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

We present an N-phase extension to the second-order conservative phase field method. The proposed phase field 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. Additionally, the boundedness properties of the two-phase model are inherited by the N-phase model. We show that this phase field model allows for variable interface thicknesses between different phases. For coupling to momentum transport, we extend the two-phase mass-momentum consistent model to N-phase flows. Using second-order central spatial schemes, the resulting N-phase flow solver is not only mass-momentum consistent, but also 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 surface tension forces. Using multiple numerical tests, we show that our fully coupled N-phase flow solver boasts high accuracy in simulating N-phase flows.