Oscillatory thermocapillary convection in open cylindrical annuli. Part 2. Simulations

Abstract

Oscillatory thermocapillary convection in open cylindrical annuli heated from the outer wall is investigated numerically. Results at fixed inner/outer radius ratio of 0.5, aspect ratios ($Ar$) of 1, 2.5, 3.33, and 8, zero Biot number, and a Prandtl number of 6.84 are obtained and compared with experiments (Part 1 of this paper). Convection is steady and axisymmetric at sufficiently low values of the Reynolds number (${\it Re}$). Transition to oscillatory states occurs at critical values of $Re$ which depend on $Ar$. With $Ar\,{=}\, 1$, 2.5 and 3.33, we observe 5, 9 and 12 azimuthal wavetrains, respectively, travelling clockwise at the free surface near the critical $Re$. With $Ar \,{=}\, 8$, there are 20 standing waves near the critical $Re$. Experimental results in Part 1 support this finding. A multi-roll structure appears beyond the critical $Re$ in shallow liquid layers with $Ar \,{=}\, 3.33$ and 8. The critical $Re$ and frequency are in qualitative but not in quantitative agreement with the experimental ones. Either heat loss from the free surface or heating from the surroundings to the free surface stabilizes the flow, and the critical $Re$ increases with increasing Biot number while the critical period goes down. The numerical results agree better with the experimental ones if the free surface is assumed to be heated as shown in Part 1. We have also computed supercritical time-dependent states and find that while the non-dimensional frequency increases with increasing $Re$ near the critical region, it approaches an asymptote at supercritical $Re$.