Convection instability in phase-change Rayleigh-Benard convection systems at a finite Stefan number

In this paper, we revisit the convection instability in phase-change Rayleigh-Benard convection systems at a finite Stefan number, where a pure solid substance confined between two horizontal walls is isothermally heated from below in order to induce melting, assuming no heat conduction in the solid phase. By establishing a connection between the heat transfer behaviors in the conduction and convection melting regimes through the jump events in the temporal evolution of the heat flux and the melted liquid fraction, two criteria (the critical average fluid temperature theta <overbar></mml:mover> f c and the critical melted liquid fraction f l c) are derived to characterize the convection onset. In contrast to the conventional instability analysis, the derivation in the present work is much more convenient and removes the limitation of a vanishing Stefan number ( Ste -> 0). The two obtained criteria are successfully validated by the data available in the literature, together with the numerical simulations conducted in this paper. The validations revealed that the proposed theta <overbar></mml:mover> f c and f l c work well at a finite Ste and that f l c is slightly less accurate than theta <mml:mo><overbar></mml:mover> f c, due to the error inherited from the employed scaling law describing the convective heat flux. With the relation between the effective and global parameters, f l c is further converted into the commonly used critical effective Rayleigh number by R a ec <mml:mo>= R a f l c 3</mml:msubsup>, which is found depending on Ste only, being the same as the criterion of <mml:msub> <mml:mover accent="true"> theta <mml:mo><overbar></mml:mover> f c, while its precision is less satisfying due to amplified error from the cubic power operation of f l c 3.