Numerical experiments have been presently conducted aiming at studying the influence of the surface energy on the crystallization process of supercooled water in terms of the supercooling degrees. The mathematical model consists primarily of the equation governing the thermal energy field solved independently in both phases in accordance with the two-scalar approach by utilizing the Stefan condition at the interface to couple both temperature fields. The computational algorithm relying on the level-set method for solid-liquid interface capturing has been appropriately upgraded aiming at accuracy level increase with respect to the discretization of the thermal energy equation and the normal-to-interface derivative of the temperature field. The model describes the freezing mechanism under supercooled conditions, relying on the physical and mathematical description of the two-phase moving-boundary approach. The relevant numerical algorithm is implemented into the open source software OpenFOAM®. The results obtained illustrate the stabilizing effect of the surface energy compared to the level of supercooling. The effect of the anisotropy of the interfacial energy on the ice nucleus growth, resulting in a six-fold crystal shape, is also illustrated.