Instability by localized disturbances in critical region in a precessing sphere

Abstract

The flow state in a precessing sphere of radius a is characterized by the spin angular velocity Ωs, the precession angular velocity Ωp and the kinematic viscosity ν of fluid. The spin and precession axes are assumed to be orthogonal to each other. When two non-dimensional parameters, e.g. the Reynolds number Re = a2Ωs/ν and the Poincaré number Po = Ωp/Ωs are in some range of values, the flow is believed to approach a steady state even if it starts from an arbitrary initial condition. Such a stable region for the steady flow in the whole parameter space (Re, Po) is, however, not known yet. Here, we investigate, by the linear stability analysis of disturbances localized in a critical region, the boundary of the stable region of the steady flow in the strong spin and weak precession limit. It is found that the boundary curve takes the power form, asymptotically for . This asymptotic form agrees well with the corresponding experimental observation (Goto et al 2011 Proc. 7th Int. Symp. on Turbulence and Shear Flow Phenomena) as well as the results by numerical simulation (Lin et al 2015 Phys. Fluids 27 046611). The critical mode is composed of two types of sinusoidal waves along the critical circle: one travels around the spin axis with phase velocity faster than Ωs and the other slower.