Chapter 4 - Structured Finite Volume Schemes

Abstract

This chapter considers the 2-D and the 3-D cases separately for a general quadrilateral or hexahedral control volume, respectively. The chapter outlines three approaches for the definition of the control volume and the location of the flow variables. The advantages and the shortcomings of the three approaches and how the convective fluxes can be approximated have also been discussed in the chapter. In a cell vertex scheme, all flow variables are associated with the nodes of the computational grid. Within the approach based on overlapping control volumes, the grid cells still represent the control volumes, just as in the case of the cell centered scheme. The cell vertex scheme with overlapping control volumes has an advantage over the dual volume scheme in the treatment of wall boundaries, but it cannot be combined with the popular upwind discretization methods such as Total Variation Diminishing (TVD), Advection Upstream Splitting Method (AUSM), or Convective Upwind Split Pressure (CUSP). The cell vertex scheme with dual control volumes and the cell centered scheme are numerically very similar in the interior of a stationary flow field. The main differences occur on distorted grids, in the boundary treatment, and for the unsteady flows. The concept of the CUSP scheme is quite similar to that of AUSM. However, the CUSP approach has the advantage of being formulated as an average of fluxes (but without weighting like within AUSM) minus a dissipation term. This feature is crucial for the implementation in an explicit, hybrid multistage scheme. Furthermore, because of the different scaling factors as compared to AUSM, the CUSP scheme behaves more favorably in the case of flow alignment. The TVD schemes are based on a concept aimed at preventing the generation of new extrema in the flow solution.