On the transition from the laminar to disordered flow in a precessing spherical-like cylinder

Abstract

We investigate, through both asymptotic analysis and direct numerical simulation, precessionally driven flow of a homogeneous fluid confined in a fluid-filled circular cylinder that rotates rapidly about its symmetry axis and precesses about a different axis that is fixed in space. A particular emphasis is placed on a spherical-like cylinder whose diameter is nearly the same as its length. At this special aspect ratio, the strongest direct resonance occurs between the spatially simplest inertial mode and the precessional Poincaré forcing. An asymptotic analytical solution in closed form describing weakly precessing flow is derived in the mantle frame of reference for asymptotically small Ekman numbers. We also construct a nonlinear three-dimensional finite element model – which is validated against both the asymptotic solution and a constructed exact solution – for elucidating the nonlinear transition leading to disordered flow in the precessing spherical-like cylinder. Properties of both weakly and strongly precessing flows are investigated with the aid of a complete inertial-mode decomposition of the fully nonlinear solution. Despite a large effort being made, the well-known triadic resonance is not found in the precessing spherical-like cylinder. The energy contained in the precessionally forced inertial mode is primarily transferred, through nonlinear effects in the viscous boundary layers, to the geostrophic flow that becomes predominant when the precessional Poincaré force is sufficiently large. It is found that the nonlinear flow evolutes gradually and progressively from the laminar to disordered as the precessional force increases.