Spatio-temporal behaviour in an enclosed corotating disk pair

Abstract

We present a numerical investigation of the flow between corotating disks with a stationary outer casing – the enclosed corotating disk pair configuration. It is known that in such a geometry, axisymmetric and three-dimensional flow regimes develop depending on the value of the rotation rate. The three-dimensional flow is always unsteady flowing to its wavy structure in the radial-tangential plane. Axisymmetric regimes exhibit first a pitchfork bifurcation, characterized by a symmetry breaking with respect to the inter-disk midplane, before a Hopf bifurcation is established. The regime diagrams for these bifurcations are given in the (Re, G)-plane, where Re(= Ωb²/ν) is the rotational Reynolds number and G(= s/(b−a)) is the gap ratio. For values of G smaller than a critical limit G_c ∼ 0.26, there exists a range of rotation rates where the motion becomes time-dependent before bifurcating to a steady symmetry breaking regime. It is shown that for G [ges ] G_c the transition to unsteady three-dimensional flow occurs after the pitchfork bifurcation, and the flow structure is characterized by a shift-and-reflect symmetry. The transition to three-dimensional flow is consistent with experimental observations made by Abrahamson et al. (1989) where multiple solutions develop (known as the intransitivity phenomenon) with the presence of quasi-periodic behaviour resulting from successive vortex pairings. On the other hand, for smaller values of gap ratio, the three-dimensional flow shows a symmetry breaking. Finally, it is found that the variation of torque coefficient as a function of the rotation rate is the same for both the axisymmetric and three-dimensional solutions.