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

TL;DR

This work analyzes the centrifugal instability of Taylor-Couette flow in stably stratified, highly diffusive fluids at low Prandtl numbers. It combines 1D local linear stability analysis, 2D bi-global stability analysis on saturated Taylor vortices, and direct numerical simulations to examine linear growth, nonlinear saturation, and secondary instabilities across $Pr\le1$ and varying $N$ and $Re_i$. A key finding is that strong thermal diffusion suppresses the stabilizing effect of stratification, leading to a unifying small-$Pr$ scaling with the parameter $P_N=N^2Pr$; axisymmetric perturbations dominate at onset, while secondary non-axisymmetric modes can be delayed or promoted depending on $Pr$ and $N$, sometimes yielding wavy or chaotic states. The Nusselt number $Nu$ reveals how angular-momentum transport increases with $Re_i$ and is modulated by secondary transitions. These results have implications for angular momentum transport in astrophysical and geophysical flows and provide a framework for predicting stability and pattern formation in stratified, diffusive rotating shear systems.

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-Väisälä 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.

Figures (18)

• Figure 1: (Left) Illustration on how the internal oscillation of a fluid parcel in stably stratified fluid is suppressed by a fast thermal diffusion process. (Right) Schematic of Taylor-Couette flow with stable temperature stratification.
• Figure 2: (a) Growth-rate curves for various sets of $(N,Pr)$ at $\mu=0$, $\eta=0.9$, $Re_{i}=200$ and $m=0$. (b) Neutral stability curves in the parameter space $(N,Re_{i})$ for different $Pr$ at $\mu=0$, $\eta=0.9$ and $m=0$. (c) The curves for different $Pr$ same as (b) but over a wider range of $N$. (d) The curves same as (c) overlapped due to the rescaled parameter $P_{N}=N^{2}Pr$ on the abscissa. An additional thick grey line is a neutral stability curve obtained from the small-$Pr$ approximation.
• Figure 3: (a,b) Real (solid) and imaginary (dashed) parts of the mode shape $\hat{v}(r)$ and rescaled mode $\hat{T}(r)/Pr$ for $Pr=1$ (red), $Pr=10^{-2}$ (blue), and $Pr=10^{-4}$ (grey) at $\mu=0$, $\eta=0.9$, $Re_{i}=200$, $N=1$, $m=0$ and $k_{d}=3.91$. (c) Perturbation temperature $T(r,z)$ reconstructed from $\hat{T}(r)$ for $Pr=10^{-2}$.
• Figure 4: (a,b) Neutral stability curves for different $m$ for (a) $Pr=1$ and (b) $Pr=10^{-4}$ at $\mu=0$ and $\eta=0.9$. (c) Neutral stability curves for different $Pr$ over the rescaled parameter $P_{N}$ at $\mu=0$, $\eta=0.9$ and $m=2$. A thick grey line denotes the neutral stability curve from the small-$Pr$ approximation.
• Figure 5: Neutral stability curves obtained from the small-$Pr$ approximation at $\mu=0$.