Morphology evolution of a melting solid layer above its melt heated from below

• Published in: Journal of fluid mechanics 2023-02, Vol.956, Article A23
• Main Authors: Yang, Rui, Howland, Christopher J., Liu, Hao-Ran, Verzicco, Roberto, Lohse, Detlef
• Format: Article
• Language: English
• Subjects: Amplitude; Amplitudes; Boundary conditions; Cell size; Cellular convection; Convection; Convective flow; Direct numerical simulation; Finite difference method; Flow structures; Fluid dynamics; Free convection; Heat flux; Heat transfer; Height; Interface roughness; JFM Papers; Lower bounds; Melting; Morphology; Phase transitions; Plumes; Rayleigh number; Roughness; Scaling; Simulation; Topography; Topology; Velocity; Wavelength
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2023.15

We numerically study the melting process of a solid layer heated from below such that a liquid melt layer develops underneath. The objective is to quantitatively describe and understand the emerging topography of the structures (characterized by the amplitude and wavelength), which evolve out of an initially smooth surface. By performing both two-dimensional (achieving Rayleigh number up to $Ra=10^{11}$) and three-dimensional (achieving Rayleigh number up to $Ra=10^9$) direct numerical simulations with an advanced finite difference solver coupled to the phase-field method, we show how the interface roughness is spontaneously generated by thermal convection. With increasing height of the melt the convective flow intensifies and eventually even becomes turbulent. As a consequence, the interface becomes rougher but is still coupled to the flow topology. The emerging structure of the interface coincides with the regions of rising hot plumes and descending cold plumes. We find that the roughness amplitude scales with the mean height of the liquid layer. We derive this scaling relation from the Stefan boundary condition and relate it to the non-uniform distribution of heat flux at the interface. For two-dimensional cases, we further quantify the horizontal length scale of the morphology, based on the theoretical upper and lower bounds given for the size of convective cells known from Wang et al. (Phys. Rev. Lett., vol. 125, 2020, 074501). These bounds agree with our simulation results. Our findings highlight the key connection between the morphology of the melting solid and the convective flow structures in the melt beneath.