Modelling of solutal dendritic growth in three dimensions

Abstract

A three-dimensional （3D） model for the simulation of solutal dendritic growth in the low Pclet number region is presented. In the model is adopted a solutal equilibrium approach previously proposed by Zhu and Stefanescu to calculate the evolution of the solid/liquid（SL） interface， which allows the accurate simulation of dendritic growth from the initial unstable stage to the steady-state stage with a good computational efficiency. In this approach， the kinetics of dendritic growth is driven by the difference between the local equilibrium composition， calculated from the local temperature and curvature， and the local actual liquid composition，obtained by solving the solutal transport equation. To describe the specific crystallographic orientation of 3D dendritic growth， a weighted mean curvature （WMC） algorithm， which is incorporated with the anisotropy of surface tension， is proposed to calculate the local SL interface curvature. This approach is convenient to be implemented and to make the transformation for WMC calculation from two to three dimensions. The model is verified by the comparison of our numerical resuets with the analytical ones. The simulated steady-state tip velocity and radius varying with the degree of undercooling of an Al-2wt%Cu alloy are found to be close to the ones predicted by the Lipton-Glicksman-Kurz analytical model. The steady-state morphology of the needle dendrite tip is analyzed. It is found that the tip is nonaxisymmetric and deviates from a paraboloid in the manner of the fourfold symmetry. Finally， the simulated 3D multi-equiaxed dendrites with various crystallographic orientations are presented.