determined critical condition, Kc=370-389 for all the said three types of
2 flows below which there is no turbulence occurrence. The theory also suggests the mechanism of instability associated with inflectional velocity profile for viscous flow, and is valid for pressure driven flow and shear driven parallel flows. It has been shown that the theory works well for the wall bounded parallel flows (plane Poiseuille flow, pipe Poiseuille flow, and plane Couette flow) [21,22]. In this study, we apply the energy gradient theory to analyze the Taylor-Couette flow between concentric rotating cylinders, and demonstrate that the mechanism of instability in Taylor-Couette flow can be explained via the energy gradient concept. Through comparison with experiments, we show that the energy gradient parameter K as a stability criterion is sufficient to describe and characterize the flow instability in Taylor-Couette flow. We also show that plane Couette flow is just the limiting case of the Taylor-Couette flow when the curvature of the walls tends to be zero. For flow between concentric rotating cylinders, the flow instability may be induced by the rotating of the inner cylinder or the outer cylinder. If it is induced by the former, Taylor vortex cell pattern will be formed when the critical condition is violated as in the experiments; if it is induced by the latter, Taylor vortex cell pattern will not occur and the flow may directly transit to turbulence when the critical condition is reached like as in plane Couette flow [1-3, 6]. In this study, the critical condition for the former situation is only studied.
2. Energy gradient theory
Recently, Dou [20] proposed an energy gradient theory with the aim to clarify the mechanism of transition from laminar to turbulence for wall bounded shear flows. Here, we give a short discussion for a better understanding of the work presented in this study. In the theory, the whole flow field is treated as an energy field. It is thought that the gradient of total energy in the transverse direction of the main flow and the viscous friction in the streamwise direction dominate the instability phenomena and hence the flow transition for a given disturbance. The energy gradient in the transverse direction has the potential to amplify a velocity disturbance, while the viscous friction loss in the streamwise direction can resist and absorb this disturbance. The transition to turbulence depends on the relative magnitude of these two roles of energy gradient amplification and viscous friction damping to the initial disturbance. Based on such, a new dimensionless parameter, K (the ratio of the energy gradient in the transverse direction to that in the streamwise direction), is defined to characterize the stability of the base flow,
∂E / ∂n K = . (1) ∂E / ∂s 1 Here, E = p + ρV 2 + ρgξ is the total energy for incompressible flows with ξ as the 2 coordinate in the direction of gravitational field, n denotes the direction normal to the streamwise direction and s denotes the streamwise direction. Furthermore, ρ is the fluid density, g is the gravity acceleration, V is the velocity, and p is the hydrodynamic pressure. For pressure driven flows, the magnitude of energy gradient ∂E / ∂s equals to the energy loss of unit volume fluid ∂H / ∂s in unit length along the streamline due to the viscous friction. In other word, the mechanism of generation of streamwise energy gradient is resulted by the energy loss due to viscous friction. Therefore, for shear driven flows, the calculation of K can be obtained by the ratio of the energy gradient in transverse direction and the energy loss in unit length along the ∂E / ∂n streamline, K = . As such, the parameter K in Eq.(1) is a field variable. Thus, the ∂H / ∂s distribution of K in the flow field and the property of disturbance may be the perfect means to describe the disturbance amplification or decay in the flow. It is suggested that the flow instability
3 can first occur at the position of K max which is construed to be the most “dangerous” position. Thus, for a given disturbance, the occurrence of instability depends on the magnitude of this dimensionless parameter K and the critical condition is determined by the maximum value of K in the flow. For a given flow geometry and fluid properties, when the maximum of K in the flow field is larger than a critical value K c , it is expected that instability can occur for certain initial disturbance [20]. Turbulence transition is a local phenomenon in the earlier stage. For a given flow, K is proportional to the global Reynolds number. A large value of K has the ability to amplify the disturbance, and vice versa. The analysis has suggested that the transition to turbulence is due to the energy gradient and the disturbance amplification, rather than just the linear eigenvalue instability type as stated in [23,24]; this also follows that the pipe Poiseuille flow is linearly stable for all the Reynolds number. Both Grossmann [23] and Trefethen et al. [24] commented that the nature of the onset-of-turbulence mechanism in parallel shear flows must be different from an eigenvalue instability of linear equations of small disturbance. In fact, finite disturbance is needed for the turbulence initiation in the range of finite Re as found in experiments [25]. Dou [20] demonstrated that the criterion obtained has a consistent value at the subcritical condition of transition determined by the experimental data for plane Poiseuille flow, pipe Poiseuille flow as well as plane Couette flow (see Table 1). From this table it can be deduced that the turbulence transition takes place at a consistent critical value of K c at about 385 for both the plane Poiseuille flow and pipe Poiseuille flow, and about 370 for plane Couette flow. This may suggest that the subcritical transition in parallel flows takes place at a value of Kc ≈ 370-385. The finding further suggests that the flow instability is likely resulted from the action of energy gradients, and not due strictly to the eigenvalue instability of linear equations. The critical condition for flow instability as determined by linear stability analysis differs largely from the experimental data for all the three different types of flows, as shown in Table 1. Using energy gradient theory, it is also demonstrated that the viscous flow with an inflectional velocity profile is unstable for both two-dimensional flow and axisymmetrical [26].
