Centrifugal instability of Taylor-Couette flow in stratified and diffusive fluids

Abstract

The linear and non-linear dynamics of centrifugal instability in Taylor-Couette flow are investigated when fluids are stably stratified and highly diffusive. One-dimensional local linear stability analysis (LSA) on cylindrical Couette flow confirms that the stabilising role of stratification on centrifugal instability is suppressed by strong thermal diffusion (i.e. low Prandtl number $Pr$). For $Pr\ll1$, it is verified that the instability dependence on thermal diffusion and stratification with the non-dimensional Brunt-V\"ais\"al\"a frequency $N$ can be prescribed by a single rescaled parameter $P_{N}=N^{2}Pr$. From direct numerical simulation (DNS), various non-linear features such as axisymmetric Taylor vortices at saturation, secondary instability leading to non-axisymmetric patterns or transition to chaotic states are investigated for various values of $Pr\leq1$ and the Reynolds number $Re_{i}$. Two-dimensional bi-global LSA on axisymmetric Taylor vortices, which appear as primary centrifugal instability saturates nonlinearly, is also performed to find the secondary critical Reynolds number $Re_{i,2}$ at which the Taylor vortices become unstable by non-axisymmetric perturbation. The bi-global LSA reveals that $Re_{i,2}$ increases (i.e. the onset of secondary instability is delayed) in the range $10^{-3}<Pr<1$ at $N=1$ or as $N$ increases at $Pr=0.01$. Secondary instability leading to highly non-axisymmetric or irregular chaotic patterns is further investigated by the 3D DNS. The Nusselt number $Nu$ is also computed from the torque at the inner cylinder for various $Pr$ and $Re_{i}$ at $N=1$ to describe how the angular momentum transfer increases with $Re_{i}$ and how $Nu$ varies differently for saturated and chaotic states.