Transitions of axisymmetric flow between two corotating disks in an enclosure

Fluid Dynamics Research 39 (2007) 193 – 208

Transitions of axisymmetric flow between two corotating disks in an enclosure Tomohito Miura, Jiro Mizushima∗ Department of Mechanical Engineering, Faculty of Engineering, Doshisha University, Kyotanabe, Kyoto 610-0321, Japan Received 30 September 2005; received in revised form 31 May 2006; accepted 19 August 2006 Communicated by T. Kambe

Abstract Transitions of flow between two corotating disks in an enclosure are investigated numerically. The outer cylindrical boundary of the flow field is fixed, whereas the inner cylinder rotates together with the two disks. The flow is not only symmetric with respect to the inter-disk midplane but also axisymmetric around the axis of rotation at small Reynolds numbers although it becomes unstable to disturbances at large Reynolds numbers. Two kinds of instability modes are known, one of which breaks the axisymmetry of the flow to yield a polygonal flow pattern in the radial–tangential plane. Such instability occurs for small length ratios, the ratio of the length of cylinders (gap between two disks) to the width of the annulus. The other instability is a symmetry-breaking instability with respect to the inter-disk midplane retaining the axisymmetry, which occurs for intermediate length ratios. We investigate the latter symmetry-breaking instability assuming the axisymmetry of the flow field, in which we obtain steady axisymmetric flows numerically and analyze their linear stability. It is found that oscillatory flow as well as steady asymmetric flow appears resulting from the first instability even in the intermediate length ratios where the flow remains axisymmetric. Numerical simulations are also performed to obtain time-periodic flows, and bifurcation diagrams of the flow are depicted for various values of length ratio. The critical Reynolds numbers for the Hopf and symmetry-breaking pitchfork instabilities are evaluated and a transition diagram is obtained. © 2006 The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. Keywords: Taylor–Couette flow; Two rotating disks; Instability; Transition; Bifurcation; Symmetry-breaking; Hysteresis

∗ Corresponding author. Fax: +81 774 65 6835.

E-mail address: [email protected] (J. Mizushima). 0169-5983/$32.00 © 2006 The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. doi:10.1016/j.fluiddyn.2006.08.006

194

T. Miura, J. Mizushima / Fluid Dynamics Research 39 (2007) 193 – 208

1. Introduction The flow between two corotating disks enclosed in a stationary outer cylinder has been investigated as a simple model of flow in computer disk storages, in which the inter-disk spacing is assumed to be very small. Though the flow is steady and axisymmetric at small Reynolds numbers, it becomes unstable to oscillatory disturbances above a critical Reynolds number, which results in an appearance of polygonal shapes in the radial–tangential plane as observed in the experiments by Lennemann (1974). The objective in his experiments was a reduction of disk flutter in computer disk storages. He observed that the polygonal patterns rotate at about 80% of the disk rotation rate. The polygonal flow patterns appear as a first bifurcation of the axisymmetric flow when the ratio of inter-disk spacing to the width of the annulus, say the length ratio , is less than about 0.3. The flow structure in the radial–tangential plane was investigated in more detail by flow visualization method by Abrahamson et al. (1989), who identified three distinct regions in the polygonal flow field, i.e. an inner and outer regions and a shroud boundary layer. The inner region is characterized by solid-body rotation near the inner cylinder and the outer region is occupied by large vortices. The inner and outer regions are separated by a detached shear layer with a polygonal shape of boundary. There are many small vortices in the shroud boundary layer. It is also reported that the number of structures decreases in a stepwise manner as the rotation speed of disks increases. The appearance of polygonal shapes and the complexity in the flow field attracted intense interest of researchers so that many experimental (Schuler et al., 1990; Humphrey and Gor, 1993) and numerical works (Humphrey et al., 1995; Herrero et al., 1999) have been devoted to investigations of the structure and transition of the flow for the case of small inter-disk spacing. In these experiments and numerical simulations, the circumferential velocity profiles is measured or calculated. A simplified analysis is given and its results are compared with their experimental data measured with a laser-Doppler velocimeter by Schuler et al. (1990). It is shown in the experiment by Humphrey and Gor (1993) that the detached shear layer between the solid-rotating inner region and the outer region oscillates above a critical Reynolds number and the unsteadiness is found to originate at the enclosure side wall. Herrero et al. (1999) clarified that the polygonal flow patterns appear due to a Hopf bifurcation from the axisymmetric flow. For very large inter-disk spacing (?1), the flow field is approximated by an infinitely long fluid layer which was studied in the celebrated work by Taylor (1923) if the end effect is neglected. The effect of end walls was investigated by Benjamin (1978) and Benjamin and Mullin (1981), in which the number of cells appearing in the meridian cross-section of flow field was studied for finite length ratios of  ∼ 3.4.4.0. Benjamin found that primary flow has two or four cells depending upon the length ratio for small Reynolds numbers and that it develops smoothly when the Reynolds number is increased until wavy modes occur. He also observed a jump from a two-cell mode to a four-cell mode and vice versa, which accompanies a hysteresis. In most works investigating the end effect of Taylor–Couette flow, the end walls are fixed, which is in contrast with the case of two rotating disks. Hence, the direction of flow is normally inward near the top and bottom boundaries. However, flows having an odd number of cells and being directed outward near the end walls were observed in the experiment of Benjamin and Mullin (1981), which they called anomalous modes. In order to explain their experimental results, numerical calculations were performed by Bolstad and Keller (1987) and Tavener et al. (1991). Bolstad and Keller found extra vortices that might have been overlooked in experiment of Benjamin and Mullin, and identified the origin of the anomalous modes.

