A conservative second order phase field model for simulation of N-phase flows

We present a novel N-phase extension to the second-order conservative phase field method, first introduced by Chiu and Lin [1]. The proposed N-phase model is in conservative form and is symmetric with respect to the phases while satisfying volume conservation. The model is reduction consistent, meaning that in the absence of M phases, the equations reduce to the equations for an N -M-phase flow. This eliminates the possibility of fictitious phases appearing. For coupling to momentum transport, we extend the two-phase mass-momentum consistent model to N-phase flows. By adopting second-order central spatial schemes, the boundedness properties of the two-phase model are inherited by the N-phase model, and the coupled mass momentum consistent N-phase flow solver inherits the conservation properties of its two-phase version, resulting in the first N-phase flow method that analytically and discretely conserves mass, momentum and kinetic energy (in the absence of capillary and viscous effects). A novel surface tension model is proposed for modeling surface tension forces in N-phase flows. The phase field model allows for variable interface thicknesses between different phase pairs. This endows the model with the property of attaining correct equilibirum configurations at triple junction points. Specifically, we analytically demonstrate that by using interfacial thicknesses that are inversely proportional with the pairwise surface tension, the coupled system achieves steady state at the correct configuration. Using several canonical and practical numerical tests, we demonstrate the accuracy of the phase field equation, the surface tension model, and the fully coupled N-phase flow solver.