A phase-field method for three-phase flows with icing

A numerical scheme to simulate three-phase fluid flows with phase change is proposed. By combining the Cahn-Hilliard model for water-air interface, Allen-Cahn equation for ice and fluid and Navier-Stokes equation for momentum, we solve the evolution of the water-air interface and water-ice interface simultaneously, including the volume expansion associated with solidification and due to the density difference between water and ice. Unlike existing schemes assuming a divergence-free flow field, the proposed continuous formulation allows for density changes while ensuring mass conservation. A Poisson equation for the pressure field is derived from mass conservation with constant coefficients, which can efficiently be solved without any pre-conditioning. The results demonstrate that the volume expansion during the ice formation and the subsequent motion of the water-air interface are successfully captured. A parametric study is carried out to examine the dependence of the icing on different physical and numerical parameters. Computations with flow disturbance of different amplitudes demonstrate the robustness of the computational scheme and the uniqueness of the solution over the parameters considered.