T. Miura, J. Mizushima / Fluid Dynamics Research 39 (2007) 193 – 208

195

To study the end effect of Taylor–Couette flow, another choice is possible for the boundary condition where the end walls rotate together with the inner cylinder, which corresponds to the case of large spacing of the flow between two corotating disks studied by Lennemann (1974). The case was studied experimentally as well as numerically by Tavener et al. (1991) mainly for 3.2  4.4. In this configuration, a driving effect is produced on the end cells which increases with Reynolds number and makes the flow to lose stability as the Reynolds number increases, because the end cells grow and squeeze out the interior ones. This leads to interesting bifurcation behavior which is different from that observed in the case of stationary ends. It is noted that the strong symmetric driving effect produces asymmetric flows more preferentially than in the case of stationary ends and that the asymmetry arises due to an inter-disk symmetry-breaking (pitchfork) bifurcation from a symmetric flow. It was found that in numerical calculations by Radel and Szeri (1997) that there exist multiple steady solutions with axisymmetry in a limited region of the Reynolds number for  ∼ 0.186. The flow between two corotating disks in a stationary enclosure has been considered as a simple model of centrifugal pumps and cyclone separators, for which the length ratio is about unity or less than 1. The transition of the flow was investigated by Randriamampianina et al. (2001) and a regime diagram of the bifurcation is obtained in a parameter space consisting of the Reynolds number and the length ratio. From the diagram, a critical length ratio is determined as c ∼ 0.26 below which the fluid motion becomes time dependent before bifurcating to an asymmetric flow. For  > c , unsteady flow occurs after the pitchfork bifurcation to an asymmetric flow is realized. It has not been clarified yet whether the axisymmetry is kept after the occurrence of a pitchfork bifurcation for a specific value of . According to Randriamampianina et al., the pitchfork bifurcation preceded the transition to unsteady three-dimensional flow for the case of =0.6. Herrero et al. investigated the transition of the flow in the range of 0.110.43 and showed that the flow becomes three-dimensional before the symmetry-breaking pitchfork bifurcation occurs in the range of  they studied. Our objective in the present paper is to clarify the bifurcation structure of the flow between the two rotating disks and study the underlying physics of the appearance of various flow patterns under the assumption of axisymmetric flow field. 2. Formulation of the problem We consider fluid motions between two corotating disks in an enclosure as illustrated in Fig. 1. The outer cylinder of radius r2 is fixed and the inner cylinder of radius r1 rotates together with the two disks with angular velocity . The gap r between the edge of disks and the outer cylinder is neglected theoretically. Taking r2  and d (≡ r2 − r1 ) as characteristic velocity and length scales, we define the Reynolds number Re as Re ≡ r2 d/, where  is the coefficient of kinematic viscosity of the fluid. The spacing between the two disks, i.e. the length of the cylinders, is  and the length ratio  is defined by  ≡ /d. Various flow patterns appear depending on a set of parameters (, Re). There is another nondimensional parameter, the radius ratio  = r1 /r2 , which is assumed to be 0.5 throughout this paper. We assume incompressible flow with the axisymmetry. The variable  is defined by  ≡ (r − r1 )/d, and the velocity u = (u, v, w) is written in terms of Stokes’ stream function  and a scalar function in a nondimensional form as 1 j r1 1 j u= (1) , v = ,  =  + . , w=−  jz  j 

d

T. Miura, J. Mizushima / Fluid Dynamics Research 39 (2007) 193 – 208

196

Fig. 1. Configuration and coordinates.

