Exploring the phase diagram of fully turbulent Taylor-Couette flow

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.