Unsteady mode of fully developed flow

In the area labelled ``Rotation'' in Fig $.\,\,\ref{fig:table}$, flows are time dependent. Unsteady flows show periodical motions.

Figures 5 and 6 show the time variation of the mean enstrophy ${\sl\Omega}$ at ${\sl\Gamma }$= 0.5 and Re = 600, 1500. ${\sl\Omega}$ is given by

$${\sl\Omega} = \frac{1}{A} \int_S \left(\frac{\partial \boldma... ...c{\partial \boldmath w}{\partial \boldmath r} \right)^2 dr dz,$$ (4)

where S is an integral domain and A is the area of a meridional section.

Figure 7 shows the contours of the stream function ${\sl\psi}$ at time points shown in Fig $.\,\,\ref{fig:ens-1500}$. While the time variation of ${\sl\Omega}$ is almost sinusoidal at Re = 600, the flow at Re = 1500 is not monotonic and ${\sl\Omega}$ is reduced suddenly from time point 4 to time point 5. The mean kinetic energy in the (r, z) plane exhibits a similar variation to that of the mean enstrophy.

The time variation of flow field at Re = 1500 is shown in Fig $.\,\,\ref{fig:rot}$. The time points from 1 to 6 correspond to those shown in Fig $.\,\,\ref{fig:ens-1500}$. At time point 1 in Fig $.\,\,\ref{fig:rot}$(a), the flow has four vortices in the radial direction. As ${\sl\Omega}$ increases, the vortex closest to the outer cylinder decays and three vortices remain (Fig $.\,\,\ref{fig:rot}$(b)). When ${\sl\Omega}$ increases further, the vortices at the two sides of the cylinder merge (Fig $.\,\,\ref{fig:rot}$(c)), and a flow field with two large vortices appears at the time point where ${\sl\Omega}$ was maximized (Fig $.\,\,\ref{fig:rot}$(d)). Then each vortex is split by the other and four vortices emerge (Figs $.\,\,\ref{fig:rot}$(e)(f)). Taking into account the fact that the pattern in Fig $.\,\,\ref{fig:rot}$(f) is the vertical reverse of the pattern in Fig $.\,\,\ref{fig:rot}$(a), we can say that the two periods in the ${\sl\Omega}$ graph correspond to one period of unsteady motion.

When Re is 600, flows are weak and each of the vortices is not split by the other.

When the aspect ratio was less than 0.6, we confirmed that the unsteady motion predicted by the present calculation is found experimentally. As well, the calculated and experimental results of the range of ${\sl\Gamma }$ and Re in which the unsteady mode appears show good agreement.