Lattice Boltzmann framework for multiphase flows by Eulerian-Eulerian Navier-Stokes equations

Although Lattice Boltzmann Method (LBM) is relatively straightforward, it demands a well-crafted framework to handle the complex partial differential equations involved in multiphase flow simulations. This document presents some potential strategies for developing an Eulerian-Eulerian LBM solver tailored for multiphase systems. The paper first states what are the starting equations governing a multiphase flow in classical CFD. Secondly, it derives a pseudo-compressible (targeting the incompressible limit) system of equations for deriving the Eulerian-Eulerian LBM framework to simulate multiphase flows. Finally, a dispersed phase volume fraction equation is provided to balance the degree of freedom less due to the pressure gradient coupling. The effectiveness of these approaches can only be confirmed through rigorous numerical experimentation.