Survey of instability thresholds of flow between exactly counter-rotating disks

Abstract

The three-dimensional linear instability of axisymmetric flow between exactly counter-rotating disks is studied numerically. The dynamics are governed by two parameters, the Reynolds number $Re$ based on cylinder radius and disk rotation speed and the height-to-radius ratio $\Gamma$. The stability analysis performed for $0.5 \,{\le}\, \Gamma \,{\le}\, 3$ shows that non-axisymmetric modes are dominant and stationary and that the critical azimuthal wavenumber is a decreasing function of $\Gamma$. The patterns of the dominant perturbations are analysed and a physical mechanism related to a shear layer instability is discussed. No evidence of complex dynamical behaviour is seen in the neighbourhood of the 1:2 codimension-two point when the $m\,{=}\,2$ threshold precedes that of $m\,{=}\,1$. Axisymmetric instabilities are also calculated; these may be stationary or Hopf bifurcations. Their thresholds are always higher than those of non-axisymmetric modes.