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me608_final_project_sterba_li - UNSTRUCTURED MESH SOLVER...

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1 UNSTRUCTURED MESH SOLVER FOR THE CONVECTION DIFFUSION EQUATION Zachary A. Sterba Department of Mechanical Engineering Purdue University West Lafayette, Indiana Email: [email protected] Hongwei Li Department of Mechanical Engineering Purdue University West Lafayette, Indiana Email: [email protected] ABSTRACT The steady-state convection-diffusion equation is solved on a non-orthogonal unstructured mesh using the upwind difference method and the finite volume scheme. A Fortran90 code loads a two-dimensional mesh of arbitrary shape containing cells of any polygonal shape, or a mixture of several polygonal shapes. Data structures used are chosen with much attention given to memory and computational efficiency. This mesh is then solved for a test case of uniform flow, where Dirichlet boundary conditions are specified, and the high Peclet number assumption is specified at the output. It is then demonstrated that the solution corresponds to a uniform mesh solution of the same problem. The solver is then tested on a large mesh with irregular boundaries, and a mixture of cell geometries to demonstrate its ability to handle highly complicated input. The convergence of this large mesh was unacceptably slow, demonstrating that any future work will demand a faster and more sophisticated solution scheme. NOMENCLATURE a p ,a nb matrix coefficient at a cell and its neighbor A f face area vector b forcing term b gradient vector resulting from least squares fit D f diffusion term Dir n keeps track of whether the normal vector of face n points into or out of the cell in question e ξ unit vector in the cell centroid to cell centroid direction f a cell’s face; all faces of a cell F f convection term G gradient coefficient matrix J flux vector h synonymous with Δ ξ i, j unit vectors in the x - and y -directions nb a cell’s neighbor; all neighbors of a cell L face “area” (i.e. length) vector in two-dimensions M geometry matrix for least squares fit S source term T p , T nb function value (i.e. temperature) at a cell and its neighbor T function value vector for use in least squares fit V velocity vector V b boundary velocity vector V end n the index of vertex at end n of a face where n is 1 or 2 — is useful in linking cells and faces X, Y centroid location Δ ξ distance between cells Δ V the volume of one cell Г diffusion coefficient η vertex-to-vertex direction ξ cell centroid to cell centroid direction ρ density secondary gradient term MOTIVATION Finite volume models generally use one of two mesh types; these types are characterized by cell connectivity. Structured meshes have a regular, predictable connectivity where all grid points have the same number of neighbors (Figs. 1a, 1b). Unstructured meshes, however, allow a grid point to have any number of neighbors (fig. 1c) (Owen).
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