Cell pattern and stagnation ring of the flow driven by the counter-rotation in a fluid-filled cylinder

For the flow driven by the counter-rotation between the top and bottom endwalls in a fluid-filled cylinder, a stagnation ring can be observed on the slower rotating endwall in experiment. Its appearance corresponds to a two-cell flow pattern in the meridional plane, where a flow separation forms in the Ekman boundary layer. In this paper we numerically show that, in addition to the single-cell and two-cell patterns previously studied, there exist more complex cell patterns, namely, three-cell and merged-cell patterns, when the flow is driven under differently counter-rotating manner that is realized between the top and bottom endwalls as a whole against the sidewall. Such a counter-rotating flow makes the stagnation ring to appear simultaneously on both top and bottom endwalls rather than just on the slower rotating endwall. Moreover, the three-cell and merged-cell patterns, which are formed by a combination of the Ekman layer separation with the “vortex breakdown bubble”, are unique characteristics. The appearance of the cell pattern and stagnation ring is primarily decided by the counter rotation-rate ratio s, but is also affected by the Reynolds number Re and height-to-radius aspect ratio Λ, so a cell-pattern zone and a stagnation ring zone are proposed numerically as a function of s and Re for a given Λ.