A conservative lattice Boltzmann model for the volume-averaged Navier–Stokes equations based on a novel collision operator

The volume-averaged Navier–Stokes (VANS) equations are at the basis of numerous models used to investigate flows in porous media or systems containing multiple phases, one of which is made of solid particles. Although they are traditionally solved using the finite volume, finite difference or finite element method, the lattice Boltzmann method is an interesting alternative solver for these equations since it is explicit and highly parallelizable. In this work, we first show that the most common implementation of the VANS equations in the LBM, based on a redefined collision operator, is not valid in the case of spatially varying void fractions. This is illustrated through five test cases designed using the so-called method of manufactured solutions. We then present an LBM scheme for these equations based on a novel collision operator. Using the Chapman–Enskog expansion and the same five test cases, we show that this scheme is second-order accurate, explicit and stable for large void fraction gradients.