Charlesworth, DJ; (2004) Solution of the incompressible Navier Stokes equations on unstructured meshes. Doctoral thesis , University of London.

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Abstract

Since Patankar first developed the SIMPLE (Semi Implicit Method for Pressure Linked Equations) algorithm, the incompressible Navier-Stokes equations have been solved using a variety of pressure-based methods. Over the last twenty years these methods have been refined and developed however the majority of this work has been based around the use of structured grids to mesh the fluid domain of interest. Unstructured grids offer considerable advantages over structured meshes when a fluid in a complex domain is being modelled. By using triangles in two dimensions and tetrahedrons in three dimensions it is possible to mesh many problems for which it would be impossible to use structured grids. In addition to this, unstructured grids allow meshes to be generated with relatively little effort, in comparison to structured grids, and therefore shorten the time taken to model a particular problem. Also, through the use adaptive refinement, the mesh generation process can be coupled to the solution algorithm to allow the mesh to be refined in areas where complex flow patterns exist. Whilst the advantages to unstructured meshes are obvious they have inherent difficulties associated with them. The computational overheads of using an unstructured grid are increased and the discretisation process becomes more complex. Also, it is inevitable that some of the discretisation methods used as standard on structured grids, do not perform as accurately when used on an unstructured mesh. Therefore, the use of unstructured meshes in computational fluid dynamics (CFD) is still an area of active research. This thesis aims to investigate the use of unstructured meshes to solve the incompressible Navier-Stokes equations using the SEMPLE algorithm. A discretisation strategy drawing on the work of others is developed, that attempts to maintain the accuracy of the solution despite the discretisation problems that unstructured grids present. Particular attention is paid to the convective term in the momentum equations, which is often the cause of inaccuracy in pressure-based solvers. High order convective models, first developed for structured meshes, are adapted for use within an unstructured discretisation to ensure stable and bounded solutions are calculated. To reduce computational costs, the discretisation is based on a pointer system that aims to minimise the amount of connectivity data stored for a particular grid. In addition an efficient multigrid algorithm accelerates the solution of the equations to achieve more realistic calculation times. As an initial test of the solver's accuracy and efficiency, calculated results are compared with standard laminar flow problems in both two and three dimensions. However, for any solution strategy to be of practical use it must be able to model turbulent flow. To that end the algorithm is extended to find solutions to the incompressible Reynolds averaged Navier-Stokes equations, using the k-c turbulence model to close the equations. Again, two and three-dimensional problems are used to test the solver's accuracy and efficiency at calculating turbulent flow. Finally the findings of the research work are summarised and conclusions drawn.

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