Rayleigh–Bénard convection with a melting boundary

Abstract

We study the evolution of a melting front between the solid and liquid phases of a pure incompressible material where fluid motions are driven by unstable temperature gradients. In a plane layer geometry, this can be seen as classical Rayleigh-Bénard convection where the upper solid boundary is allowed to melt due to the heat flux brought by the fluid underneath. This free-boundary problem is studied numerically in two dimensions using a phase-field approach, classically used to study the melting and solidification of alloys, which we dynamically couple with the Navier-Stokes equations in the Boussinesq approximation. The advantage of this approach is that it requires only moderate modifications of classical numerical methods. We focus on the case where the solid is initially nearly isothermal, so that the evolution of the topography is related to the inhomogeneous heat flux from thermal convection, and does not depend on the conduction problem in the solid. From a very thin stable layer of fluid, convection cells appears as the depth-and therefore the effective Rayleigh number-of the layer increases. The continuous melting of the solid leads to dynamical transitions between different convection cell sizes and topography amplitudes. The Nusselt number can be larger than its value for a planar upper boundary, due to the feedback of the topography on the flow, which can stabilize large-scale laminar convection cells. †

Bistability in Rayleigh-Bénard convection with a melting boundary

“…This thermodynamical effect is nevertheless the starting point when deriving a diffuse-interface method called the phase-field method [6]. The problem described above is solved numerically by using a mixed pseudo-spectral fourth-order finite-difference method [22,24] and the particular phase-field model which has been discussed and validated in [23]. For several cases, we also checked our results by using the open-source pseudo-spectral solver Dedalus [9,10] (more information at http://dedalus-project.org).…”

“…The interaction between a convective flow and a melting solid has recently received some attention [21,23,51] where the gradual melting of a pure isothermal solid is investigated considering a standard Rayleigh-Bénard configuration. The melting process causes a vertical growth of the liquid layer until the critical height is reached and convective instabilities set in.…”

“…The numerical study by [23] shows that, as the convection cells are stretched, due to the vertical growth of the solid-liquid boundary, convection cells merge creating wider ones, thus respecting the aspect ratio one would observe in classical Rayleigh-Bénard convection [13]. During this slow evolution, the convective heat flux has been showed to be consistent in first approximation with that of classical Rayleigh-Bénard [21,23]. The case where a material, confined between two horizontal boundaries, is heated from below and cooled from above has also been studied experimentally by Davis et.…”

“…The existence of a bi-stability regime is discussed in section IV. We finally conclude in section V. A brief description of the numerical method is given in Appendix A which is identical to the one proposed by [23] and thus for a more detailed description, interested readers are referred to that particular paper.…”

Energetics of Radiatively Heated Ice‐Covered Lakes

“…For a discussion in the context of the global ocean, see, for example, Saenz et al (), and references therein, particularly Tailleux (). The cold, freshwater, Boussinesq framework employed here is directly applicable to flows near‐ice (see, e.g., Favier et al, ; Mondal et al, ; Toppaladoddi & Wettlaufer, ), radiatively driven convection in ice‐free water (Bouillaut et al, ; Lepot et al, ) and is particularly apt for ice‐covered inland waters (Kirillin et al, ; Leppäranta et al, ; Ulloa et al, ; Volkov et al, ) and supraglacial lakes (Christoffersen et al, ; Leeson et al, ).…”