Symmetry-preserving discretization of Navier–Stokes equations on collocated unstructured grids

A fully-conservative discretization is presented in this paper. The same principles followed by Verstappen and Veldman (2003) [3] are generalized for unstructured meshes. Here, a collocated-mesh scheme is preferred over a staggered one due to its simpler form for such meshes. The basic idea behind this approach remains the same: mimicking the crucial symmetry properties of the underlying differential operators, i.e., the convective operator is approximated by a skew-symmetric matrix and the diffusive operator by a symmetric, positive-definite matrix. A novel approach to eliminate the checkerboard spurious modes without introducing any non-physical dissipation is proposed. To do so, a fully-conservative regularization of the convective term is used. The supraconvergence of the method is numerically showed and the treatment of boundary conditions is discussed. Finally, the new discretization method is successfully tested for a buoyancy-driven turbulent flow in a differentially heated cavity.