In a frame of reference rotating at an angular velocity , dynamical equations for  and are written in a nondimensional form as   1 4 j 2 j 1 2 j 2 2 j = D − + D  + 2(1 − ) , (2) J (, D ) − 2 D  jt 

j z j Re jz 

j jt

−

1 

J (, ) =

j 1 2 D − 2(1 − ) , Re jz

(3)

where D2 =

j2 j2

−

1 j  j

j2

+

jz 2

and

J (f, g) ≡

j(f, g) j(, z)

=

jf jg j jz

−

jf jg jz j

.

The boundary conditions for  and are given by j j

=

j jz

=  = 0,

=0

(4)

at the inner cylindrical surface ( = 0) and the conditions j j 1 =  = 0, = − = (5) j jz 1− are applied on the fixed enclosure wall ( = 1). Since we assume that the two disks rotate together with the inner cylinder, the boundary conditions at the top and bottom boundaries (z = 0, ) are written as j jz

=  = 0,

= 0.

(6)

Steady-state solutions (¯ , ¯ ) can be obtained directly irrespective of their stability or symmetry by solving steady-state equations, which are obtained by dropping the terms with time derivative in Eqs. (2) and (3) as   ¯ ¯ ¯ 1 j 1 4 2 j 2¯ 2 ¯ j ¯ ¯ + D ¯ − 2(1 − ) , (7) =− J (, D ) + 2 D  

jz j Re jz 

1 

J (¯ , ¯ ) = −

¯ j 1 2 D ¯ + 2(1 − ) , Re jz

under boundary conditions (4)–(6).

(8)

T. Miura, J. Mizushima / Fluid Dynamics Research 39 (2007) 193 – 208

197

For stability analysis of the steady-state solutions (¯ , ¯ ), we consider a disturbance ( ,  ) added to the steady-state solution and express the stream function  and the scalar function as  = ¯ +  , = ¯ +  , respectively. Substituting these expressions into Eqs. (2) and (3), subtracting steady-state equations (7) and (8), and neglecting the nonlinear terms of the disturbance, we obtain linear disturbance equations as   ˆ ¯ ˆ ¯ 2 j j 2ˆ 2 ¯ j 2 ˆ j +D  + ¯ + ˆ D  − 2 D  jz jz jz jz 

−

1 

{J (¯ , D 2 ˆ ) + J (ˆ , D 2 ¯ )} =

ˆ− D 2

1 

{J (¯ , ˆ ) + J (ˆ , ¯ )} =

ˆ j 1 4 D ˆ + 2(1 − ) , Re jz

ˆ j 1 2 D ˆ − 2(1 − ) , Re jz

(9)

(10)

where  is a complex linear growth rate of the disturbance and the time dependence of the disturbance is assumed to be  = ˆ exp(t) and  = ˆ exp(t). The boundary conditions for the disturbance ˆ and ˆ are given by ˆ j j

=

ˆ j jz

= ˆ = ˆ = 0

(11)

on the cylindrical surface ( = 0, 1). The nonslip boundary conditions on the bottom and top surfaces (z = 0, ) are written as ˆ j jz

= ˆ = 0,

ˆ = 0.

(12)

The stability of the steady solutions is determined by the real part Re() of the complex linear growth rate . The steady-state solution is stable if Re() < 0, or unstable if Re() > 0. The imaginary part Im() of  shows the oscillation frequency of the disturbance. It is a stationary disturbance which grows due to the instability if Im() = 0, or an oscillatory disturbance if Im() = 0 when Re() > 0. 3. Numerical methods 3.1. Numerical simulation Numerical simulation is performed to obtain stationary and periodic-state solutions. For the convenience of numerical calculation, we introduce new variables  = 2 − 1 and z = 2z/ − 1, and the stream function ( , z , t) and the scalar function ( , z , t) are expanded as ( , z , t) = ( , z , t) =

N M   m=0 n=0 N M+2  m=0 n=0

amn (t)T˜˜ m ( )T˜˜ n (z ),

(13)

bmn (t)T˜m ( )T˜n (z ),

(14)

T. Miura, J. Mizushima / Fluid Dynamics Research 39 (2007) 193 – 208

198

where M and N are the truncation numbers of the expansions. The function Tm (x) is the mth order Chebyshev polynomial and T˜˜ m (x) = (1 − x 2 )2 Tm (x),

