Simulation of spiral instabilities in wide-gap spherical Couette flow

We numerically study the wide-gap spherical Couette flow between two concentric spheres with the inner sphere rotating and the outer one stationary. Two wide-gap clearance ratios, β=(R₂-R₁)/R₁ =0.33 and 0.50, are chosen to investigate the transition scenarios of the spiral instabilities with increasing Reynolds number (Re). For β = 0.33, we first obtain the steady 1-vortex flow at Re = 700 by using the 1-vortex flow for a medium gap β = 0.18 at Re = 700 as the initial condition. The 1-vortex flow for β = 0.33 exists for Re ϵ [450, 2050]and it collapses back to the basic flow when Re > 2050. We then detect spiral instabilities by increasing the Reynolds number gradually. The basic flow becomes unstable at Re_c1 = 2900 where spiral waves of wavenumber m = 6 appear first. Increasing the Reynolds number further, the wavenumber decreases to 5 and 4 at Re_c2 = 3000 and Re_c3 = 4000 respectively. The flow becomes turbulent when Re > 4500. For β = 0.50, no Taylor vortices are found. The basic flow becomes unstable at Re_c1 = 1280 where spiral waves of wavenumber m = 5 occur first. As the Reynolds number is increased, the wavenumber becomes 4 at Re_c2 =1700, 5 again at Re_c3 = 1800, 4 at Re_c4 = 2000, and becomes 3 at Re_c5 = 2200 while the flow becomes turbulent for Re > 2200. The computed rotational frequencies as a function of the Reynolds number for β = 0.33 spiral waves of wavenumber m = 5, 4 and 3 are in good agreement with previous experimental results. The present transition scenario of the spiral wavenumber with increasing Reynolds number for is the same as that of Egbers and Rath (1995 Acta Mech. 111 125-40), while for β = 0.50, it is only partially similar to those of Wulf et al (1999 Phys. Fluids 11 1359-72) and Egbers and Rath (1995 Acta Mech. 111 125-40).