A phase field-finite difference lattice Boltzmann method for modeling dendritic growth solidification in the presence of melt convection

Abstract

In this study, an alternative numerical method is proposed to predict the two-phase flow during dendritic growth under melt convection. It consists of two main parts. The phase-field method (PFM) is used for modeling the solid phase and tracking the morphological changes during dendritic solidification, and the finite-difference lattice Boltzmann method (FDLBM) is used for predicting the fluid flow of the liquid phase. The no-slip boundary condition to take into account the interaction between liquid and solid phases at the interface is modeled as a diffusive drag force. This method is referred to as phase field-finite difference lattice Boltzmann method (PF-FDLBM). In the numerical formulation, a collision term with single relaxation time (SRT) is used for simplicity, and a negative viscosity term is included to improve numerical stability. The proposed method can be implemented on regular and non-regular grids. Two-dimensional benchmark computations of flows past a circular cylinder show that the no-slip boundary condition is well satisfied. The flow patterns and drag coefficients are appropriately predicted even for flows at low Reynolds numbers, in which viscous forces commonly deteriorate numerical stability. Then, simulations of dendritic growth under melt convection are carried out with the present method. The effect of melt convection on the solidification patterns is well predicted for an alloy with its actual physical properties. Finally, the efficiency of the proposed method is evaluated considering an increase of the relaxation time, and its influence related to the computational resources. As a result, faster and stable computations are possible by appropriately setting the relaxation time and the negative viscosity term in PF-FDLBM formulation.