A Numerical Investigation on the Onset of the Various Flow Regimes in a Spherical Annulus

The spherical Couette system, consisting of the flow in the annular gap between two concentric rotating spheres, is a convenient problem for studying the laminar–turbulent transition. Many of the transitional phenomena encountered in this flow are of fundamental relevance for the understanding of global processes in the planetary atmospheres as well as in astrophysical and geophysical motions. Furthermore, the study of spherical Couette flow (SCF) is of basic importance in the field of hydrodynamic stability. This paper focuses principally on the numerical prediction of various transitions between flow regimes in a confined spherical gap between a rotating inner sphere and a fixed outer spherical shell. The finite-volume-based computational fluid dynamics, FLUENT software package, is adopted to investigate numerically the flow of a viscous incompressible fluid in the closed spherical gap. Two important dimensionless parameters completely define the flow regimes: the Reynolds number, Re = Ω₁R₁²/ν, for the rotation of the inner sphere and the gap width, β = (R₂ − R₁)/R₁ = 0.1, for the geometry. The numerical calculations are carried out over a range of Reynolds number from two until 60,000. The numerical results are compared with the experimental data available in the literature, and the agreement between the two approaches is very good. The laminar–turbulent transition, the onset of different instabilities, the formation mechanisms of various structures, and the flow behavior are examined and described in detail by the pressure field, meridional streamlines, circumferential velocity, and skin friction coefficient. In addition, the velocity time series and the corresponding power spectral density are considered and analyzed over a large range of Reynolds number. Three kinds of fundamental frequencies expressed by F₀, F₁, and F₂ are obtained corresponding to the spiral mode associated with the wavy mode (SM + WM), the wavy mode (WVF), and the chaotic fluctuation (CF), respectively. However, no sharp fundamental frequency components are observed for the turbulent regime.