Three-dimensional instability of axisymmetric flow in a rotating lid–cylinder enclosure

Abstract

The axisymmetry-breaking three-dimensional instability of the axisymmetric flow between a rotating lid and a stationary cylinder is analysed. The flow is governed by two parameters – the Reynolds number Re and the aspect ratio γ (=height/radius). Published experimental results indicate that in different ranges of γ axisymmetric or non-axisymmetric instabilities can be observed. Previous analyses considered only axisymmetric instability. The present analysis is devoted to the linear stability of the basic axisymmetric flow with respect to the non-axisymmetric perturbations. After the linearization the stability problem separates into a family of quasi-axisymmetric subproblems for discrete values of the azimuthal wavenumber k . The computations are done using the global Galerkin method. The stability analysis is carried out at various densely distributed values of γ in the range 1 < γ < 3.5. It is shown that the axisymmetric perturbations are dominant in the range 1.63 < γ < 2.76. Outside this range, for γ < 1.63 and for γ > 2.76, the instability is three-dimensional and sets in with k = 2 and k = 3 or 4, respectively. The azimuthal periodicity, patterns, characteristic frequencies and phase velocities of the dominant perturbations are discussed.