Numerical Methods for Flows with Evolving Interfaces
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- Institutt for marin teknikk 
The present thesis addressed two important aspects of Computational Hydrodynamics: solution methods for following evolving interfaces, and stabilisation techniques for finite element method sin the context of convection-dominated, or pure convective flows. There basically exist two main types of solution methods for following sharp evolving interfaces in fluid flow problems: interface tracking and interface capturing methods. While the techniques belonging to the former type are usually more accurate and efficient, those belonging to the latter type are more robust and can handle more complicated physical situations. One method of each type was used in the present thesis: the height function method and the level set method, respectively. The former is implemented using an Arbitrary-Lagrangian-Eulerian (ALE) framework for the bulk flow equations, and surface-fitted mesh. The interface problem is solved on a mesh containing one spatial dimension less then the mesh used for solving bulk flow problems. This latter mesh is regenerated at every time step using a linear blending technique. It is shown how an algebraic constraint can be imposed on the linear system of equations obtained in order to enforce numerical mass conservation. When it comes to the level set method, an Eulerian framework is used for the computation of the flow. The level set problem is solved on a mesh having the same spatial dimension as the one used for the bulk flow problem. No re-meshing is done. A reinitialisation step is implemented in order to prevent break-down of the algorithm. Galerkin finite element methods are well-known for their lack of stability when solving convection-dominated flow problems. The easiest and most direct way of counteracting this misbehaviour is to sufficiently refine the computational mesh. This technique can, however, easily lead to problems that become so large that they are unrealisable on today’s computers. Many methods have therefore been devised for enhancing stability properties of finite element methods. In the present thesis, the finite calculus method is used for both spatial and temporal stabilisation. It is shown to be an efficient spatial stabilisation method. It is, in addition, shown that used in combination with the forward Euler scheme, the finite calculus method leads to an explicit scheme that can be used for Courant numbers larger than one.