Andrew Wells

Abstract:

© 2019 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.

Abstract:

We present studies of the solidification of binary aqueous solutions that undergo time-periodic cooling from below. We develop an experiment for solidification of aqueous NH4Cl solutions, where the temperature of the cooling boundary is modulated as a simple periodic function of time with independent variations of the modulation amplitude and frequency. The thickness of the mushy layer exhibits oscillations about the background growth obtained for constant cooling. We consider the deviation given by the difference between states with modulated and fixed cooling, which increases when the modulation amplitude increases but decreases with increasing modulation frequency. At early times, the deviation amplitude is consistent with a scaling argument for growth with quasi-steady modulation. In situ measurements of the mush temperature reveal thermal waves propagating through the mushy layer, with amplitude decaying with height within the mushy layer, whilst the phase lag behind the cooling boundary increases with height. This also leads to phase lags in the variation of the mushy-layer thickness compared to the boundary cooling. There is an asymmetry of the deviation of mushy-layer thickness: during a positive modulation (where the boundary temperature increases at the start of a cycle) the peak thickness deviation has a greater magnitude than the troughs in a negative modulation mode (where the boundary temperature decreases at the start of the cycle). A numerical model is formulated to describe mushy-layer growth with constant bulk concentration and turbulent heat transport at the mush–liquid interface driven by compositional convection associated with a finite interfacial solid fraction. The model recovers key features of the experimental results at early times, including the propagation of thermal waves and oscillations in mushy-layer thickness, although tends to overpredict the mean thickness.