Crystallization of supercooled water: A level-set-based modeling of the dendrite tip velocity

'It is well-known that solidification front of a supercooled liquid is unstable; consequently, this instability leads to the appearance of an array of dendrites of sub-micron diameter. The shape and the velocity of the dendrite propagation are determined by the thermodynamic properties of the liquid and solid phases, including interfacial energy as well as the initial temperatures of both. Accordingly, the numerical simulation of solidification process is a rather challenging problem which requires an accurate prediction of high temperature gradients near the moving solidification front. In this study a relevant level set formulation has been developed enabling correct determination of the position and the curvature of the liquid/solid interface. At this interface a Dirichlet boundary condition for the temperature field is imposed by applying a ghost-face method. For the purpose of updating the level set function and optimizing computing time a narrow-band around the interface is introduced. Within this band, whose width is temporally adjusted to the maximum curvature of the interface, the normal-to-interface velocity is appropriately expanded. The computational model is firstly validated along with the analytical solution of stable freezing. The tip velocity of dendritic patterns (pertinent to unstable freezing) is investigated by performing two-dimensional simulations. The computational results exhibit excellent qualitative and quantitative agreement with the marginal stability theory as well as with the available experiments in the heat-diffusion-dominated region.