T˜m (x) = (1 − x 2 )Tm (x)

(15)

are modified Chebyshev polynomials which satisfy boundary conditions (4)–(6). In order to avoid the discontinuity in the circumferential velocity between the two disks and the fixed enclosure, we assume the profile of the scalar function at  = 1 as = (1/(1 − ))(1 − z k ) where k is an even integer. We confirmed that the numerical results are almost independent of the specific value of k if k  16 and adopt the value of k = 16 in numerical calculations throughout the present paper. Substituting expansions (13) and (14) into Eqs. (2) and (3) and applying the collocation and tau methods, we obtain a set of ordinary differential equations for time development of the coefficients amn (t) and bmn (t), which are written in a matrix form as da + N1 (a, b) = L1 (a), (16) dt db (17) + N2 (a, b) = L2 (b), dt where a = t [a00 (t), a01 (t), . . . , aMN (t)] and b = t [b00 (t), b01 (t), . . . , b(M+2)(N+2) (t)], and Ni and Li (i = 1, 2) represent nonlinear and linear terms, respectively. We take j = cos(j − 1) /(M + 2) (j = 1, 2, . . . , M + 3) and zj = cos j /(N + 2) (j = 1, 2, . . . , N + 1) as collocation points. The resultant ordinary differential equations are solved as an initial value problem by the Runge–Kutta method with second-order accuracy. 3.2. Numerical calculation of steady-state solutions For direct numerical calculation of steady-state solutions (¯ , ¯ ), we expand the stream function ¯ and the scalar function ¯ in the same way with those in numerical simulation and use the collocation and tau methods. Then, we obtain a set of algebraic equations for the expansion coefficients, which we solve by Newton–Raphson’s method. In order to obtain steady symmetric flow, we put amn = 0 for even number of index n and bmn = 0 for odd number of n in expansions (13) and (14), respectively, because (, z) = −(,  − z) and (, z) = (,  − z) hold for symmetric flow. Solutions for asymmetric flow are explored by adding appropriate small values for those coefficients amn with even n and bmn with odd n as an initial guess. 3.3. Numerical method for linear stability analysis For numerical calculation in the linear stability analysis, the disturbance functions ˆ ( , z ) and ˆ ( , z ) are expanded as ˆ ( , z ) =  ˆ ( , z ) =

N M   m=0 n=0 N M   m=0 n=0

amn T˜˜ m ( )T˜˜ n (z ),

(18)

bmn T˜m ( )T˜n (z ).

(19)

T. Miura, J. Mizushima / Fluid Dynamics Research 39 (2007) 193 – 208

199

Algebraic equations for the coefficients amn and bmn are derived by substituting expansions (18) and (19) into Eqs. (9) and (10) and using the collocation method, where we take i = cos i /(M + 2) (i = 1, 2, . . . , M + 1) and zi = cos i /(N + 2) (i = 1, 2, . . . , N + 1) as collocation points. A set of algebraic equations constitute an eigenvalue problem in a matrix form as Aa = B a + C b,

(20) (21)

D b = E a + F b,

which we solve numerically by QZ decomposition (see, for example, Golub and Van Loan, 1983). 4. Numerical results 4.1. Accuracy assessment We shall discuss the convergence of the numerical results as the truncation numbers M and N of the expansion are increased. As a representative value to assess the accuracy of the numerical calculations, we adopt the critical Reynolds number Rec at which the flow field becomes asymmetric for the case of  = 0.3 and 0.6. The critical values Rec are evaluated by analyzing the data obtained in numerical calculations and tabulated in Table 1. We confirmed that the relative error in the critical value is about 0.1% between the values with (M, N) = (20, 30) and (30, 40) for  = 0.6, and 0.3% between the values with (M, N) = (20, 30) and (30, 40) for  = 0.3. Therefore, we adopted (M, N) = (20, 30) for 0.4   0.8 and (M, N) = (30, 40) for  = 0.3 in all numerical calculations in the present paper. 4.2. Flow pattern We performed numerical simulation and calculation of steady-state solutions in the range of the Reynolds number Re  1300 and the length ratio 0.3  0.8, and obtained various flow patterns, among which we show typical flow patterns for  = 0.6, 0.694, and 0.72. The flow is steady and symmetric with respect to the inter-disk center line irrespective of the length ratio  if the Reynolds number is small enough. For  = 0.6, the flow pattern in a meridian cross-section consists of two symmetric vortices having the same extent for Re = 900 as shown in Fig. 2(a), in which stream lines are depicted. The direction of the flow is outward along the disks and inward along the inter-disk center line. Such a steady flow is obtained also by solving steady-state equations (7) and (8) under boundary conditions (4)–(6), and can be attained by numerical simulation if it is stable. It is asymmetric at Re = 1200 as seen in Fig. 2(b), in which one vortex occupies a larger portion of the section at the expense of the other. Table 1 Critical Reynolds number Rec Truncation numbers M ×N 18 × 24 20 × 30 30 × 40

