Transition to turbulence in swirl created by a vertical magnetic field in liquid metal electrodes

Flow transitions in electromagnetically-driven swirl in a cylindrical domain are investigated using three-dimensional simulations for a liquid metal battery electrode. The parameter space considered: applied current $I=11.11-100A$; Current density, $J_0 \approx 4,000-35,000 A/m^2$, domain radius, $R=3-9cm$, domain height, $H=3$ and $4.5cm$). The background magnetic field, $B_0$, strength is varied systematically from 0.1 to 20 G to characterize the route to chaos. \cite{davidson2022swirling} found the onset of chaos occurs around $1-1.5 G$ for aspect ratio $1/6$ and current of 100A, however, this is shifted to $16-17 G$ for a lower current of 16A and aspect ratio $1/2$. Across a range of geometrical configurations and forcing conditions, including variations in aspect ratio, current and current density, a consistent sequence of dynamical transitions is evident from phase portraits, FFT spectrum, 0-1 chaos test, autocorrelation function and correlation dimension. The flow evolution is by the Ruelle--Takens-Newhouse route to chaos, with the strange non-chaotic attractor (SNA) regime providing an intermediate state prior to the onset of chaos. In our results, it is observed that, for a fixed aspect ratio, the transition is governed by a dimensionless group, $Q_m$, representing the ratio of applied Lorentz force to viscous force. The critical value of $Q_m$ at the onset of oscillatory flow is found to be nearly invariant except when the domain height is varied. Moreover, our results follow the $Re \sim Q_m^{2/3}$ scaling law expected from torque balance moderately-forced electro-magnetically driven swirl flows.