Flow type Re expression Eigenvalue Experiments, Kmax at Rec analysis, Rec Rec (from experiments), ≡ Kc Pipe Poiseuille Re = ρUD / µ Stable for all Re 2000 385 Plane Poiseuille Re = ρUL / µ 7696 1350 389 5772 1012 389 Re = ρu0 h / µ Plane Couette Re = ρUh / µ Stable for all Re 370 370
Table 1 Comparison of the critical Reynolds number and the energy gradient parameter Kmax for plane Poiseuille flow and pipe Poiseuille flow as well as for plane Couette flow [20]. U is the averaged velocity, u0 the velocity at the mid-plane of the channel, D the diameter of the pipe, h the half-width of the channel for plane Poiseuille flow (L=2h) and plane Couette flow. For Plane Poiseuille flow and pipe Poiseuille flow, the Kmax occurs at y/h=0.5774, and r/R=0.5774, respectively. For Plane Couette flow, the Kmax occurs at y/h=1.0.
For plane Poiseuille flow, this said position where Kmax> Kc should then be the most dangerous location for flow breakdown, which has been confirmed by Nishioka et al’s experiment [27]. Nishioka et al's [27] experiments for plane Poiseuille flow showed details of the outline and process of the flow breakdown. The measured instantaneous velocity distributions indicate that the first oscillation of the velocity occurs at y/h=0.50~0.62.
4 For pipe flow, in a recent study, Wedin and Kerswell [28] showed that there is the presence of the "shoulder" in the velocity profile at about r/R=0.6 from their solution of the traveling waves. They suggested that this corresponds to where the fast streaks of traveling waves reach from the wall. It can be construed that this kind of velocity profile as obtained by simulation is similar to that of Nishioka et al's experiments for channel flows [27]. The location of the "shoulder" is about same as that for K max . According to the present theory, this "shoulder" may then be intricately related to the distribution of energy gradient. The solution of traveling waves has been confirmed by experiments recently [29].
3. Energy Gradient Theory for Taylor-Couette Flow
3.1 Velocity distribution for Taylor-Couette Flow
The solution of velocity distribution between concentric rotating cylinders can be found ∂ in some books [1-3]. Because of u = 0 and = 0 , the Navier-Stokes equations in radial and r ∂θ circumferential directions for steady flows reduce to
u 2 dp ρ = , (2) r dr and ∂ ∂u u + = 0 . (3) ∂r ∂r r Integrating Eq.(3) and using the boundary conditions gives the solution of the velocity field as, B u = Ar + (4) r and 2 (η − λ) 2 (1− λ) A = ω1 and B = ω1R1 , (5) η 2 −1 1−η 2
where η = R1 / R 2 and λ = ω2 /ω1 . R1 is the radius of the inner cylinder and R2 is the radius of the outer cylinder. ω1 and ω2 are the angular velocities of the inner and outer cylinders, respectively.
3.2 Energy gradient in the transverse direction
The energy gradient in the transverse direction is ∂E ∂( p +1/ 2ρu 2 ) du u 2 = = ρu + ρ . (6) ∂r ∂r dr r
Introducing Eq.(4) and Eq.(5) into Eq.(6), the energy gradient in transverse direction therefore is
5 2 ∂E B B 1 B = ρ Ar + A − + Ar + ∂r r r 2 r r . (7) B = 2ρA Ar + r
3.3 Energy Loss Distribution for Taylor-Couette Flow
The following equation for calculating the energy loss distribution along the radius for Taylor-Couette Flow is obtained as [30],
dH τ du τ ≡ − . (8) ds u dr r This equation is applicable to flows for one cylinder rotating and the other at rest, and cylinders rotating in counter directions. For cylinders rotating in the same direction, a different equation must be used [30] dH τ du τ ≡ a a − a , (9) ds ua dr r where ua is the velocity in the flow field expressed by ua = u − rωb and ωb = ω2 assuming that ω1 > ω2 and τ a is the shear stress in the velocity field expressed by ua = u − rωb . The details of the derivation for dE/ds can be found in [30] and is not repeated here.
