Chapter 5 - Unstructured Finite Volume Schemes

Abstract

There are two popular approaches for the definition of the control volume and for the location of the flow variables: the cell-centered scheme and the median-dual scheme. Both the approaches have been presented in detail in the chapter. Attention is particularly devoted to the definition of the various types of control volumes together with spatial discretization methodologies for convective and viscous fluxes. The 3-D finite volume formulations of the most popular central and upwind schemes are presented in detail in the chapter. Within the cell-vertex scheme, the flow variables are associated with the grid nodes (vertices). The median-dual control volumes are formed by connecting the centroids, the face- and edge-midpoints, of all cells sharing the particular node. The relative advantages and disadvantages of the cell-centered and the median-dual scheme are the subject of controversial debates. The main reason for the controversy is the lack of fair comparisons of the two methodologies with respect to accuracy, computational time, and memory for realistic configurations. Apart from boundary treatment, both the methods are computationally equivalent on hexahedral grids, where the number of faces equals the number of edges. Considering the memory requirements, the cell-centered scheme has to store about six times more flow variables on tetrahedral and about three times more variables on usual mixed grids as compared to the median-dual scheme.