Rotating Taylor–Green flow

Abstract

The steady state of a forced Taylor–Green flow is investigated in a rotating frame of reference. The investigation involves the results of 184 numerical simulations for different Reynolds numbers $\mathit{Re}_{F}$ and Rossby numbers $\mathit{Ro}_{F}$. The large number of examined runs allows a systematic study that enables the mapping of the different behaviours observed to the parameter space ($\mathit{Re}_{F},\mathit{Ro}_{F}$), and the examination of different limiting procedures for approaching the large $\mathit{Re}_{F}$ small $\mathit{Ro}_{F}$ limit. Four distinctly different states were identified: laminar, intermittent bursts, quasi-two-dimensional condensates and weakly rotating turbulence. These four different states are separated by power-law boundaries $\mathit{Ro}_{F}\propto \mathit{Re}_{F}^{-{\it\gamma}}$ in the small $\mathit{Ro}_{F}$ limit. In this limit, the predictions of asymptotic expansions can be directly compared with the results of the direct numerical simulations. While the first-order expansion is in good agreement with the results of the linear stability theory, it fails to reproduce the dynamical behaviour of the quasi-two-dimensional part of the flow in the nonlinear regime, indicating that higher-order terms in the expansion need to be taken into account. The large number of simulations allows also to investigate the scaling that relates the amplitude of the fluctuations with the energy dissipation rate and the control parameters of the system for the different states of the flow. Different scaling was observed for different states of the flow, that are discussed in detail. The present results clearly demonstrate that the limits of small Rossby and large Reynolds numbers do not commute and it is important to specify the order in which they are taken.