After Benjamin’s [1] study, the Taylor vortex flow between two concentric rotating cylinders with finite length has been investigated from various viewpoints.　stigated from various viewpoints.　 The aspect rati? and the Reynolds number Re are considered as the main parameters that determine the modes of the Taylor vortex flow [2][3][4].　 are considered as the main parameters that determine the modes of the Taylor vortex flow [2][3][4].　 The aspect rati? is defined as the ratio of the cylinder length to the gap between cylinders, and the Reynolds number Re is based on the rotation speed of the inner cylinder.　 is based on the rotation speed of the inner cylinder.　 Alziary de Roquefort and Grillaud [5] analyzed numericallthe Taylor vortex flow by using steady axisymmetric Navier-Stokes equations formulated by the finite difference method.　 Lucke et al. [6][7] used the unsteady equations, and they reported that Ekman vortices that develop on the　cke et al. [6][7] used the unsteady equations, and they reported that Ekman vortices that develop on the　stationarend walls cause the bulk Taylor vortex flow when the inner cylinder started to rotate suddenly from rest.　rotate suddenly from rest.　 K and Ball [8] took the effect of the buoyancy into account and conducted the three dimensional numerical simulation.　buoyancy into account and conducted the three dimensional numerical simulation.　 They found the steady modein the case that the rotational speed of the inner cylinder was gradually increased from zero. The Taylor vortex flow has a normal mode and an anomalous mode.　tex flow has a normal mode and an anomalous mode.　 When thend walls of the cylinders are fixed, the normal mode has a normal cell that gives an inward flow in the region adjacent to the end wall.　.　 The anomalous mode has anomalous cell(s) on either or bh end walls.　s.　 The anomalous cell gives an outward flow near the end wall, which is opposite to the flow directiofound in the normal mode.　in the normal mode.　 Bielek et al. [9] observed experimentally thmode formation processes of the Taylor vortices, and　Taylor vortices, and　they have concded that the anomalous three-cell mode never formed directly after sudden start of the inner cylinder but originated from the decay of the anomalous four-cell mode or the six-cell mode.

One major character of the Taylor vortex flow is its non-uniqueness.　One major character of the Taylor vortex flow is its non-uniqueness.The difference of the inner cylinder acceleration rate causes various modes, though the aspect ratio and the Reynolds number are fixed.　The difference of the inner cylinder acceleration rate causes various modes, though the aspect ratio and the Reynolds number are fixed.As mention above, a large number of researches have been carried out into the Taylor vortex flow, but little is known about the non-uniqueness of the mode formation processes of the Taylor vortex flow. The effect of the acceleration rate of the inner cylinder on the mode formation processes has not been reported in detail.　.　 When the flow pattern changes,　unsteady variatns of property values such as torque and kinetic energy arise. The unsteady variation carries the potential for the unexpected system destruction. 　Therefore, the analysis and prediction of the mode formation processes is important from the engineering viewpoint. In this study, the two-dimensional flow visualization numerically demonstrates the mode formation processes of the Taylor vortex flow between two concentric rotating cylinders.　cylinders.　 Under the condition that the aspect ratio and the Reynolds number e fixed, the rotation velocity of the inner cylinder is increased from zero at various acceleration rates.　of the inner cylinder is increased from zero at various acceleration rates.The non-unique mode formation processes from the sudden-start of the inner cylinder have also been investigated by us [10].

All physical parameters are made dimensionless by using the characteristic length that is the gap between cylinders, the characteristic velocity that is the maximum circumferential rotation speed attained during each calculation, and the characteristic time defined as the ratio of the characteristic length to the characteristic velocity.　fined as the ratio of the characteristic length to the characteristic velocity.　 The governing equations are the unsteady axisymmetric Navier-Stokes equations and the equation of continuity, which are expressed in the cylindrical coordinatesr, ?, z),

where u is velocity vector with components (u, v, w) in each direction, p is pressure and t is time. Staggered grid is used in this calculation.　Staggered grid is used in this calculation.　 The number of grid pots in the radial direction is 41, and the number of grid points in the axial direction is determined by the proportionality to the cylinder length with 42 points for an aspect ratio of unity.　ts in the radial direction is 41, and the number of grid points in the axial direction is determined by the proportionality to the cylinder length with 42 points for an aspect ratio of unity.　 The number of grid points is estimatelarge enough not to influence results, and the time interval fills the CFL condition.　uence results, and the time interval fills the CFL condition.　 The basic solution procedure is the MAC method, and the time integration is the Euler explicit method.　 The spatial differentiation is the QUICK method for convection terms and the second-orr central difference method for other terms.　 central difference method for other terms.　 The inner cylinder is accelerated linearly from zerduring the non-dimensional accelerating time T.　.　 The initial conditions are that all velocity components are zero in the entire domain.　 The outer cylinder de, the upper end wall and lower end wall are stationary.　de, the upper end wall and lower end wall are stationary.　 The boundary conditions at the cylinder walls and both end walls are no-slip conditions for velocity components, and Neumann conditions for pressure that are obtained from momentum equations.　 T Stokes stream function is determined as follows, which is used for visualization of the calculated results.

