Modulated and unmodulated travelling azimuthal waves on the toroidal vortices in a spherical Couette system

Abstract

We have investigated the modulated and unmodulated travelling azimuthal waves appearing on the toroidal Taylor–Gortler (TG) vortices in a fluid contained between two concentric spheres with the inner sphere rotating. For smaller-clearance cases, toroidal TG vortices appear at the equator, just as in the flow between two concentric cylinders with the inner cylinder rotating. When the Reynolds number of the flow increases quasi-statically, spiral TG vortices appear in addition to toroidal TG vortices, and no modulation occurs, even if the Reynolds number further increases quasi-statically. However, when the Reynolds number is increased from zero to a particular value with a specific acceleration of the inner sphere, modulated wavy toroidal TG vortices appear. We found that the necessary condition for occurrence of modulation is the prevention of spiral TG vortices. Using simultaneous flow-visualization and spectral techniques, and measuring the fluctuation of sinks and sources of vortex boundaries, we obtained the frequency f 1 of travelling azimuthal waves passing a fixed point in the laboratory and the modulation frequencies f 2 and f ′ 2 of these waves, as determined by an observer in the laboratory and an observer fixed in a reference frame that rotates in phase with the travelling azimuthal waves, respectively. The relationship among the characteristic frequencies, f 1 , f 2 and f ′ 2 , obtained by modal analysis and the experimental results, is ( f ′ 2 + kf 1 / m )/ f 2 = − 1, where k and m are a modulation parameter and the wavenumber of travelling azimuthal waves, respectively.