With the velocity gradient obtained from Eq.(4), the shear stress is therefore,
∂u u B 1 B 2B τ = µ − = µ A − − Ar + = −µ , (10) ∂r r r 2 r r r 2 where µ is the dynamic viscosity. Thus, we have τ 2B = −µ (11) r r 3 and −1 τ du 2B B B = −µ Ar + A − . (12) u dr r 2 r r 2
Introducing Eqs.(11) and Eq.(12) into Eq.(8), the energy loss is
−1 dH τ du τ 2B B B 2B ≡ − = −µ Ar + A − + µ ds u dr r r 2 r r 2 r 3 . (13a) −1 −1 2B 1 B B 4B 2 B = µ − Ar + A − = µ Ar + 2 2 4 r r r r r r
For cylinders rotating in same direction, using the same procedure as that in against Eq.(13a), the equation can be obtained from Eq.(9) as,
6 −1 dH τ du τ 4B 2 B ≡ a a − a = µ a A r + a , (13b) 4 a ds ua dr r r r where 2 (η − λa ) 2 (1− λa ) Aa = ω1a and Ba = ω1a R1 , (14) η 2 −1 1−η 2
and η = R1 / R 2 , λa = ω2a /ω1a , ω1a = ω1 − ωb , ω2a = ω1 − ωb , and ωb = ω2 .
3.4 The K parameter
Introducing Eq.(7) and (13a or 13b) into Eq.(1), the ratio of the energy gradients in the two directions, K, can be written as,
du u 2 B ρu + ρ 2A Ar + ∂E / ∂r 1 r K = = dr r = , (15) ∂H / ∂s τ du τ ν 2 −1 − − 4Ba Ba Aa r + u dr r r 4 r where ν is the kinematic viscosity. In this equation, the calculations of A and B are carried out using Eq.(5). The evaluations of Aa and Ba are different for counter rotating and same direction rotating of cylinders. For cylinders rotating in opposite directions, Aa = A and Ba = B
(calculated using Eq.(5); For cylinders rotating in same direction, Aa = Aa and Ba = Ba (calculated using Eq.(14)).
Introducing Eqs.(4) and (5) or (14) into Eq.(15), then simplifying and rearrangement, the ratio of the energy gradient in the two directions, K, can be written as,
4 2 2 2 1 r ω η − λ η −1 ω (η − λ) 1 2 (1− λ) K = 1 ()( ) 1 r − ω R 4 2 2 2 1 1 2 2ν R1 ω1a ω1 ()1− λa ()η −1 r ()1−η . (16) 2 ω ()η − λ 1 2 ()1− λ 1a a r − ω R a 2 1a 1 2 ()η −1 r ()1−η
Eq.(16) is valid for same direction rotating of cylinders. For cylinders rotating in opposite directions, λa and ω1a should be replaced by λ and ω1 , respectively.
Re-arranging, Eq.(16) can be rewritten as
7 1 ω R 2 ω r 4 η 2 − λ K = 1 1 1 ( ) 4 2 2 2 ν ω1a R ()1− λ ()η −1 1 a . (17) r 2 R1 r 2 R1 ()η − λ − ()1− λ ()η − λa − ()1− λa R1 r R1 r ω R h Using a more appropriate form by explicitly showing the Reynolds number, Re = 1 1 , ν Eq.(17) can be expressed as 1 ω R h R ω r 4 η 2 − λ K = 1 1 1 1 ( ) 4 2 2 2 ν h ω1a R ()1− λ ()η −1 1 a , (18) r 2 R1 r 2 R1 ()η − λ − ()1− λ ()η − λa − ()1− λa R1 r R1 r
where h = R2 − R1 is the gap width between the cylinders.
If the outer cylinder is at rest (ω2 = 0 ), and only the inner cylinder is rotating (ω1 ≠ 0 ),
ω1 then λ = 0 , λa = 0 , and = 1. By simplifying Eq.(18), we obtain ω1a
2 1 ω R h R 2 R r 2 r 2 K = 1 1 1 1 1− . (19) 2 ν 2 R + R 2 2 h 1 2 R1 R2
Next, by letting r = R2 − y , Eq.(19) is rewritten as
1 ω R h R 2 R (R − y)2 (2R − y)2 y 2 K = 1 1 1 1 2 2 2 2 2 2 2 ν h R1 + R2 R R R 2 2 1 ω R h y 2 R ()R − y 2 ()2R − y 2 = 1 1 1 2 2 2 2 2 ν h 2(R1 + R2 ) R2 R2 2 2 ω R h y 2 R y h y h = 1 1 1 1− 2 − . (20) 2 ν h 2(R1 + R2 ) h R2 h R2
This equation easily relates to the plane Couette flow. It can be seen that K is proportional to Re ω R h ( ≡ 1 1 ), in any location in the field. K is an eighth order function of distance from the outer ν cylinder across the channel, which is related to the value of relative channel width h / R2 . The distribution of K along the channel width between cylinders calculated using Eq. (20) is depicted in Fig.2 for various values of h/R2 with the inner cylinder rotating while the outer cylinder is kept at rest (ω2 = 0 ). For a given Re and h/R2, it is found that K increases with increasing y/h and the maximum of K is obtained at y / h = 1 for low values of h/R2 (h/R2<0.43). That is, it reaches its
8 maximum at the surface of the inner cylinder. For higher value of h/R2 (h/R2>0.43), the location of Kmax moves to within the flow located between y/h=0 and y/h=1. The cases studied in the literature are usually for low gap width. We shall focus for the case of h/R2<0.43 in this study. The maximum of K for h/R2<0.43 can be expressed as
2 2 ω R h R h h 1 1 1 K max = 1− 2 − . (21) ν 2(R1 + R2 ) R2 R2
As we will see below, the critical condition of the stability will be determined by this equation. When the radii of cylinders tend to infinite, we have
2 2 R1 1 y h y h → , 1− → 1, and 2 − → 4 . (22) 2(R1 + R2 ) 4 h R2 h R2
Then, Eq.(21) reduces to
ω R h y 2 K = 1 1 . (23) ν h2
This equation at the limit of infinite radii of cylinders is the same as that for plane Couette flow [21]. The corresponding maximum of K at y=h is
