Towards a level set reinitialisation method for unstructured grids
Edwards, William Vincent
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Interface tracking methods for segregated flows such as breaking ocean waves are an important tool in marine engineering. With the development in marine renewable devices increasing and a multitude of other marine flow problems that benefit from the possibility of simulation on computer, the need for accurate free surface solvers capable of solving wave simulations has never been greater. An important component of successfully simulating segregated flow of any type is accurately tracking the position of the separating interface between fluids. It is desirable to represent the interface as a sharp, smooth, continuous entity in simulations. Popular Eulerian interface tracking methods appropriate for segregated flows such as the Marker and Cell Method (MAC) and the Volume of Fluid (VOF) were considered. However these methods have drawbacks with smearing of the interface and high computational costs in 3D simulations being among the most prevalent. This PhD project uses a level set method to implicitly represent an interface. The level set method is a signed distance function capable of both sharp and smooth representations of a free surface. It was found, over time, that the level set function ceases to represent a signed distance due to interaction of local velocity fields. This affects the accuracy to which the level set can represent a fluid interface, leading to mass loss. An advection solver, the Cubic Interpolated Polynomial (CIP) method, is presented and tested for its ability to transport a level set interface around a numerical domain in 2D. An advection problem of the level set function demonstrates the mass loss that can befall the method. To combat this, a process known as reinitialisation can be used to re-distance the level set function between time-steps, maintaining better accuracy. The goal of this PhD project is to present a new numerical gradient approximation that allows for the extension of the reinitialisation method to unstructured numerical grids. A particular focus is the Cartesian cut cell grid method. It allows geometric boundaries of arbitrary complexity to be cut from a regular Cartesian grid, allowing for flexible high quality grid generation with low computational cost. A reinitialisation routine using 1st order gradient approximation is implemented and demonstrated with 1D and 2D test problems. An additional area-conserving constraint is introduced to improve accuracy further. From the results, 1st order gradient approximation is shown to be inadequate for improving the accuracy of the level set method. To obtain higher accuracy and the potential for use on unstructured grids a novel gradient approximation based on a slope limited least squares method, suitable for level set reinitialisation, is developed. The new gradient scheme shows a significant improvement in accuracy when compared with level set reinitialisation methods using a lower order gradient approximation on a structured grid. A short study is conducted to find the optimal parameters for running 2D level set interface tracking and the new reinitialisation method. The details of the steps required to implement the current method on a Cartesian cut cell grid are discussed. Finally, suggestions for future work using the methods demonstrated in the thesis are presented.