Rec ( = 0.6)

Rec ( = 0.3)

964 966 967

2870 2516 2525

200

T. Miura, J. Mizushima / Fluid Dynamics Research 39 (2007) 193 – 208

a

z

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1.0



0.2

0.4

0.6

0.8

1.0



b

z

0.6

0.4

0.2

0

Fig. 2. Flow patterns in the meridian section (stream lines). Solid line: anticlockwise circulation. Dashed line: clockwise circulation.  = 0.6. (a) Re = 900. (b) Re = 1200.

For  = 0.694, three different steady solutions are found by numerical calculation of the steady-state equations, which we show in Figs. 3(a)–(c). The asymmetric flow depicted in Fig. 3(b) is obtained in numerical simulation, whereas the other two depicted in Figs. 3(a) and (c) are obtained in numerical calculation of the steady-state solutions by adopting an appropriate initial guess for each flow. Though it is speculated that the asymmetric flow shown in Fig. 3(b) is stable, we will confirm it by the linear stability analysis in the next subsection. We observed oscillatory flow for  = 0.72, whose instantaneous flow patterns are depicted in Figs. 4(a)–(c) in every 18 period of the oscillation. The period of oscillation is about 300 in nondimensional time, which should be compared to the turnaround time 4 = 12.5 for the disks to rotate once around the axis. It is rather unexpected that such oscillatory flow is realized even in the assumption of axisymmetric flow field. 4.3. Linear stability We have obtained oscillatory flow as well as steady symmetric and asymmetric flows in the preceding subsection. The steady symmetric flow exists for small Reynolds numbers, but loses its stability at a critical Reynolds number. As a consequence of the instability it takes various flow patterns. In order to

T. Miura, J. Mizushima / Fluid Dynamics Research 39 (2007) 193 – 208

a

201

z

0.694 0.6 0.4 0.2

0

b

0.2

0.4

0.6

0.8

1.0



0.2

0.4

0.6

0.8

1.0



0.2

0.4

0.6

0.8

1.0



z

0.694 0.6 0.4 0.2

0

c

z

0.694 0.6 0.4 0.2

0

Fig. 3. Flow patterns in the meridian section (stream lines). Solid line: anticlockwise circulation. Dashed line: clockwise circulation.  = 0.694. Re = 1200. (a) Stable steady symmetric flow. (b) Stable steady asymmetric flow. (c) Unstable steady asymmetric flow.

examine the linear stability of the flow, we calculate the steady symmetric flow (¯ , ¯ ) numerically by solving Eqs. (7) and (8) and solve the eigenvalue problem consisting of Eqs. (9) and (10). Numerical solution of the eigenvalue problem revealed that the linear growth rate  has a real value for  = 0.6. Hence an exchange of stabilities occurs and induces a pitchfork bifurcation for  = 0.6. The

202

T. Miura, J. Mizushima / Fluid Dynamics Research 39 (2007) 193 – 208

a

z

0.72 0.6 0.4 0.2

0

b

0.2

0.4

0.6

0.8

1.0



0.2

0.4

0.6

0.8

1.0



0.2

0.4

0.6

0.8

1.0



z

0.72 0.6 0.4 0.2

0

c

z

0.72 0.6 0.4 0.2

0

Fig. 4. Snapshots of oscillatory flow in the meridian section (stream lines). Solid line: anticlockwise circulation. Dashed line: clockwise circulation.  = 0.72. Re = 1200. (a) t = t0 . (b) t0 + 1/8T . (c) t0 + 1/4T . T = 300: nondimensional period.

value of r is depicted in Fig. 5. The critical Reynolds number Rec at which r vanishes is determined as Rec = 966.5 from this figure. The flow pattern of the corresponding eigen function is shown in Fig. 6. The flow pattern consists of a sole vortex which extends in the whole meridian cross-section and resembles the disturbance observed in the Taylor–Couette flow for  =∞ though the vortex is skewed in the present case.

