Permeability Tensor for Columnar Dendritic Structures: Phase-Field and Lattice Boltzmann Study

Permeability is a very important multiscale parameter providing information of dendritic structures in macroscale simulations of solidification with liquid flow during casting. In our previous study, it was demonstrated that the permeability of liquid flow normal to a columnar dendritic structure can be well approximated by the Kozeny-Carman (KC) equation with a coefficient kc = 9. In this study, the permeability of liquid flow parallel to a columnar dendritic structure is investigated using a method that combines both phase-field and lattice Boltzmann simulations. It is shown that the permeability of parallel flow can also be approximated by the KC equation with kc = 3. A permeability tensor is then proposed for a columnar dendritic structure. Validity of the permeability tensor is confirmed by simulations of liquid flow in arbitrary directions of a columnar dendritic structure.