Mode bifurcation

When a fully developed flow at a specific Reynolds number is decelerated by the gradual decrease of Re, the transition to an other mode may occur. We investigated this phenomenon at the moderate aspect ratio, and compared our calculated results with experimental results (Watanabe and Maeda, 1997). In order to extend earlier results, we observed the transition among steady modes N2, A1 and TWIN in Fig $.\,\,\ref{fig:table}$when Re is gradually decreased.

Figure 8 shows the time variation of the ${\sl\psi}$ contour during the mode transition process from A1 to N2 when ${\sl\Gamma }$ is 0.8 and Reis decreased from 500 to 100. A similar transition was observed when Re was decreased from 350 to 340 and the reduction was 10. At t=1500, an anomalous cell accompanies with extra cells at both cylinder sides. As rotation speed decreases, the extra cells merge with each other and form one new normal cell, and the flow becomes the normal two-cell mode.

Figure 9 shows the mode transition process from N2 to A1 when ${\sl\Gamma }$ is 1.0 and Reis decreased from 800 to 500. As rotation decelerates, one cell enlarges. The other cell is pushed aside and is divided into two cells: one on the inner cylinder and the other on the outer cylinder. Finally the flow becomes the anomalous one-cell mode.

The ranges of Re in which the mode changes from N2 to A1 and from A1 to N2 are shown in Fig $.\,\,\ref{fig:bif}$, where the difference between two Reynolds numbers is narrowed to about 10 and deceleration is slowed. Cliffe's result (1983), obtained by using the steady equation, is also given in Fig $.\,\,\ref{fig:bif}$. The upper loci of transition point from normal two-cell mode to anomalous one-cell mode are in quantitative agreement. However, a difference is found among lower loci. This is because our results are not those for a steady state but have been obtained from an unsteady process.

Figure 11 shows the mode transition process from TWIN to A1 when ${\sl\Gamma }$ is 0.8 and Reis reduced from 1000 to 600. At t = 3000, an extra cell is formed around the inner corner of the cylinder where r = 0.0, z = 0.0. As rotation speed decreases, the separation point on the wall opposite to the end wall on which the extra cell attaches gradually shifts outward and moves onto the outer cylinder. Then the flow becomes the anomalous one-cell mode.