T. Miura, J. Mizushima / Fluid Dynamics Research 39 (2007) 193 – 208

5.0

203

×10−5

2.5

λr

Rec = 966.5 0.0

−2.5 −5.0 966.2

966.5 Re

966.8

Fig. 5. Linear growth rate r .  = 0.6.

z 0.6 0.4 0.2 0 0.2

0.4

0.6

0.8

1.0



Fig. 6. Flow pattern of disturbance (eigen function) in the meridian section (stream lines).  = 0.6. Re = 966.5.

Although it is very difficult to explain the reason why such instability occurs, several explanations for the mechanism of the instability in the Taylor–Couette flow have been proposed. If a fluid particle at an inner location with larger angular momentum moves to an outer site keeping its angular momentum, the particle is pushed to the outer side and comes to a further outer site. The reverse movement of a particle at an outer location is possible. This is a simplified explanation of the mechanism of instability in the Taylor–Couette flow. In the present problem, there are two large vortices symmetrical with respect to the inter-disk center line and fluid flows between them like a jet. The jet is pulled in two opposite directions by these two large voritices because the pressure inside the vortices are low. If the force from one vortex happens to be stronger than the other, the jet is attracted to the stronger vortex, and hence an asymmetric flow appears. On the other hand, the linear growth rate  has a complex value for  = 0.72, which shows overstability and induces oscillatory flow due to a Hopf bifurcation. We show the value of r in Fig. 7, from which we determine the critical Reynolds number as Rec = 1172.2. The imaginary part i , the oscillation frequency of the disturbance, is about 0.0243 at the criticality, from which the Strouhal number St is evaluated as St = i /(2 ) = 0.00387. Since the time is made nondimensional by d/r2  = (d/r2 )(1/f )(1/2 ) = 0.0796/f , the oscillation in the flow is 0.05 times per one round of the disks, which is very slow. The corresponding eigen function is also a complex function, whose real and imaginary parts are depicted in

204

T. Miura, J. Mizushima / Fluid Dynamics Research 39 (2007) 193 – 208

5.0

×10−5

2.5

λr

Rec=1172.2 0.0 −2.5 −5.0 1171.8

1172.2 Re

1172.6

Fig. 7. Linear growth rate r .  = 0.72.

Figs. 8(a) and (b). Both the flow patterns differ little from the one for the case of exchange of stabilities depicted in Fig. 6. The only difference is the existence of two vortices in the corners of the outer side of the meridian sections. The instability is thought to occur because the jet-like flow between the two vortices collide with a rigidly rotating inner fluid and the collision of the jet on the rigid-rotating fluid induces oscillation in the flow field. 4.4. Bifurcation diagram In order to analyze the bifurcation structure of the flow, we adopt the velocity w1 in the z-direction at the point (, z) = (0.8, 0.5) (P1 in Fig. 1) as a characteristic quantity which manifests the magnitude of asymmetry in the flow field. The bifurcation diagrams obtained are shown in Fig. 9, where solid and dashed lines indicate stable and unstable steady-state solutions, respectively. For  = 0.6 (Fig. 9(a)), we can see that the steady symmetric flow with respect to inter-disk midplane (AD in this figure) is stable in the range Re < Rec = 966.5 (AP), and unstable for Re > Rec (PC). There occurs a supercritical pitchfork bifurcation at Re=Rec , from which two steady asymmetric solutions (PD and PD ) appear. The bifurcation structure is almost independent of  in the range of 0.3   0.68. However, as the gap ratio  increases up to about 0.69, the bifurcation structure changes as shown in Fig. 9(b). The first bifurcation is a subcritical pitchfork bifurcation at the point P where the steady symmetric solution becomes unstable to stationary disturbance. When the Reynolds number increases, the steady symmetric solution restabilizes at Q due to a supercritical pitchfork bifurcation. In the ranges ReS (ReS )  Re  ReP and ReQ  Re  ReH , there exist multiple steady solutions. The steady symmetric solution loses stability again at the point where a Hopf bifurcation occurs. Steady asymmetric flow or oscillatory flow is realized in the range Re  ReH depending upon its initial condition. As the gap ratio  increases, the critical value indicated by P (the first subcritical pitchfork bifurcation point in Fig. 9(b)) becomes larger, while the value indicated by Q (the second supercritical pitchfork bifurcation point in Fig. 9(b)) becomes smaller. After the two bifurcation points merge together, these points disappear to yield two saddle-node branches DSE and D S E . On the other hand, the Hopf bifurcation point H remains almost unmoved as shown in Fig. 9(c). The bifurcation structure is similar to that for  = 0.694 in the range of 0.694 <   0.71.

