Projection Methods for Interdendritic Flows

"In spite of recent advances in the mathematical modeling of fluid dynamics for materials processing applications, no significant advances have been made in the numerical discretization of these equations. In this work, the application of two-step projection methods for the numerical simulation of interdendritic flows is discussed. Unlike previous methods, the methods presented here are constructed for the exact equations which are characterized by variable density and volumetric fraction of the liquid. The drag terms, which describe the momentum loss due to the flow around and through the dendrite structures, are treated implicitly. Numerical examples for shrinkage-induced flow during solidification of an Al-4.5% Cu alloy bar is used to illustrate the effectiveness of the proposed algorithm.IntroductionLiquid contraction during solidification plays an important role in the formation of casting defects such as shrinkage voids, microporosity, and macrosegregation. Owing to their importance, the study of transport processes during solidification have received considerable attention during the past years [1, 2]. By the use of volume-averaging techniques, the mass and momentum equations for flow through a stationary dendritic structure were derived [3]. By using the volume-averaging techniques for multi-phase systems, two-phase transport equations for solidification problems were derived [4]."