Abstract
The solidification of a binary alloy results in the formation of a porous mushy layer, within which spontaneous localisation of fluid flow can lead to the emergence of features over a range of spatial scales. We describe a finite volume method for simulating binary alloy solidification in two dimensions with local mesh refinement in space and time. The coupled heat, solute, and mass transport is described using an enthalpy method with flow described by a Darcy-Brinkman equation for flow across porous and liquid regions. The resulting equations are solved on a hierarchy of block-structured adaptive grids. A projection method is used to compute the fluid velocity, whilst the viscous and nonlinear diffusive terms are calculated using a semi-implicit scheme. A series of synchronization steps ensure that the scheme is flux-conservative and correct for errors that arise at the boundaries between different levels of refinement. We also develop a corresponding method using Darcy's law for flow in a porous medium/narrow Hele-Shaw cell. We demonstrate the accuracy and efficiency of our method using established benchmarks for solidification without flow and convection in a fixed porous medium, along with convergence tests for the fully coupled code. Finally, we demonstrate the ability of our method to simulate transient mushy layer growth with narrow liquid channels which evolve over time.
Highlights
A wide variety of physical processes involve binary alloy solidification
We develop an adaptive mesh refinement (AMR) code to simulate the solidification of binary alloys
We consider a series of transient solutions to the fully coupled problem on both uniform and adaptive meshes, which reveal the computational savings provided by the implementation of AMR
Summary
A wide variety of physical processes involve binary alloy solidification. Within industrial settings, understanding this phenomena is relevant to metal casting [1] whilst, in a geophysical context, notable applications include the study of the Earth’s core [2] and sea ice formation [3]. Heat diffuses away from the solid-liquid interface faster than the solute, causing liquid adjacent to the freezing interface to be constitutionally supercooled [4] Under these conditions, the solid-liquid interface is unstable to dendritic growth [5] and a mushy layer will form: a porous solid matrix bathed in its melt. A binary alloy of salt and water, these regions are known as brine channels and efficiently transport brine into the ocean [3]. In industrial settings, such as metal casting, these ‘chimneys’ or ‘freckles’ are undesirable defects [1].
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