T. Miura, J. Mizushima / Fluid Dynamics Research 39 (2007) 193 – 208

a

205

z

0.72 0.6

0.4

0.2

0

b

0.2

0.4

0.6

0.8

1.0



0.2

0.4

0.6

0.8

1.0



z

0.72 0.6

0.4

0.2

0

Fig. 8. Flow pattern of disturbance in the meridian section (stream lines).  = 0.72. Re = 1172.2. (a) Real part of eigen function. (b) Imaginary part of eigen function.

The bifurcation diagram for  = 0.72 is shown in Fig. 9(d), in which the points H and S (S ) indicate a Hopf bifurcation and saddle-node points, respectively. The steady symmetric solution first becomes unstable not to stationary disturbance, but to oscillatory one in this case. It is found that there exists a critical gap ratio c between 0.71 <  < 0.72, at which the steady symmetric flow becomes oscillatory due to a Hopf bifurcation before it makes a transition to a steady asymmetric flow when the Reynolds number is increased. Randriamampianina et al. (2001) reported the appearance of similar oscillatory flow with axisymmetry for much smaller length ratio   0.26. The oscillation period in the flow is about 1000 in nondimensional time which is about four times longer than that of the present case. However, for such a small length ratio the first bifurcation is axisymmetry breaking so that the oscillatory flow with axisymmetry may not be observed in experiment. 4.5. Transition diagram As shown in the preceding subsection, the supercritical pitchfork bifurcation occurs in the range of 0.3  0.68, whereas the bifurcation structure changes from supercritical to subcritical, and the flow becomes unstable not only to stationary disturbance but also to oscillatory one for 0.69   0.8.

T. Miura, J. Mizushima / Fluid Dynamics Research 39 (2007) 193 – 208

206

a

b

0.04 D

0.03

D

0.02

0.02 S

0

A

C

P

w1

w1

0.01

1100 Re

1300

d D

1200

S

H

−0.01

S'

0

D'

1100

1200 Re

1300

E

F

A

−0.01

−0.03 1000

C

H F' S

−0.02

−0.02 −0.03 1000

w1

E C E'

D

S

0.01

A

1300

0.03 0.02

0.02

w1

D'

1100 Re

0.03

0.01

E C E'

QH

S

−0.03 1000

−0.04 900

P

−0.02

D'

0

A

−0.01

−0.02

c

0

1100

1200

E' D'

1300

Re

Fig. 9. Bifurcation diagram. w1 : axial velocity at the point (, z) = (0.8, 0.5). Solid line: stable steady-state solution. Dashed line: unstable steady-state solution. (a)  = 0.6. (b) 0.69. (c) 0.694. (d) 0.72.

We evaluated the critical Reynolds numbers for the pitchfork, Hopf bifurcations, and saddle-node point in the range of 0.3  0.8 and obtained a transition diagram as summarized in Fig. 10. In this figure, the critical Reynolds number for the pitchfork bifurcation is indicated by circle and that for Hopf bifurcation by triangle and saddle-node points by square. As shown in this figure, the flow is always steady and symmetric with respect to the inter-disk midplane in region P, whereas in region Q the steady symmetric flow is unstable and two steady asymmetric flows are stable. In region T, the steady asymmetric flow is stable, and oscillatory flow is also possible. The critical values for the pitchfork bifurcation evaluated by Randriamampianina et al. (2001) are indicated by crosses. There is a significant difference between our evaluation and theirs. Although every author asserts validity of his own data, the judging should be remitted to third work in the future. Fig. 10(b) is an enlargement of Fig. 10(a). It is seen in this figure that not only the steady asymmetric flows but also the symmetric flow are stable in region R, and we can observe hysteresis due to the occurrence of the subcritical pitchfork bifurcation in region R1 . The critical Reynolds number for the Hopf bifurcation becomes smaller than that for the saddle-node bifurcation for  > c ∼ 0.712 (shown in Fig. 10(b)). Consequently, only the oscillatory flow is realized in the region S.

