Flow topology of vortex breakdown in spherical Couette flow

Abstract

This study investigates the bubble-type vortex breakdown within the azimuthal vortices of a spherical Couette flow, where a viscous fluid is trapped between a stationary outer sphere and a rotating inner sphere. The study examined flow parameters with gap ratios(β) of 0.5, 1, and 1.5, where the gap ratio is defined as the ratio of the gap between the spheres to the radius of the inner sphere. The Reynolds numbers were up to 3000, based on the inner sphere's rotation. The system exhibits various steady and unsteady breakdown topologies depending on the gap ratio. The non-dimensionalized governing equations are solved in spherical coordinates using Dedalus, which solves partial differential equations using a sparse spectral method. The three-dimensional flow is decomposed into a two-dimensional velocity field in the r-θ plane and an out-of-plane velocity vector, known as two-dimensional three-component (2D3C) decomposition. Here, r is the radial coordinate, θ is the polar coordinate, and ϕ is the azimuthal component. Helicity is similarly decomposed into planar helicity and an out-of-plane component. The analysis identifies a correlation between planar helicity and the vortex breakdown bubble. The topology of the breakdown bubble is reconstructed using the iso-surface of planar helicity for axisymmetric, steady, and unsteady cases. This detailed investigation provides insights into the complex dynamics of vortex breakdown in spherical Couette flow and contributes to understanding breakdown mechanisms.