A mass-weighted upwind-based control volume finite-element method for steady two-dimensional viscous compressible flows

Abstract

The formulation and implementation of a mass-weighted upwind-based control volume finite element method (CVFEM) for steady, two-dimensional, viscous compressible internal flows is reported in this thesis. In the development of this method, a CVFEM for steady, quasi-one-dimensional, viscous compressible flows was also formulated. The proposed method is a colocated shock capturing formulation. Polygonal control volumes are constructed around each node in the finite-element mesh, and discretized forms of the governing equations are obtained by deriving algebraic approximations to integral conservation equations for each control volume. -- The proposed methods are formulated using the velocity components, pressure, and temperature as the dependent variables: density is calculated from an equation of state. Linear interpolation is applied to pressure and diffused scalars, and a mass weighted upwind function is applied to the convected scalars. An interpolation function incorporating a pseudo-velocity and a pressure gradient is used to represent mass conserving velocities; this allows the development of a colocated method valid for compressible flows. The discretized forms of the governing equations are solved using an iterative algorithm. In this algorithm, linearized forms of the discretized momentum and continuity equations are solved in a segregated manner by using a tridiagonal matrix algorithm. -- The proposed quasi-one- and two-dimensional CVFEM's are applied to several inviscid and viscous compressible fluid flow problems, and the solutions generated are compared with theoretical, numerical, and experimental results available in the literature. This comparison shows that the proposed CVFEM's can generate solutions that are in agreement with the expected physical behaviour of some compressible flows and with the available results. The results suggest further research is required in the evaluation and enhancement of the mass-weighted interpolation functions currently used, however, as the shock smearing in the proposed method is excessive, and the accuracy of solution is not satisfactory.