T. Miura, J. Mizushima / Fluid Dynamics Research 39 (2007) 193 – 208

a

207

3000 2500

Re

2000 Q 1500

T P

1000 500 0.3

0.4

0.5

0.6

0.7

0.8



b

1300 T S

Re

1200 R

Q

c∼ 0.712

1100

1000

900 0.68

R1

P

0.70 

0.72

Fig. 10. Transition diagrams. Circle: pitchfork bifurcation point (present results). Cross: pitchfork bifurcation point Randriamampianina et al., 2001. Triangle: Hopf bifurcation point. Square: saddle-node bifurcation point. Region P: steady symmetric flow. Region Q: steady asymmetric flow. Region R, R1 : stable steady symmetric and asymmetric flows. Region S: oscillatory flow. Region T: steady symmetric and oscillatory flows. (a) Whole diagram. (b) Enlargement.

5. Concluding remarks Transitions of flow between two corotating disks in an enclosure were investigated by numerical simulations and linear stability analyses of the axisymmetric flow. At small Reynolds numbers, the flow is steady and symmetric with respect to inter-disk midplane, irrespective of the value of the gap ratio. The flow field becomes asymmetric due to instability above a critical Reynolds number. For small gap ratios (0.3   0.68), the first instability is a supercritical pitchfork bifurcation, which leads to a transition of flow to an asymmetric flow. The bifurcation structure changes to a subcritical pitchfork bifurcation when the gap ratio increases up to about 0.69, which inevitably accompanies a hysteresis. For large gap ratios (0.712 <  0.8), the steady symmetric flow makes a transition directly to an oscillatory flow due to a Hopf bifurcation. The finding of these three kinds of axisymmetric flow is expected to play an important role in clarifying the bifurcation structure of the flow to nonaxisymmetric flows with a polygonal flow pattern, because there are two types of polygonal-shape flow, i.e. an asymmetric flow in the meridional

208

T. Miura, J. Mizushima / Fluid Dynamics Research 39 (2007) 193 – 208

plane and a flow with shift-and-reflect symmetry. The former is expected to stem from the axisymmetric flow which is asymmetric with respect to the inter-disk midplane and the latter appears due to a Hopf bifurcation from the flow with the symmetry. Acknowledgments This paper is dedicated to the late Prof. Isao Imai. One of the authors (J.M.) is grateful to him for his interest and invaluable comments on my works and for continuous encouragement. References Abrahamson, S.D., Eaton, J.K., Koga, D.J., 1989. The flow between shrouded corotating disks. Phys. Fluids A 1, 241–251. Benjamin, T.B., 1978. Bifurcation phenomena in steady flows of a viscous fluid. Part A and Part B. Proc. R. Soc. Ser. A 359, 1–26 27–43. Benjamin, T.B., Mullin, T., 1981. Anomalous modes in Taylor–Couette flow. Proc. R. Soc. Ser. A 377, 221–249. Bolstad, J.H., Keller, H.B., 1987. Computation of anomalous modes in the Taylor experiment. J. Comput. Phys. 69, 230–251. Golub, G.H., Van Loan, C.F., 1983. Matrix Computations. Johns Hopkins Univ. Press, Baltimore, MD (Section 7.7). Herrero, J., Giralt, F., Humphrey, J.A.C., 1999. Influence of the geometry on the structure of the flow between a pair of corotating disks. Phys. Fluids 11, 88–96. Humphrey, J.A.C., Gor, D., 1993. Experimental observations of an unsteady detached shear layer in enclosed corotating disk flow. Phys. Fluids A 5, 2438–2442. Humphrey, J.A.C., Schuler, C.A., Webster, D.R., 1995. Unsteady laminar flow between a pair of disks corotating in a fixed cylindrical enclosure. Phys. Fluids 7, 1225–1240. Lennemann, E., 1974. Aerodynamic aspects of disk files. IBM J. Res. Dev. 18, 480–488. Radel, V.S., Szeri, A.Z., 1997. Symmetry breaking bifurcation in finite disk flow. Phys. Fluids A 9, 1650–1656. Randriamampianina, A., Schiestel, R., Wilson, M., 2001. Spatio-temporal behaviour in an enclosed corotating disk pair. J. Fluid Mech. 434, 39–64. Schuler, C.A., Usry, W., Weber, B., Humphrey, J.A.C., Greif, R., 1990. On the flow in the unobstructed space between shrouded corotating disks. Phys. Fluids A 2, 1760–1770. Tavener, S.J., Mullin, T., Cliffe, K.A., 1991. Novel bifurcation phenomena in a rotating annulus. J. Fluid Mech. 229, 483–497. Taylor, G.I., 1923. Stability of a viscous liquid contained between two rotating cylinders. Philos. Trans. R. Soc. London Ser. A 223, 289–343.