Towards modelling of initial and final stages of supercooled water solidification

It is well known from experimental studies (e.g., Jung et al., 2012, [13]), that solidification of supercooled water occurs in two stages: the initial rapid, recalescent stage of crystallization and the final slower, quasi-isothermal stage. To date, it is not well understood how these stages can be modelled. In the present study it is demonstrated that both freezing stages of supercooled water can be mathematically modelled by utilizing exclusively the minimal (pertinent to the heat-diffusion description) model of solidification: the first unstable stage is physically well described by the dendritic-growth-approach, whereas the second stage of the freezing process is qualitatively and quantitatively defined by a stable planar solidification. The computational model, introduced initially in [7,31], has been appropriately upgraded with respect to the high-fidelity discretization of the thermal-energy equation as well as the normal-to-the-interface temperature gradient determination. At supercooling degrees ΔT higher than ≈5 K (corresponding to high supercooling range), the previously obtained numerical results related to the unstable freezing exhibit a distinct deviation from available experimental data. Possible reasons for this departure can be either the non-consideration of the thermal interaction between the growing needle-like dendrites or neglect of the kinetic effects influencing the growth at such high supercooling conditions. Two-dimensional computations concerning the growth of a needle-like dendrite surrounded by an array of needles indicate that they do not considerably influence the steady-state tip velocity of an isolated needle at higher supercooling. Therefore, the thermal energy equation has been extended through an appropriately derived term mimicking the kinetic undercooling phenomenon. The relevant computational validation results in a very good qualitative and quantitative agreement with experiments in the high-supercooling range as well, illustrating the model's predictive capability of describing both stages of crystallization: the first rapid, dendritic growth (pertinent to unstable freezing) and the second planar growth stage (corresponding to the stable freezing).