The level set method for capturing interfaces with applications in two-phase flow problems

Loch, Eva (Author); Reusken, Arnold (Thesis advisor)

Aachen / Publikationsserver der RWTH Aachen University (2013) [Dissertation / PhD Thesis]

Page(s): 174 S. : Ill., graph. Darst.

Abstract

Moving interfaces occur in many scientific and engineering problems. For example, in combustion problems the interface is the boundary between burned and unburned regions, and in multi-phase flow problems the interface is the boundary between immiscible fluids. There are many methods to detect the moving interface. In this thesis we focus on the so-called level set method. In a level set method one defines the interface implicitly as the zero level of the so-called level set function. The movement of the interface is implicitly given by the movement of the level set function, which is described by the so-called level set equation. This approach renders the method robust with respect to topological changes of the interface. Despite the drawback that it often suffers from mass loss, which is caused by the numerical discretization schemes, the level set method is a very popular method in multi-phase computational fluid dynamics (CFD). There are many approaches to solve the level set equation numerically. For instance, finite differencing methods such as ENO and WENO schemes are often used. But also discretizations of the level set equation by means of finite volume schemes or finite element methods can be found in the literature. In this thesis we focus on the spatial discretization of the level set equation by finite element methods combined with a time stepping scheme. That means, we use the method of lines, i.e. first the spatial discretization of the level set equation is performed and then the resulting system of ordinary differential equations (ODEs) is discretized by means of a time stepping scheme. A main objective of this thesis is to compare two different spatial finite element discretization methods. The first method, the Streamline-Upwind-Petrov-Galerkin (SUPG) method, is based on continuous finite elements and is a common stabilized finite element method. An analysis of the SUPG method applied to the level set equation can be found in recent literature [Burman2009]. The second method is a Discontinuous Galerkin (DG) method with upwind flux. In this method the finite element functions are allowed to be discontinuous across the element boundaries. DG methods have already been used for the spatial discretization of the level set equation. In this thesis we present an analysis of the DG method with upwind flux applied for the spatial discretization of the level set equation on a bounded domain in $R^d$, $d=2,3$, combined with the Crank-Nicolson scheme for the time discretization. To our knowledge there is no literature on this particular problem so far. As the objective of the level set equation is to capture the moving interface, the error between the exact interface and the approximate interface is of major interest. For both method an interface error bound is derived. A second important quantity related to the level set method is the volume that is enclosed by the interface. In case of the SUPG method we also derive an error bound for the volume error. The SUPG method and the DG method are systematically compared by means of the theoretical error bounds and experimental results. The theoretical error bounds have the same order. However, in numerical experiments the measured errors and the order differ. A significant difference between the two methods is the number of unknowns. To be able to draw a conclusion we compare the measured errors relative to the number of unknowns. Using this criterion the SUPG method yields slightly better results in most test cases. Furthermore, both methods are used to capture the interface in a two-phase flow simulation of a rising butanol droplet in water. Therefore, both methods are implemented in the two-phase flow solver DROPS, which is developed at the Chair for Numerical Mathematics at RWTH Aachen University. We compare the final rise velocity and the droplet shape. The results of the simulations with the SUPG method and the DG method are very similar. However, during the computation there was the need to re-initialize the level set function from time to time and to use a volume correction strategy in every time step to obtain physically reasonable results. Thus, a direct comparison of the results is difficult. We run the simulations again without volume correction to compare the volume conservation properties of the two methods. Here, the DG method yielded a better result.