Exploring the phase diagram of fully turbulent Taylor-Couette flow

TL;DR

This paper uses direct numerical simulations to explore the phase diagram of fully turbulent Taylor-Couette flow, analyzing the transition to the ultimate regime and the influence of geometric and rotational parameters on flow regimes and torque scaling.

Contribution

It provides new insights into the transition to the ultimate turbulence regime and the effects of radius ratio, aspect ratio, and rotation on flow behavior and torque scaling in Taylor-Couette flow.

Findings

Transition to ultimate regime is independent of rotation ratio and aspect ratio but depends on radius ratio.

Identified two main flow regimes based on Coriolis force magnitude: co-rotating and counter-rotating.

Different aspect ratios lead to crossing torque scaling branches within 15% of the transition Reynolds number.

Abstract

Direct numerical simulations of Taylor-Couette flow (TC). Shear Reynolds numbers of up to $3\cdot10^5$, corresponding to Taylor numbers of $Ta=4.6\cdot10^{10}$, were reached. Effective scaling laws for the torque are presented. The transition to the ultimate regime, in which asymptotic scaling laws (with logarithmic corrections) for the torque are expected to hold up to arbitrarily high driving, is analysed for different radius ratios $\eta$, different aspect ratios $\Gamma$ and different rotation ratios $Ro$. It is shown that the transition is approximately independent of $Ro$ and $\Gamma$, but depends significantly on $\eta$. We furthermore calculate the local angular velocity profiles and visualize different flow regimes that depend both on the shearing of the flow, and the Coriolis force originating from the outer cylinder rotation. Two main regimes are distinguished, based on the magnitude of the Coriolis force, namely the co-rotating and weakly counter-rotating regime dominated by Rayleigh-unstable regions, and the strongly counter-rotating regime where a mixture of stable and unstable regions exist. Furthermore, an analogy between $\eta$ and outer-cylinder rotation is revealed, namely that smaller gaps behave like a wider gap with co-rotating cylinders, and that wider gaps behave like smaller gaps with weakly counter-rotating cylinders. Finally, the effect of $\Gamma$ on the effective torque versus $Ta$ scaling is analysed and it is shown that different branches of the torque-versus-$Ta$ relationships associated to different aspect ratios are found to cross within $15%$ of the $Re$ associated to the transition to the ultimate regime. The paper culminates in phase diagram in the inner vs outer $Re$ number parameter space and in the $Ta$ vs $Ro$ parameter space, which can be seen as the extension of the Andereck \emph{et al.} phase diagram towards the ultimate regime.