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

TL;DR

This study explores how stratification and thermal diffusion influence centrifugal instability in Taylor-Couette flow, revealing the stabilizing effects, nonlinear behaviors, and transition thresholds through linear stability analysis and direct numerical simulations.

Contribution

It introduces a combined parameter $P_{N}$ to describe instability dependence and analyzes secondary instability thresholds and nonlinear flow patterns in stratified, diffusive fluids.

Findings

Stratification's stabilizing effect is suppressed by strong thermal diffusion at low Prandtl numbers.

The critical Reynolds number for secondary instability increases with decreasing Pr or increasing stratification.

Nonlinear simulations show transition from Taylor vortices to chaotic flow states depending on parameters.

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.