Analysis and improvement of the accuracy of unstructured mesh operators with applications to an unsteady incompressible flow solver

• Main Author: de Foy, B.
• Published: University of Cambridge 1998
• Subjects: 530.44
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.598447

A new three dimensional unsteady incompressible flow solver is developed for unstructured meshes based on the pressure correction method. Collocation of variables at cell vertices is employed in conjunction with a consistent discretisation of the Poisson equation. This is solved using the Biconjugate Gradient Stabilised matrix solver with Incomplete Lower Upper factorisation as a preconditioner and renumbering of nodes to improve the matrix structure. The discretisation of the boundary conditions on the Poisson equation is developed so as to automatically satisfy the compatibility relation. Validation of the code is presented for standard cases and results are shown for calculations in low speed aerodynamics, bridge flow and the flow in a model lung. Taylor series analysis and Fourier behaviour analysis are developed for gauging the accuracy of any type of operator on arbitrary meshes in three dimensions. Operators are written in a standardised edge weight multiplication system such that they can all be analysed in the same manner. The new methods are applied to the analysis of common Finite Volume operators believed to be second order accurate on regular meshes, describing the degradation in accuracy with increasing mesh irregularity. Second and fourth order edge sum smoothing operators are shown to have particularly poor accuracy. A novel method is developed based on these methods of accuracy analysis to obtain more accurate operators on unstructured meshes. Edge weights are determined on stencils consisting of an arbitrary collection of points by setting up a Least-Squares optimisation problem based on a set of constraints and minimisations. This lends great flexibility in specifying the type of operator desired and the type of errors to be tolerated. Optimised viscous and smoothing operators are determined in two and three dimensions yielding much improved solutions for inviscid flow over a hump and unsteady vortex shedding behind a cylinder.