Volume weighted interpolation for unstructured meshes in the finite volume method

Abstract

The finite volume method is widely used for the numerical simulation of fluid flow because of its rigorous local conservation properties and its compatibility with arbitrary unstructured meshes for meshing complex domains. Interpolation plays an integral role in the finite volume method. Variables are located at cell centres but are also required at other positions such as cell faces. Variable values at these positions must be interpolated from cell values. In this thesis volume weighted interpolation is introduced as an alternative method of interpolation for the finite volume method. The main advantage of volume weighted interpolation is that variables can be interpolated conservatively between overlapping meshes. The accurate evaluation of convective fluxes on complex meshes remains a central issue in the finite volume method. While existing convection schemes perform well on structured orthogonal meshes, the use of orthogonal meshes is limited to simple domains. The application of volume weighted interpolation for convection modelling is investigated in this thesis in order to improve solutions on skew and non-orthogonal meshes. The method involves the construction of three-point interpolation stencils orthogonal to cell faces. A conservative interpolation is performed between the original mesh and the orthogonal stencil cells. The stencil is then used for the interpolation of face values of variables. Test cases are presented to test the interpolation stencil by using high-resolution convection schemes. Promising results are obtained with the stencil on unstructured meshes. Volume weighted interpolation also finds application as a pre- and post-processing tool for the finite volume method. Examples are presented to demonstrate how volume fraction fields can be initialised for two-phase flow simulations. Volume weighted interpolation can be used as a post-processing tool to map results from one mesh onto another as well as to calculate mass flows through surfaces. The applications described and examples presented in this thesis establish the potential of volume weighted interpolation as a valuable tool for the finite volume method.