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Rayleigh–Bénard convection with a melting boundary

Published online by Cambridge University Press:  06 November 2018

B. Favier Open the ORCID record for B. Favier [Opens in a new window] ,
J. Purseed  and
L. Duchemin
B. Favier*
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, Marseille, France
J. Purseed
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, Marseille, France
L. Duchemin
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, Marseille, France
*
†Email address for correspondence: favier@irphe.univ-mrs.fr
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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 appear 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.

JFM classification

Phase change: Solidification/melting Geophysical and Geological Flows: Topographic effects

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Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 858 , 10 January 2019 , pp. 437 - 473
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© 2018 Cambridge University Press 

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Favier et al. supplementary movie

Visualizations of the total numerical domain for case C in Table 1. The temperature is shown on the left (dark red corresponds to θ=1 while dark blue corresponds to θ=θM) while vorticity is shown on the right (blue and red colors correspond to ±0.25ωmax respectively). The grey line corresponds to the interface defined by the isosurface φ=1/2.

Download Favier et al. supplementary movie(Video)
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