ω R h K = 1 1 = Re . (24) max ν As discussed in [22], the development of the disturbance in the flow is subjected to the mean flow condition and the boundary and initial conditions. The mean flow is characterized by energy gradient parameter K. Therefore, the flow stability depends on the distribution of K in the flow field and the initial disturbance provided to the flow. In the area of high value of K, the flow is more unstable than that in the region of low value of K. The first sign of instability should be associated with the maximum of K (Kmax) in the flow field for a given disturbance. In other words, the position of maximum of K is the most dangerous position. For given flow disturbance, there is a critical value of Kmax over which the flow becomes unstable. It is not trivial to directly predict this critical value by theory as in parallel flows [20] since it is obviously a strongly nonlinear process. Nevertheless, it can still be observed in experiments. The Kmax can be taken as a criterion for instability; if Kmax > Kc , the flow will become unstable. Thus, the study of distribution of K in the flow field can help to locate the region where the flow is inclined to be unstable. In Fig.2, K increases with increasing y/h for given h/R2 (at low value of h/R2), and its maximum occurs at the inner cylinder. Thus, the flow at the outer cylinder is most stable and the flow at the inner cylinder is most unstable. Therefore, a small disturbance can be amplified at the inner cylinder if the value of K reaches its critical value for the given geometry. In other words, the location at the inner cylinder is a possible place for occurrence of instability occurrence first, as generally observed in the experiments [5,15]. In Fig.2, the line for h/R2=0 corresponds to plane Couette flow, which is a parabola [21]. It can be found that there is little difference in the distribution of K for h/R2=0.01 and h/R2=0. In terms of that view, one may expect that the critical conditions of instability for these two values of h/R2 are very near. When h/R2 increases, Kmax decreases. This does not, however, imply that the flow becomes more stable as h/R2 increasing. This is because that the critical value of Kmax is
9 different for different h/R2. It will be shown by experiments in later sections that Kc decreases with the increasing h/R2.
4. Comparison with Experiments at Critical Condition
Taylor [5] used the chart of ω1 /ν versus ω2 /ν to present the results of critical condition for the primary instability. In order to use the same chart as Taylor for ease of reference, the comparison of theory with experiments is also plotted in such a way. Rewriting Eq.(17), we have
2 2 2 K (1− λ ) (η −1) η − λ = a . (25) () 2 4 1 ω1R1 ω1 r r 2 R1 r 2 R1 ()η − λ − ()1− λ ()η − λa − ()1− λa 4 R r R r 2 ν ω1a R1 1 1
Rearrangement Eq.(25), the following equation (26) is obtained,
ω ω R 2 2K ω R 4 (1− λ )2 (η 2 −1) 1 = 2 2 + 1a 2 a . (26) 2 2 4 ν ν R1 R2 ω1 r r 2 R1 r 2 R1 ()η − λ − ()1− λ ()η − λa − ()1− λa R1 r R1 r
Thus, the critical condition for a given geometry is given by Kc. That is
ω ω R 2 2K ω R 4 (1− λ )2 (η 2 −1) 1 = 2 2 + c 1a 2 a . 2 2 4 ν c ν R1 R2 ω1 r r 2 R1 r 2 R1 ()η − λ − ()1− λ ()η − λa − ()1− λa R1 r R1 r (27) In Equation (27), Kc is the critical value of Kmax at the primary instability condition, which can be determined from experiments. For a given flow geometry, Kc is treated a constant for the initiation of instability as described before. It is found that the first term in the right hand side of Eq.(27) is that for Rayleigh’s inviscid criterion, and the second term in the right hand side of Eq.(27) is due to the effect of viscous friction. If Kc is zero, Eq.(27) degrades to Rayleigh’s equation. The theory is compared with available experimental data in literature [5][11][12][15] concerning the primary instability condition of Taylor-Couette flow. Figures 3 to 4 show the comparison of theory with Taylor’s experiments [5] for two parametric conditions, while Fig. 5, Fig.6 and Fig.7 show the comparisons of theory with Coles’ experiments [11], Snyder’s experiments [12], and Andereck et al’s experiments [15], respectively. The critical value of the energy gradient parameter K (Kc) is determined by the experimental data at ω2 = 0 and ω1 ≠ 0 (the outer cylinder is fixed, the inner cylinder is rotating). Using the determined value of Kc for a given set of geometrical parameters, the critical value of ω1 /ν versus ω2 /ν is calculated for the range of the ω2 /ν as in the experiments using Eq.(27). In Figs.3-7, Rayleigh’s inviscid 2 criterion ( ()ω1 c = (R2 / R1 )ω2 ) is also included for comparison. It can be seen from Fig.3-7 that when the cylinders rotate in same directions, the theory obtains very good concurrence with all the experimental data. When cylinders rotate in opposite directions, the theory obtains good agreement with the experimental data for small relative gap
