The mid-point Green-Gauss gradient method and its efficient implementation in a 3D unstructured finite volume solver

To achieve consistent second-order spatial accuracy in cell-centered finite volume methods using the divergence theorem, the gradient computation at the cell-centers should at least be first-order accurate, which is not assured by the Traditional Green-Gauss (TGG) method on irregular meshes. The Circle Green-Gauss (CGG) method, however, achieves this by constructing a new auxiliary volume (AV) around the cell of interest using a circle and then employs linear interpolation to obtain estimates of the solution at the face centers of this AV, which are then used to compute the cell-center gradients. We propose a modified method, which we call the mid-point Green-Gauss method (mpGG), that alleviates the complexity in the implementation while maintaining the same accuracy as the CGG method. We offer detailed construction of the three-dimensional AV for the most commonly used multi-hydral elements in unstructured meshes, that is, hexahedral, pyramidal, prism, and tetrahedral cells, using an in-house finite volume solver that reads and writes data based on the computational fluid dynamics (CFD) General Notation System (CGNS). Accuracy analysis of the proposed method is carried out in comparison with the TGG, CGG, and weighted least squares methods. The proposed method is further validated using the case of laminar flow over a sphere at a Reynolds number of 100. An inviscid vortex dissipation study is performed to compare the level of numerical dissipation produced by these methods. Further, we study the performance of these methods in a problem that involves turbulent shock-wave boundary layer interaction.