Transition of Taylor–Görtler vortex flow in spherical Couette flow

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Abstract
The critical Taylor number, phenomena accompanying the transition to turbulence, and the cellular structure of Taylor–Görtler vortex in the flow between two concentric spheres, of which the inner one is rotating and the outer is stationary, are investigated using three kinds of flow-visualization technique. The critical Taylor number generally increases with the ratio β of clearance to inner-sphere radius. For β [les ] 0.08, the critical Taylor number in spherical Couette flow is smaller than in circular Couette flow, but vice versa for β > 0.08. A pair of toroidal Taylor–Görtler vortices occurs first around the equator at the critical Reynolds number Rec (or critical Taylor number Tc). More Taylor–Görtler vortices are added with increasing Reynolds number Re. After reaching the maximum number of vortex cells, as Re is increased, the number of vortex cells decreases along with the various transition phenomena of Taylor–Görtler vortex flow, and the vortex finally disappears for very large Re, where the turbulent basic flow is developed. The instability mode of Taylor–Görtler vortex flow depends on both β and Re. The vortex flows encountered as Re is increased are toroidal, spiral, wavy, oscillating (quasiperiodic), chaotic and turbulent Taylor–Görtler vortex flows. Fourteen different flow regimes can be observed through the transition from the laminar basic flow to the turbulent basic flow. The number of toroidal and/or spiral cells and the location of toroidal and spiral cells are discussed as a means to clarify the spatial organization of the vortex. Toroidal cells are stationary. However, spiral cells move in relation to the rotating inner sphere, but in the reverse direction of its rotation and at about half its speed. The spiral vortices number about six, and the spiral angle is 2–10°.