To analyze the property of the Taylor vortex flow, we adopt the mean kinetic energy E and the mean enstrophy ? which are defined by

where S is an integral domain and A is the area of a meridional section.　 is the area of a meridional section.　 We run the calculations 1210 times a? from 2.6 to 4.6, Re from 100 to 1000 and 10 different Ts.

Table 1 shows the final modes of the Taylor vortex flow at ? = 4.0 and Re = 700 in the case that the inner cylinder is accelerated linearly from zero during T.　.　 Each column of the table indicates the non-dimensional acceleration timT, flow mode, the mean kinetic energy E and the mean enstrophy ?.　.Depending on the acceleration time of the inner cylinder, three different modes appear: the anomalous four-cell mode (A4), the normal two-cell mode (N2) and the normal four-cell mode　ur-cell mode (A4), the normal two-cell mode (N2) and the normal four-cell mode　 (N4).At the same final mode, the energy E and the enstrophy ? are the same.　.　 Table shows that the various modes appear even when ? and Re are fixed. This has been known as the non-uniqueness of the Taylor vortex flow [1].

Figure 1 illustrates the time variations of counters in　 counters in　ther, z) plane to shows the mode formation processes of the Taylor vortex flow at some Ts.　.　 In the figure, the rotating ier cylinder is on the left and the stationary outer cylinder is on the right.　er cylinder is on the left and the stationary outer cylinder is on the right.　　The warm color area shows a vortex rotating in a clockwise directi, and a vortex rotating in a counter-clockwise direction is shown cold color area.　 and a vortex rotating in a counter-clockwise direction is shown cold color area.　 Figure 1 (ais the AVI animation that shows the mode formation process of the anomalous four-cell mode.　the mode formation process of the anomalous four-cell mode.　 First, two vortices are generatearound the mid-plane in the axial direction and they develop in the radial direction.　the mid-plane in the axial direction and they develop in the radial direction.　 On the other hand, anoth vortices are formed at the inner-lower and　lower and　the inner-upper corns.　.After a while, the flow field becomes the normal six-cell mode.　the flow field becomes the normal six-cell mode.Then, the vortex on the cylinder end wall is pushed aside by the second vortex from the cylinder end wall, and the end-wall cell is divided into two cells: one on the inner cylinder and the other on the outer cylinder. Finally, the boundaries of the interior vortices reach to the cylinder end wall, and the flow field becomes the stable anomalous four-cell mode.

Figure 1 (d) shows the mode formation process of the normal four-cell　ell　mode.　 The vortices are generateat the inner-lower and the inner-upper corners.　.First, the normal ten-cell mode is formed. Then, the second vortices from the end walls weaken and disappear.　weaken and disappear.The vortices adjacent to the disappearing vortices rotate in the same direction, and they merge into one vortex.　merge into one vortex.Next, the flow field becomes the normal six-cell mode, and the second vortex and the third vortex from the lower cylinder end wall weaken and disappear.　he flow field becomes the normal six-cell mode, and the second vortex and the third vortex from the lower cylinder end wall weaken and disappear.The final mode is the normal four-cell mode.　

To determine the number of vortices quantitatively, we use the time variation of the wave number components of the power spectrum of integrated in the radial direction, which is determined by

Table 3 presents the mode formation processes at ? = 4.8 and Re = 1000.　 = 1000.Depending on the acceleration rate of the　the acceleration rate of the　inner cylind, the flow has five different modes: the anomalous five-cell mode (A5), the anomalous four-cell mode (A4), the anomalous six-cell mode (A6), the normal six-cell mode (N6) and the normal four-cell mode (N4). At ? = 4.8 and Re = 1000, the non-uniqueness of the Taylor vortex flow is found, as is seen at ?　 = 4.0 and Re = 700.　 = 700.　 The normal six-cell mos are not stable, and their kinetic energy and mean enstrophy change with time.　 Figure 3 and 4 show the mode formation processes and the time variation of S_k, respectively.

