Simulation of directional solidification, thermochemical convection, and chimney formation in a Hele-Shaw cell

We have developed fully resolved, two-dimensional, finite volume simulations of directional solidification of a binary alloy in a Hele-Shaw cell. Use of Darcy’s law and the Enthalpy Method throughout the computational domain allows us to avoid prescribing internal boundary conditions on the interfaces between solid, mushy, and liquid regions. We present a description of the theoretical model, computational approach, two reduced benchmark calculations, and simulations of the full governing equations. In simulations with parameter values that approximate experiments, boundary-layer-mode convection produces corrugations in the mush–liquid interface. Some of these corrugations become chimneys that grow and interact within the mushy layer. We consider two porosity–permeability relations and examine their consequences for chimney spacing and mushy layer height. Our results are broadly similar to experiments on directional solidification of NH 4 Cl [S.S.L. Peppin, H.E. Huppert, M.G. Worster, Steady-state solidification of aqueous ammonium chloride, J. Fluid Mech. 599 (2008) 465–476; S.H. Whiteoak, H. Huppert, M.G. Worster, Conditions for defect-free solidification of aqueous ammonium chloride in a quasi 2d directional solidification facility, J. Cryst. Growth (2008)]. We describe other simulations that are tuned to suppress boundary layer mode convection and that, instead, go unstable by the mushy layer mode [M.G. Worster, Instabilities of the liquid and mushy regions during solidification of alloys, J. Fluid Mech. 237 (1992) 649–669]. We investigate the morphological evolution of the mush well beyond the linear instability regime.