10 width (h/R1). For larger relative gap width, the theory has some deviations from the experimental data with increasing negative rotating speed of the outer cylinder. The reason can be explained as follows. When the gap is large and the cylinders are rotating in opposite directions, the flow in the gap is more distorted compared to plane Couette flow. This distortion of velocity profile has an effect on the flow energy loss and energy gradient. On the other hand, if the rotating speed of the outer cylinder is high, the flow layer near the outer cylinder may earlier transit directly to turbulence if the disturbance is sufficiently large [6][11], which has not been the focus of research before. This will obviously alter the velocity profile of the flow and influence the distribution of the energy gradient parameter K and the maximum of K (more discussion later). For example, in
Andereck et al’s experiments [15], when ω2 /ν is -100 and the inner cylinder is at rest, the
Reynolds number based on the rotating speed of the outer cylinder Re2 ( Re2 = R2hω2 /ν ) reaches 416. At this value of 416, the plane Couette flow has already become turbulence (Rec=325--370). For opposite rotation of two cylinders in which there is a curvature in the streamlines, the transition must occur earlier than that in plane Couette flow because of the influence of the radial pressure gradient which increases the radial energy gradient near the outer cylinder. The same type of deviation in prediction is also observed in the comparison of Taylor’s mathematical theory with his experiments when cylinders rotate in opposite directions at large negative rotating speed of outer cylinder; in particular, if the relative gap is large [5]. Therefore, when cylinders rotate in opposite directions, more works are still needed to study the occurrence of turbulence as induced by shear flow near the outer cylinder (convective inertia resulted). In Fig.8, we show the distribution of K along the channel width at the critical condition of Kc=77 as shown in Fig.4. It can be seen in Fig.8a that K increases monotonically from the outer cylinder to the inner cylinder, when the inner cylinder is rotating while the outer cylinder is at rest. The maximum of K occurs at the inner cylinder, so the stability of the flow is dominated by the Kmax at the inner cylinder. In Fig.8b, it can be seen that K increases monotonically from the outer cylinder to the inner cylinder, when the two cylinders are rotating in same direction and ω1 /ν is larger than ω2 /ν . The maximum of K also occurs at the inner cylinder, so the stability of the flow is dominated by the Kmax at the inner cylinder too. In these two pictures, the base flow in the gap is laminar flow. Taylor vortex cell pattern are found in these cases as shown in experiments [5,15]. When the two cylinders rotate in opposite directions, the distribution of K generates two maximum respectively at the inner cylinder and the outer cylinder. In Fig.8c, it can be seen that the maximum at the outer cylinder is not high since the speed of the outer cylinder is small. In this case the base flow in the full gap may be still laminar, and the stability of the flow is still completely dominated by the Kmax at the inner cylinder. If the speed of the outer cylinder becomes high and exceeds a critical value, the flow near the outer cylinder may become turbulence provided that the disturbance is sufficiently large [6][11]. As shown in Fig.8d, the value of K at the outer cylinder (K=367) is about or higher than the critical value for plane Couette flow, the flow layer near the outer cylinder may be already in turbulence. Thus, the base flow is laminar near the inner cylinder and is turbulent near the outer cylinder at the critical condition of primary instability dominated by the rotating of inner cylinder. Therefore, Taylor vortex cell pattern could not be formed for such a case, but spiral vortices are generated. This is because that the generation of turbulence near the outer cylinder altered the velocity distribution from its laminar behaviour. The circulation of fluid particle between the two cylinder surfaces (alternatively laminar and turbulence) forms an intermittent and spiral pattern. This may be a good explanation for the reason for the generation of spiral pattern as found in experiments [15][11]. As reproduced in Fig.9, Andereck et al [15] plotted regimes of the flows in terms of Ro and Ri as coordinates (shown as Fig.1 in their paper). Here Ro and Ri are the Reynolds number based on the rotating speed of outer and inner cylinders, respectively. The behaviour of the flow may be better explained using the distribution of K along the gap width.
