Numerical modelling of liquid/solid phase transitions: Analysis of a gallium melting test

Abstract

We present the results of the numerical analysis of a gallium melting experiment. In the literature authors, who worked on the same experiment, computed different configurations of the melt flow. We obtained a multicellular structure similar to that reported by Dantzig in 1989 and by Le Quéré and Couturier and Sadat in 1999 using different mathematical model and numerical treatment. The mathematical model here adopted is based on the Navier–Stokes equations with buoyancy source term for the melt and the equation for the energy balance law for the whole material coupled with the Stefan condition at the moving phase front. This model was recently proposed by the authors and here is extended by using a time dependent non-dimensional form especially suitable for the treatment of melting processes. The numerical discretization includes a second order ENO scheme. The moving boundary is described via a fixed-grid technique by introducing a time dependent coordinate transformation in order to solve the equations on a fixed rectangular domain. In this paper we confirm that the absence of multicellular structures in the flow of the melted gallium, that was peculiar of the numerical results obtained by several authors, is due to poor resolution, that is either to a low order numerical scheme or to a coarse space grid of discretization. Just in this context we want to stress that our numerical method allowed us to face the experiment with a very coarse space grid compared to that used by other authors that obtained a multicellular flow.