A semi-implicit conservative sharp-interface method for liquid-solid phase transition

The solidification of an undercooled liquid is physically unstable. The dominating instability modes are affected by both the evolving temperature field in the solid and liquid phases, and characteristics of the phase interface such as the curvature and the propagation velocity. To capture the instability mode, therefore both the temperature field and the interface have to be represented accurately in a numerical model of the phase-change process. In this work, we develop conservative interface exchange terms for a sharp-interface formulation of liquid-solid phase transition. Conservation at the interface is maintained by explicit formulation of interface fluxes into both solid and liquid phases. We propose a semi-implicit level-set formulation to evolve the phase interface. A new formulation for the interface surface in a cut cell is derived, which includes the Stefan condition. We achieve low numerical dissipation by an explicit third-order Runge-Kutta scheme for time discretization, and a novel WENO-like (Weighted Essentially Non-Oscillatory) interface-gradient reconstruction. This distinguishes our level-set based sharp-interface model from previous level-set based approaches, which rely on finite-difference based interface treatment, and thus do not ensure discrete conservation at the interface. The flux terms in our approach take into account surface-tension and kinetic effects on the interface temperature (Gibbs-Thomson relation). The Stefan condition provides a relation between interface fluxes of mass and energy, and the interface-propagation velocity. Computational efficiency is maintained by a multiresolution approach for local mesh adaptation, and an adaptive local time-stepping scheme. We present one- and two-dimensional simulation results for the growth of a planar solidification front and a single parabolic dendrite affected by surface tension. The results agree well with experimental and analytical reference data, showing that the model is capable to capture both stable (planar) and unstable (dendritic-like) growth processes in the heat-diffusion dominated regime. The convergence order for successively finer meshes in the one-dimensional case is one for the interface location and the temperature field, outperforming previously reported level-set based approaches. We present numerical data of a growing crystal with four-fold symmetry. Our results indicate that the artificial dissipation of the underlying numerical scheme affects its capability to reproduce consistently physical tip-splitting instabilities. The proposed low-dissipation scheme is able to resolve such instabilities. Finally, we demonstrate the capability of the method to simulate multiple growing crystals with anisotropic surface-tension and kinetic effects. (C) 2020 The Authors. Published by Elsevier Ltd.