11 In Figs.10 and 11, we show the isolines of the Kmax along the side of inner cylinder in the plane of ω1 /ν versus ω2 /ν which occurs at the surface of the inner cylinder. Because the energy gradient dominates the flow behaviour and controls the mechanism of the flow instability and transition, the classification of regimes may be better understood using the isolines shown in Figs.10-11. By comparing Figs.10-11 and Fig.9, the regimes of the flow from experiments may appear to be aptly characterized by the isolines of Kmax along the inner cylinder in ω1 /ν and
ω2 /ν plane. It should be noticed that the isolines for Kmax
Authors R1 R2 h h/R1 (ω1/ν)c Rec Kc (cm) (cm) (cm) (cm-2) Taylor (1923) 3.80 4.035 0.235 0.06184 189.2 169 139 3.55 4.035 0.485 0.1366 70.7 120 77 3.00 4.035 1.035 0.345 30.5 95 33 Coles (1965) 10.155 11.52 1.365 0.1343 8.4 116 75 Snyder (1968) 6.023 6.281 0.258 0.0428 139.9 217 188 5.032 6.281 1.249 0.248 15 94 44 Gollub & 2.224 2.540 0.316 0.142 182. 128 80 Swinney (1975) Andereck et al 5.25 5.946 0.696 0.1326 33. 120 78 (1986) Hinko (2003) 29.54 29.84 0.30 0.01 39.5 350 338 Prigent & 4.909 4.995 0.0863 0.01752 758 320 301 Dauchot (2004) Table 2 Collected data for the detailed geometrical parameters for the experiments and the critical condition determined for the case of the outer cylinder at rest (ω2 = 0 ) and the inner cylinder rotating (ω1 ≠ 0 ).
In Table 2, experimental data are collated for the critical condition of the primary instability in the Taylor-Couette flows. A most interesting result for small gap flow was obtained by Hinko [17] recently. This result is useful to clarify how the Taylor-Coutte flow is related to
12 plane Couette flow. Hinko obtained Rec=350 for the flow in small gap of concentric rotating 2 cylinders with h/R1=0.01. Under this critical condition, the Taylor number is T=350 Χ0.01=1225. This value is quite different from the generally acceptable theoretical value of 1708. For this experiment, Kc=338 is obtained using Eq.(21). This value approaches the critical value for plane Couette flow of 325-370. All the experimental data for the primary instability in Taylor-Couette flows are depicted in Fig.12 by plotting Kc versus the relative gap width h/R1. The critical value Kc of Kmax for plane Couette flow, plane Poiseuille flow and pipe Poiseuille flow are also included. It can be seen that there should be a good correlation of Kc for all said wall bounded shear flows including parallel flows and rotating flows. It is noted that Kc decreases with increasing h/R1, which depends on Re and h/R1 as also shown by Eq.(21). When h/R1 tends to zero, Kc gets very good agreement with the wall bounded parallel flows. For all the wall bounded parallel flows, Kc=325—389, which are calculated from the experimental data [20][21]. It can be further observed from Fig.12 that Kc decreases with the reduced h/R1. When h/R1 tends to zero, the value of Kc tends to the value of plane Couette flow. Therefore, this may suggest that the energy gradient parameter K is a very reasonable parameter to describe the instability in Taylor-Couette flow. A correlation for the K curve for all types of wall bounded parallel flows (including Taylor-Couette flows, Plane Couette flow, plane Poiseuille flow, pipe Poiseuille flow) may be inferred from this picture. Nevertheless, it should be mentioned that the critical condition from experiments for Taylor-Couette flow corresponds to the infinitesmall disturbance (linear instability), while those for parallel flows were obtained at finite amplitude disturbance. This problem needs to be further investigated. For Taylor-Couette flow, Snyder has given a semi-empirical equation for the collected data [12]. Esser and Grossmann have also given an analytical equation for the critical condition [37]. ω R h The Taylor number at ω = 0 is, T = Re2 (h / R ) , and Re ≡ 1 1 . The critical value 2 1 ν for instability is Tc=1708 from linear stability calculation [1-2]. When h/R1 tends to zero, the flow reduces to plane Couette flow. In terms of the Taylor number, when h/R1 tends to zero (R1 tends to infinite), T=0 and T 6. Conclusion In this paper, the energy gradient theory is applied to Taylor-Couette flow between concentric rotating cylinders. Derivation for the calculation of