# Rayleigh-B\'enard convection with a melting boundary

## TL;DR

This paper investigates how melting boundaries influence Rayleigh-Bénard convection, revealing that melting-induced topography feedback can enhance heat transfer and alter convection patterns, using a coupled phase-field and Navier-Stokes numerical approach.

## Contribution

It introduces a novel numerical method coupling phase-field and Navier-Stokes equations to study melting boundaries in Rayleigh-Bénard convection, highlighting the impact of topography feedback on flow dynamics.

## Key findings

- Melting leads to dynamic transitions in convection cell size.
- Topography feedback can increase Nusselt number beyond planar boundary values.
- Convection patterns are stabilized by topography-induced flow modifications.

## 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-B\'enard 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.