A Runge-Kutta discontinuous-Galerkin level-set method for unsteady compressible two-fluid flow

A typically interesting type of flow problem is that of flows involving multiple fluids. Especially two-fluid flows, where two non-mixing fluids are separated by a sharp fluid interface, find many applications in both engineering and physics. Although experimental and analytical results have provided us a solid foundation for two-fluid dynamics, it is the research area involved with numerically solving the governing flow equations, better known as computational fluid dynamics (CFD), that is presently leading the way in two-fluid research. Unfortunately, there is reason to believe that, in contrast to the simulation of single-fluid flows, there is still no sufficiently accurate and efficient simulation method available for general two-fluid flows. This is mainly due to difficulties that arise when treating the interface between the two fluids and the lack of accuracy of the numerical methods that can cope with these interface problems. It is therefore that this thesis governs the development of a highly accurate numerical solver for the simulation of compressible, unsteady and inviscid two-fluid flows described by the two-dimensional Euler equations of gas dynamics. The two-fluid flow solver that is developed in this thesis is based on the level-set (LS) method. This so-called interface-tracking method describes the evolution of the two-fluid interface in the flow and as such is able to distinguish between both fluids. The LS method has been applied to two-fluid flow simulation regularly and corresponding results have effectively shown its competence. The novelty of the solver developed here is the application of a highly accurate Runge-Kutta discontinuous Galerkin (RKDG) method for the temporal and spatial discretization of the governing equations. The possibility of obtaining very high orders of accuracy and the relatively easy implementation of mesh and order refinement (hp-refinement) techniques makes the RKDG method an attractive method for solving fluid flow problems. Applying a RKDG discretization to a two-fluid flow solver based on the LS method, combines the accuracy of the former with the efficiency and easy implementation of the latter, and as such results in an attractive numerical solver for two-fluid flows. Since the combination of a RKDG method for the Euler equations and a level-set method is a novelty, several aspects of both methods required further investigation. It is shown that the level-set equation has to be used in its advective form since this approach, as opposed to the frequently used conservative form, does not generate an erroneous off-set in the interface location. A simple fix is applied to prevent the solution from becoming oscillatory near the two-fluid interface. Application of this fix requires the development of a special two-fluid slope limiter for the RKDG method. Numerical results of several one- and two-dimensional problems show the competence of the developed method.