Non-inverted skew upwind scheme for numerical heat transfer and fluid flow simulations

Abstract

This thesis studies advection modeling for heat transfer and fluid flow problems using a new Non--Inverted Skew Upwind Scheme (called NISUS). Variants of the new scheme are formulated and developed with 8-noded hexahedral elements using the Finite Element Method (FEM)and rectangular elements based on a Finite Volume Method (FVM). A new method of mass weighting to predict convective fluxes of each scalar from the nodal point values is developed. Due to an explicit representation in terms of nodal variables, local inversion of the upwind coefficient matrix is not needed. Also, this thesis evaluates two variants of the new scheme (i.e., 3-node / 3-point and 4-node / 8-point formulations) within a 3--D FEM and a third variant within a 2--D FVM. The 3--D FEM variants are applied to a variety of test problems involving the transport of a scalar variable, while the 2--D FVM variant is applied to fluid flow problems including natural convection in an enclosure and micro--channel flow simulations. The promising performance of NISUS, as compared with exact and previous solutions, is demonstrated both in terms of accuracy and stability. Furthermore, a new data storage format called Compressed Banded Data (CBD) is developed for sparse banded matrices generated by the control volume finite element method (CVFEM). The platform of the new CBD structure permits dynamic switching between various solvers, without any procedural change in the implementation of existing simulation software. The performance of different Krylov techniques with an ILU(0) preconditioner is observed and compared in three test problems with a direct solver.