LBM simulations of dispersed multiphase flows in a channel: Role of a pressure Poisson equation

In recent years, Lattice Boltzmann Methods (LBM’s) have emerged as a popular class of paradigms for the simulation of multiphase flows. These methods rely on discretized Boltzmann equations to represent the individual multiphase species. Among LBM’s advantages is its ability to explicitly account for interfacial physics and its local streaming/collision operations which make it ideally suited for parallelization. However, one drawback of LBM is in the simulation of incompressible multiphase flow, whereby the density should remain constant along material characteristics. Because LBM uses a state equation to relate pressure and density, incompressibility cannot be enforced directly. This is true even for incompressible single-phase LBM calculations, in which a finite density drop is needed to drive through the flow. This is also the case for compressible Navier-Stokes algorithms when applied to low Mach number flow. To mitigate compressibility effects, LBM can be used in low Mach regimes which should keep material density variation small. In this work, we demonstrate that the assumption of low Mach number is not sufficient in multiphase internal flows. In such flows, in the absence of a Pressure Poisson constraint to enforce incompressibility, LBM predicts a compressible solution whereby a density gradient must develop to conserve mass. Imposition of inflow/outflow boundary conditions or a mean body force can ensure that mass is conserved globally, thereby quelling density variation. The primary numerical problem we study is the deformation of a liquid droplet immersed in another fluid. Though LBM is not typically conducted with a pressure Poisson equation, we incorporate one in this work and demonstrate that its inclusion can significantly lower the density variation in view of maintaining an incompressible flow.