Introduction

After Benjamin's (1979) and Benjamin and Mullin's (1981) studies which revealed the presence of the primary, normal secondary and anomalous secondary modes, many numerical investigations have been carried out on Taylor vortex flows between two concentric cylinders of finite length. Cliffe (1983, 1988) and Bolstad and Keller (1987) used steady Navier-Stokes equations and determined the bifurcation loci between the primary and secondary modes in the $(Re, {\sl\Gamma})$ plane, where Re is the Reynolds number based on the rotation speed of the inner cylinder and ${\sl\Gamma}$ is the aspect ratio defined as the ratio of the cylinder length to the gap length between cylinders. Bolstad and Keller also pointed out that, even in an anomalous mode with an even number or odd number of cells, a hidden vortex appears and the flow is not at all outward in the region adjacent to the end wall. Pfister and Smith (1988) and Ball and Farouk (1988) used unsteady equations and revealed the boundaries separating regions of different modes. In those studies, fully developed Taylor flows were examined, and flow with a given Reynolds number and aspect ratio was calculated and the mode was determined.

Even when the Reynolds number is less than the critical value at which steady Taylor cells arise, it is known that Ekman vortices develop on the stationary end walls of an annulus of finite length (Kuo and Ball, 1997). The numerical investigation of the time-dependent problem of flow after a sudden increase in the rotation speed from zero showed that Ekman vortices growing near the end walls caused an imperfection breaking the translational symmetry of the system, and then a Taylor vortex flow propagated into the bulk of the annulus (Lucke et al., 1985). Similar unsteady phenomena were experimentally investigated by Takeda et al. (1990), though the Taylor vortex flow induced by Ekman vortices was not confirmed. In those studies, the aspect ratio was not small, i.e., 20 or 25, and no multiple modes appeared. On the other hand, Alziary de Roquefort and Grillaud (1978) conducted a numerical study of time-dependent flow with increasing rotation speed and found that the Ekman vortices at low Reynolds number and nonunique states were formed.

When the Reynolds number decreases gradually, the Taylor vortex flow with the secondary mode is attracted to the primary mode. Nakamura et al. (1989) investigated these phenomena experimentally and limited the range of the Reynolds number in which the mode changes. They also clarified the bifurcation process from the secondary mode, occasionally via another secondary mode, to the primary mode.

The mode transitions in Taylor vortex flows are interesting from the engineering and nonlinear dynamics points of view. In this work, we think of the mode change processes developing between the rotating inner cylinder and the stationary outer cylinder as time-dependent phenomena and attempt to predict them. That is, we examine the unsteady transformation from the secondary mode to the primary mode during the slow reduction of the Reynolds number. Then, comparing the numerical results with the experimental results, we clarify the mechanisms of the phenomena which are difficult to reveal experimentally.

In the following, Section 2 describes the numerical method used in this study and Section 3 presents the numerical results. Section 4 gives a discussion on mode exchange and Section 5 gives conclusions.