On the use of the method of manufactured solutions for the verification of CFD codes for the volume-averaged Navier-Stokes equations

The volume-averaged Navier-Stokes (VANS) equations are a key constituent of numerous models used to study complex problems such as flows in porous medias or containing multiple phases (e.g. solid-liquid flows). These equations solve the mesoscopic scale of the flow without taking into account explicitly each individual solid particles, therefore greatly reducing computational cost. However, due to a lack of analytical solutions, the models using the VANS equations are generally validated directly against experimental data or empirical correlations. In this work, a framework to design analytical solutions and verify codes that solve the VANS equations via the method of manufactured solutions is presented for the first time. Three test cases of increasing complexity are designed with this method and used to assess the second-order convergence of a finite volume solver developed in Open∇FOAM. The proposed approach is suitable for the verification of any code that solves the VANS equations with any CFD technique such as the finite element method or the lattice Boltzmann method.