the energy gradient parameter K is given for Taylor-Couette flow, which is also related to plane Couette flow. The limit of infinite cylinder radii of Taylor-Couette flow corresponds to the plane Couette flow. The theoretical results for the critical condition found have very good concurrence with the experiments in the literature. The conclusions are: (1) The energy gradient theory is valid for rotating flows. The critical value of Kmax is a constant for a given geometry is confirmed by the experimental data. 13 (2) The isolines chart on the plane of ω1 /ν versus ω2 /ν may provide a basic physical explanation of the flow regimes of vortices found in the experiments of Andereck et al.[15] (3) All of wall bounded shear flows share the same mechanism for the instability initiation based on the relative dominance between energy gradient and energy loss in the flow. The limit of Taylor-Couette flow becomes that of plane Couette flow. A correlation can be obtained for Kc versus h/R1 for all the wall bounded shear flows included in Fig.12. (4) The K parameter is useful for relating plane Couette flow to Taylor-Couette flow. It has a clear physical concept and meaning. On the other hand, Taylor number is not valid or appropriate in the limiting case of Taylor-Couette flow when the radii of cylinders tend towards infinity. (5) The energy gradient theory can function as a universal theory for flow instability and turbulent transition and which is valid for both pressure and shear driven flows in both parallel flow and rotating flow configurations. Acknowledgements The authors thank Dr. HM Tsai for the helpful discussions. References 1 S. Chandrasekhar, Hydrodynamics and Hydromagnetic Stability, Dover, New York, 1961, 272-381. 2 P. G. Drazin and W. H. Reid, Hydrodynamic stability, Cambridge University Press, 2nd Ed., Cambridge, England, 2004, 69-123. 3 P. Chossat and G. Iooss, The Couette-Taylor Problem, Springer-Verlag, 1994. 4 H. Schlichting, Boundary Layer Theory, (Springer, 7th Ed., Berlin, 1979), 83-111; 449-554. 5 G. I. 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Kerswell, F.Waleffe, Experimental observation of nonlinear traveling waves in turbulent pipe flow, Science, 305 (2004), Issue 5690, 10 September 2004, 1594-1598. 30 H.-S. Dou, B. C. Khoo, N. Phan-Thien, and K. S. Yeo, Energy loss distribution in the Taylor-Couette flow between concentric rotating cylinders, http://arxiv.org/abs/physics/0501151 31 A. Lundbladh and A. Johansson, Direct simulation of turbulent spots in plane Couette flow, J. Fluid Mech. 229 (1991), 499--516. 32 N. Tillmark and P. H. Alfredsson, Experiments on transition in plane Couette flow, J. Fluid Mech. 235, 89 –102, 1992. 33 F. Daviaud, J. Hegseth, and P. Berge′, Subcritical transition to turbulence in plane Couette flow, Phys. Rev. Lett. 69 (1992), 2511-2514. 34 S. Malerud, K. J. Malfy, and W.I. Goldburg, Measurements of turbulent velocity fluctuations in a planar Couette cell, Phys. Fluids, 7 (1995) 1949--1955. 35 O. Dauchot and F. 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Fluids, 8(7), 1996, 1814-1819. 15 Fig.1 Taylor-Couette flow between concentric rotating cylinders 1 h/R =0.0 2 Inner cylinder h/R2=0.01 0.8 h/R2=0.05 h/R2=0.10 h/R2=0.20 h/R =0.30 0.6 2 h/R2=0.40 h/R2=0.50 K/Re h/R2=0.60 0.4 Outer cylinder 0.2 0 0 0.25 0.5 0.75 1 y/h Fig.2 K/Re versus the channel width between the cylinders at various h/R2 for ω2 = 0 and ω1 ≠ 0 (the outer cylinder is fixed and the inner cylinder is rotating). 16 Fig.3 Comparison of the theory with the experimental data for the instability condition of Taylor- Couette flow (Taylor (1923)’s experiments, R1=3.80 cm, R2=4.035 cm). The relative gap width is h/R1=0.06184. Fig.4 Comparison of the theory with the experimental data for the instability condition of Taylor- Couette flow (Taylor (1923)’s experiments, R1=3.55cm, R2=4.035 cm). The relative gap width is h/R1=0.1366. 17 Fig.5 Comparison of the theory with the experimental data for the instability condition of Taylor- Couette flow (Coles (1965)’ experiments, R1=10.155 cm, R2=11.52 cm). The data are taken from Fig.2c in the paper [11]. The relative gap width is h/R1=0.1343 Fig.6 Comparison of the theory with the experimental data for the instability condition of Taylor- Couette flow (Snyder (1968)’s experiments, R1=6.023 cm, R2=6.281 cm). The data are taken from his Table III [12]. The relative gap width is h/R1=0.0428. 