A unified-grid finite volume formulation for computational fluid dynamics

A new extremely flexible finite volume framework is presented. It is based on updating cell average values of the dependent variables. It is embedded in a unified-grid framework that unifies the treatment of structured and unstructured grids, single and multi-block grids, patched-aligned, patched-non-aligned and overset grids, and various cell shapes, including hexahedra, triangular prisms and tetrahedra in three dimensions, quadrilaterals and triangles in two dimensions and the linear element in one dimension. A novel discretization approach has been developed to deal with unified-grid topologies. It includes a piecewise-linear multi-dimensional non-oscillatory reconstruction procedure that is based on synergistic utilization of polynomials at cell vertices. Least-squares reconstruction is used when necessary. The concept of generalized neighborhoods is introduced to account for cell neighborhoods that include cells that are not logically connected through common nodes, faces, etc. This helps in the automatic treatment of all types of multi-block meshes. To go with such a general discretization procedure, new implicit relaxation procedures are introduced that help achieve fast convergence to steady state solutions. The framework has been implemented in a new CFD code (CFD++) using a finite volume formulation. The paper presents the methodology with the help of several annotated examples taken from various compressible and incompressible flows. Copyright © 1999 John Wiley & Sons, Ltd.