Abstract
The natural convection of a liquid metal in an annular enclosure with a square cross section in the presence of a toroidal static magnetic field was investigated by linear stability analysis. Three-dimensional steady disturbances were obtained in an annular enclosure where the walls parallel to the gravitational field were heated and cooled. The Prandtl number Pr was set to 0.025 and the radius ratio of the enclosure κ was set to 0.5, while the Rayleigh number Ra, the Hartmann number Ha, and the angular wavenumber m were considered as parameters. The dimensionless governing equations were discretized by the finite difference method. Since the newly developed dual staggered grid was employed, the interpolation in the outer product terms was not required. The linear growth rate for a standing wave mode was amplified by increasing Ra and attenuated by increasing Ha. For any integer m, the neutral Rayleigh number Ran at Ha = 0 and the neutral Hartmann number Han for Ran ≤ Ra ≤ 200 000 were identified. Based on these, the neutral lines for a given m were obtained as Han ∼ (Ra − Ran)1/3. Furthermore, the critical values Rac, Hac, and mc were obtained, where mc was not necessarily limited to an integer. The distribution of mc was approximated by the power of Ra. The disturbances had symmetry in the azimuthal direction and constituted a pair of vortices rotating in opposite directions. These vortices were elongated along the main stream.
Highlights
Thermal convection occurring in a closed container is an important phenomenon often encountered in industrial equipment as well as in nature
The other is the case that boundary surfaces parallel to the gravity are heated and cooled. In the former case, natural convection occurs when the temperature difference between the upper and lower plates exceeds a critical value governed by the Rayleigh number
In natural convection of low Prandtl number fluids, such as liquid metals, an unsteady state tends to appear at high Ra
Summary
Thermal convection occurring in a closed container is an important phenomenon often encountered in industrial equipment as well as in nature. The other is the case that boundary surfaces parallel to the gravity are heated and cooled. In the former case, natural convection occurs when the temperature difference between the upper and lower plates exceeds a critical value governed by the Rayleigh number. The issue of determining the critical point has been studied as a basic but important problem.. For natural convection in a vertical annular enclosure heated from below, it was shown that many roll cells exist side by side in the azimuthal direction.. Yoo et al investigated this natural convection with a linear stability analysis and determined the critical Rayleigh numbers and the preferred cell patterns.. The issue of determining the critical point has been studied as a basic but important problem. For natural convection in a vertical annular enclosure heated from below, it was shown that many roll cells exist side by side in the azimuthal direction. Yoo et al investigated this natural convection with a linear stability analysis and determined the critical Rayleigh numbers and the preferred cell patterns. The results in Ref. 10 agreed well with the experiment of Ref. 8
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