18 Andereck et al(1986)'s Experiments 400 Energy gradient theory Rayleigh inviscid criterion 300 Energy gradient theory, Kc=78; ν Andereck et al (1986)'s experiments: / 1 R1=5.25 cm, R2=5.946 cm ω 200 100 0 -300 -200 -100 0 100 200 300 ω2/ν Fig.7 Comparison of the theory with the experimental data for the instability condition of Taylor- Couette flow (Andereck et al (1986)’s experiments, R1=5.25 cm, R2=5.946 cm). The data are taken from their Fig.2 and Fig.18 [15]. The relative gap width is h/R1=0.1326. 19 R =3.55 cm, R =4.035 cm R =3.55 cm, R =4.035 cm 1 -2 2 1 2 100 ω /ν=0.0 cm 100 ω /ν=200 cm-2 2 -2 2 -2 ω1/ν=70 cm ω /ν=278 cm K =77 1 c Kc=77 Outer cylinder K K 50 50 Inner cylinder (a) (b) 0 0 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 (r-R )/h 1 (r-R1)/h R1=3.55 cm, R2=4.035 cm R1=3.55 cm, R2=4.035 cm 100 ω /ν=-30 cm-2 ω /ν=-220 cm-2 2 -2 300 2 -2 ω1/ν=87.4 cm ω1/ν=148 cm Kc=77 Kc=77 200 K K 50 100 (d) (c) 0 0 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 (r-R )/h (r-R )/h 1 1 Fig.8 Distribution of K along the channel width at the critical condition Kc=77 corresponding to Fig.4. (a) The inner cylinder is rotating while the outer cylinder is at rest; (b) Two cylinders are rotating in same direction; (c) Two cylinders are rotating in opposite directions and the speed of the outer cylinder is low. (d) Two cylinders are rotating in opposite directions and the speed of the outer cylinder is high. 20 Fig.9 Regimes of the flow behaviour as identified by Andereck et al [15]. The ordinate and abscissa are the Reynolds number based on the channel width and the circumferential velocities of inner and outer cylinder, respectively. (Use permission by Cambridge University Press). 21 Isoline of Kmax;R1=3.80 cm, R2=4.035 cm Thick blue solid line: Instability critical line, Kc=139 Black dash line: Rayleigh inviscid criterion Green dash dot ine: ω1/ν=ω2/ν, rigid body rotation 1000 K>139 800 K=0 600 K=139 ν / 1 ω 400 K<139 200 0 -2000 -1600 -1200 -800 -400 0 400 800 ω2/ν Fig.10 Isoline of the maximum of the parameter K (Kmax) along the inner cylinder at the plane of the rotating speeds of inner and out cylinders (R1=3.80cm, R2=4.035 cm), corresponding to Fig.3. The critical value of the energy gradient parameter K (Kc) is indicated in the figure by the thick blue line, which is calculated as shown in Figure 3. 22 Isoline of Kmax;R1=3.55 cm, R2=4.035 cm Thick blue solid line: Instability critical line, Kc=77 Black dash line: Rayleigh inviscid criterion Green dash dot ine: ω1/ν=ω2/ν, rigid body rotation 300 K>77 K=0 200 ν / 1 K=77 ω 100 K<77 0 -200 -100 0 100 200 ω2/ν Fig.11 Isoline of the maximum of the parameter K (Kmax) along the inner cylinder at the plane of the rotating speeds of inner and out cylinders (R1=3.55cm, R2=4.035 cm), corresponding to Fig.4. The critical value of the energy gradient parameter K (Kc) is indicated in the figure by the thick blue line, which is calculated as shown in Figure 4. 23 Pipe Poiseuille flow, Kc=385 Plane Poiseuille flow, K =389 400 c PlaneCouetteflow,Kc=325--380 Taylor-Couette flow, Taylor(1923) Taylor-Couette flow, Coles (1965) Taylor-Couette flow, Snyder (1968) 300 Taylor-Couette flow, Gollub & Swinney (1975) Taylor-Couette flow, Andereck et al (1986) Taylor-Couette flow, Hinko (2003) Taylor-Couette flow, Prigent & Dauchot (2004) c K Taylor (1923): Kc=139, 77, and 33 for 200 h/R1=0.0618, 0.1366, and 0.345, respectively. Coles (1965): Kc=75 for h/R1=0.1343 Snyder (1968): Kc=188 and 44 for h/R1=0.0428 and 0.248, respectively. Gollub & Swinney (1975): Kc=80 for h/R1=0.142 Andereck et al (1986): Kc=78 for h/R1=0.1326 Hinko (2003): K =338 for h/R =0.01 100 c 1 Prigent & Dauchot(2004):Kc=301 for h/R1=0.01752 0 0 0.1 0.2 0.3 0.4 h/R1 Fig.12 Critical value of the energy gradient parameter Kmax (Kc) versus parameter h / R1 for Taylor-Couette flows. An artificial dash line to connect the data is drawn for visual convenience. The data for wall bounded parallel flows (plane Poiseuille flow, pipe Poiseuille flow and plane Couette flow) are also shown, which are determined using the energy gradient theory in conjunction with the experimental data [20][21]. 24