Numerical calculation of the permeability in a dendritic mushy zone

Abstract

An averaging procedure applied to the local Stokes equations in a columnar dendritic-like porous structure yields the heterogeneous form of Darcy’s law, with three Brinkman correction terms explicitly involving gradients of the liquid volume fraction. Because of these extra terms, the associated closure problem leading to the determination of the permeability becomes very complex, and the full solution is still out of reach. However, in some cases, a simplified closure problem can be used to physically describe the spatial evolution of the permeability components in the columnar dendritic mushy zone. From digitized images of dendritic structures observed experimentally during solidification of a 26 wt pct solution of aqueous NH₄Cl and succinonitrile-4 wt pct acetone, components of the permeability tensor in a direction parallel and normal to the primary dendrite arm are numerically calculated. Comparisons to experimental correlations and physical models show that the closure problem provides a more realistic physical description of the structure, especially in the vicinity of the tip of the dendrites. Finally, numerical calculations performed on a schematic dendritic structure point out that the permeability for flow parallel to the primary dendritic arms is hardly dependent on the secondary arm spacing (d ₂).