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.

Special topic on precession, nutation, and libration driven flows

“…In this device, the free surface coincides with an isopotential, which, at low rotation rates, remains planar and perpendicular to the local gravity. Hence, viewed from the frame rotating with the drum, the horizontal free surface exhibits a retrograde precession-like motion, similar to the experiment of Subbotin et al 7 The mixing of a blob of dye in the bulk of a rotating and tilted cylinder has been measured by Meunier, 14 for an annulus by Scollo et al, 9 and for a sphere by Watanabe and Goto, 15 all demonstrating the high mixing efficiency of such homogeneous mixers.…”

“…This is very similar to the case of a precessing annulus 8 or a partially filled rotating annulus tilted with respect to the vertical, the free surface acting as a precessing lid. 9,10 For larger forcing amplitudes, the flow exhibits wave turbulence and then transitions suddenly to a ring of vortices, as in Boury et al 11 It has been suggested that these vortices, located at the radius of the zonal jet, can be interpreted as Rossby waves.…”