Criteria and limits for flow modes of the spherical Taylor-Couette problem in medium and wide gaps

Abstract

Motivated by recent theoretical and experimental work, we study the spherical Taylor-Couette flow with a view to setting qualitative criteria of the formation of the various vortex modes in relation to certain physical variables. We work on medium and large aspect ratios, namely σ = 0.154, 0.18, 0.38 and 0.48 and we are able to establish in all these cases the following criterion: Transition from one vortex mode to another occurs at those Reynolds numbers at which the torque acting on the outer sphere, the work performed on each rotation by the tangential force acting on the inner sphere and the kinetic energy of the fluid vary abruptly. We propose a physical interpretation for the conditions at which the basic vortex flow is converted into 1-vortex flow. Additionally, we determine the upper limiting value of the aspect ratio, which is σ = 0.51, beyond which Taylor vortices do not exist in annular spherical flow and we numerically obtain for the first time Taylor vortices at large aspect ratios σ = 0.48-0.51. We also show that the range [Re_1, Re_2] of the Reynolds numbers within which such vortices exist decreases with increasing aspect ratio and we point out that its span is affected by three factors. The aspect ratio σ , the angular velocityω _o of the outer sphere and the Grashof number Gr of the flow. The simulation of Taylor vortices in spherical flow is achieved using an efficient numerical technique and routes leading to multiple flow states are designed for large gaps. Results can be useful for the solution of such kind of boundary value physical flow problems within shell domains in several geometries.