The major issue addressed in this thesis is the development of a Mach-uniform method that uses a staggered scheme on planar unstructured grids with triangular cells, that has a superlinear rate of convergence. In our staggered grid, the normal velocity components are associated with cell faces, whereas scalar variables are assigned to cell centers. In order to obtain a superlinearly convergent scheme, it is necessary to interpolate vector fields from staggered components with sufficient accuracy in a robust way. Several methods to achieve this objective are investigated. Like all higher order methods, the superlinearly convergent scheme obtained suffers from unphysical oscillations near discontinuities such as shocks. This difficulty is overcome by combining two approaches. We use an upwind-biased superlinearly convergent scheme, so that oscillations are damped in smooth parts of the flow. Furthermore, we follow the usual flux limiting approach to combat oscillations near discontinuities, extended to unstructured staggered schemes. An oscillation detecting limiter function is used to switch between the superlinearly convergent scheme and a first order scheme that generates (almost) no oscillations. For this we adapt the limiter of Barth and Jespersen.

Abstract The subject of this thesis is the computation of bubbly flows. Bubbly flows occur for example in chemical reactors, boiling, fuel injectors and coating. The bubbles and the surrounding fluid are modeled directly without phase averaging. This is the most fundamental approach of describing multi-phase flows. Every bubble is modeled in great detail. Since the model is so demanding, an efficient numerical approach has to be taken. This work aims to develop an efficient, robust method for the direct numerical simulation of multi-phase flows.There exist two incompressible fluids (e.g. air and water) that are separated by an interface. The interface is a moving, internal boundary, where density and viscosity change discontinuously and surface tension forces act. When the governing equations are discretized, the question arises how to deal with this interface. Various methods have been put forward to treat moving boundary problems. The two methods that are of most interest are the so-called Volume-of-Fluid method and the Level-Set method. With both methods a coloring function is used to identify the individual phase. The interface is implicitly defined, which allows arbitrarily shaped interface.The advection of the interface is the key part of this thesis, for which the Level-Set approach is chosen. The major disadvantage of the Level-Set method is that the mass of each individual phase is not conserved when the interface is advected. Additional effort is necessary and in this research the Volume-of-Fluid function is used to conserve mass.The method is applied to a falling drop and a rising bubble in two and three dimensions, respectively. Merging of rising bubbles is studied for two aligned and two misaligned bubbles. Comparison with other numerical work and experimental data is made.