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UBC Theses and Dissertations

High-order finite-volume discretisations for solving a modified advection-diffusion problem on unstructured triangular meshes Van Altena, Michael

Abstract

A fourth-order accurate diffusive flux calculation scheme has been developed which can be incorporated into any existing reconstruction-based unstructured mesh inviscid flow solver. The flux scheme makes use of a Laplace problem to model the viscous terms and of the existing least-squares reconstruction code used in Dr. Carl F. Ollivier-Gooch's inviscid flow solver. Boundary conditions on the solution are enforced during the reconstruction, which requires boundary constraints to be included in the least-squares matrix. The Taylor Constraints method for generating boundary constraints is developed and then discarded due to its drawbacks. The Gauss Constraints method is then developed which is simpler and avoids the drawbacks of the Taylor Constraints method. The flux integration is validated to demonstrate fourth-order accuracy on square, circle and cardioid meshes with a combination of Dirichlet and Neumann boundary conditions. The flux integration scheme is shown to be fourth-order accurate for a modified advection-diffusion problem of the same form as the Navier-Stokes' momentum equations. The diffusive flux calculation scheme can later be combined with an existing inviscid flow solver to solve the viscous compressible Navier-Stokes' equations.

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