In the mode formation processes described in Section 3.1, the normal two-cell mode is generated in two ways.　two ways.In one way, the flow field changes from the normal six-cell mode to the normal two-cell mode (Fig. 1 (b)).　.In the other way, the flow field changes from the normal ten-cell mode to the normal two-cell mode (Fig. 1 (e)).　.　 Similarly,　the anomalous four-cell mod(Fig. 1 (a) and (c)) and the normal four-cell mode (Fig. 1 (d) and (f)) have two different mode formation processes, respectively.　.　 Non-uniquenesof the Taylor vortex flow has been used to mean that the flow field has different modes at constant ? and Re.　.　 Now we fina new non-uniqueness of the Taylor vortex flow, which means that the mode formation processes are different even though the flow field has the same final mode.　.We can also find a new non-uniqueness in the formation processes mentioned in Section 3.2 (Fig. 3 (b) and (e).

Bielek et al. [9] experimentally concluded that the anomalous three-cell mode never formed directly after sudden start of the inner cylinder, but the mode originated from the decay of the anomalous four-cell mode or from the six-cell mode.　originated from the decay of the anomalous four-cell mode or from the six-cell mode.　 In this study, the anomalous three-cell mode appearat Γ= 2.6, Re= 900 and T= 10.8.　 10.8.In this case, the flow mode shifts from the normal six-cell mode, to the anomalous four-cell mode, then to the anomalous three-cell mode.　the anomalous three-cell mode.　 The result in this study agrs with Bielek et al.’s experimental observation.　observation.

At the fixed ? and Re, the order of the values of the mean enstrophy is listed as A2 < A3, A4 < A6 < N2 < N4 < N6.　A2 < A3, A4 < A6 < N2 < N4 < N6.In general, the mean enstrophy of the normal mode is larger than that of the anomalous mode, and the larger number of vortices the flow field has, the larger the mean enstrophy is.　.　 Howeversome exceptions are found, for example, the enstrophy of the normal two-cell mode (N2) is less than that of the anomalous three-cell mode (A3). Therefore the relation of the mean　 the relation of the mean　enstrophy between mos is not concluded clearly.　clearly.The relation of the mean kinetic energy between modes is not determined, neither.

The Taylor vortex flow between two concentric rotating cylinders with finite length has been investigated using the numerical flow visualization.　ation.　 The non-uniquenesof the Taylor vortex flow, which has been found in previous studies, is confirmed in the present study. Though the same final mode at constant ? and Re is formed, we found that the mode formation processes may be depend on the acceleration rates of the inner cylinder.　 That is, another modes may be taken during the flow development. At the same final mode, the mean kinetic energy is identical, and so is the mean enstrophy.　the mean enstrophy.　　Thtime variation of the power spectrums of the integrated in the radial direction help us to determine the number of vortices quantitatively.　 integrated in the radial direction help us to determine the number of vortices quantitatively.The first vortices appear around the mid-plane as well as at the inner-lower and the inner-upper corners at the beginning of the mode formation, and the location depends on the acceleration rate of the inner cylinder. In the mode formation processes of the anomalous mode, the second vortex from the end wall divides the normal cell on the end wall into the inner and the outer regions of the annulus. Then, the second vortex reaches the cylinder end wall, and the anomalous mode is formed.

1. Benjamin T. B.: Bifurcation phenomena in steady flows in a viscous flow - II. Experiment-. Proceedings of the Royal Society of London, Series A, Vol. 359, pp. 27-43, 1979.
2. Cliffe K.A.: Numerical calculation of two-cell and single cell Taylor flow. Journal of Fluid Mechanics, Vol. 135, pp. 219-233, 1983.
3. Cliffe K. A.: Numerical calculation of the primary flow exchange process in the Taylor problem. Journal of Fluid Mechanics, Vol. 197, pp. 57-79, 1988.
4. Bolstad J. H. and Keller H. B.: Computation of anomalous modes in the Taylor experiment. Journal of computational Physics, Vol. 69, pp. 230-251, 1987.
5. Alziary de Roquefort, T. and Grillaud, G.: Computation of Taylor vortex flow by a transient implicit method. Computers and Fluids, Vol. 6, pp. 259-269, 1978.
6. Lücke M., Mihelicic M. and Wingerath, K.: Flow in a small annulus between concentric cylinders. Journal of Fluid Mechanics. 140, pp. 343-353, 1984.
7. Lucke M., Mihelcic M. and Wingerath, K.: Front propagation and pattern formation of Taylor vortices growing into unstable circular couette flow. Physical Review, A, Vol. 31, No. 1, pp. 396-409, 1985.
8. Kuo D.C. and Ball K.S.: Taylor-Couette flow with buoyancy: onset of spiral flow. Physics of Fluids, Vol. 9, No. 10, pp. 2872-2884, 1997.
9. Bielek C.A. and Koschmieder E.L.: Taylor vortices in short fluid columns with large radius ratio. Physics of Fluids, A 2-9, pp. 1557-1563, 1990.
10. Watanabe T., Furukawa H., Aoki M. and Nakamura I.: Visualization of sudden-start flow between two concentric rotating cylinders. Proc. 9^th Int. Symp. Flow Visualization (this volume).