Resonances and instabilities in a tilted rotating annulus

Abstract

The flow inside a rotating annulus tilted with respect to gravity is characterized experimentally and theoretically. As in the case of a tilted rotating cylinder the flow is forced by the free surface, maintained flat by gravity. It leads to resonances of global inertial modes (Kelvin modes) when the height of fluid is a multiple of half the wavelength of the mode. The divergence of the mode is saturated by viscous effects at the resonance. The maximum amplitude scales as the Ekman number to the power $-1/2$ when surface Ekman pumping is dominant, and to the power $-1$ when volumic damping is dominant. An analytical prediction is given with no fitting parameter, in excellent agreement with experimental results. At lower Ekman numbers, the flow destabilizes with respect to a triadic resonance instability, as already observed by Xu & Harlander (Phys. Rev. Fluids, 2020). We provide here a linear stability analysis leading to the viscous threshold of the instability for small tilt angles. For large tilt angles, a centrifugal instability is observed due to the acceleration of the flow by the inner cylinder. Finally, the features of the turbulent flow and its mixing efficiency are characterized experimentally. We underline the potential interest of this configuration for bioreactors.