Transitions of axisymmetric flow between two corotating disks in an enclosure

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.