Subcritical instability of finite circular Couette flow with stationary inner cylinder

Abstract

The presence of endwalls in Taylor–Couette flows has far reaching effects, leading to dynamics that are qualitatively different from those associated with the idealized situation involving infinitely long cylinders. This is well known in the classical situation where the inner cylinder is rotating and the outer cylinder is stationary. The effects of endwalls in the centrifugally stable situation with stationary inner cylinder and rotating outer cylinder have not been previously considered in detail. The meridional flows induced by the endwalls lead to the formation of a thin sidewall boundary layer on the inner cylinder wall if the endwalls are rotating, or on the outer cylinder wall if they are stationary. At sufficiently high Reynolds numbers (non-dimensional rotation rate of the outer cylinder), the sidewall boundary layer has concentrated shear, the pressure gradient in the azimuthal direction (which is the streamwise direction for the boundary layer flow) is zero (the flow is axisymmetric) and the boundary layer thickness is constant. At a critical Reynolds number, the sidewall boundary layer loses stability at a subcritical Hopf bifurcation, breaking the axisymmetry of the basic state flow, and for Reynolds numbers slightly above critical, the basic state is unstable to a packet of Hopf modes with azimuthal wavenumbers clustered about the critical wavenumber. The early time evolution of the critical Hopf mode is a rotating wave whose behaviour is analogous to a Tollmien–Schlichting wave. As the Hopf modes grow with time, nonlinear interactions lead to modulations in the waves, localization of the disturbances and the evolution of concentrated streamwise vortical streaks which become very intense via vortex stretching.