This thesis deals with the construction of a numerical method for solving two-dimensional elliptic interface problems, such as fully developed stratified flow in pipes. Interface problems are characterized by its non-smooth and often discontinuous behaviour along a sharp boundary separating the fluids or other materials. Classical numerical schemes are not suitable for these problems due to the irregular geometry of the interface. Standard finite difference discretization across the interface violates the interfacial boundary conditions; therefore special care must be taken at irregular grid nodes.
In this thesis a decomposed immersed interface method is presented. The immersed interface method is a numerical technique formulated to solve partial differential equations in the presence of an interface where the solution and its derivatives may be discontinuous and non-smooth. Componentwise corrections terms are added to the finite difference stencil in order to make the discretization well-defined across the interface. A method that approximates the correction terms is also proposed. Results from numerical experiments show that the rate of convergence is approximately of second order.
Moreover, the immersed interface method is applied to stratified multiphase flow in pipes. The flow is assumed to be fully developed and in steady-state. For turbulent flow, both a low Reynolds number turbulence model and a two-layer turbulence model are adopted in order to imitate turbulence in the flow field and in the vicinity of the boundaries. The latter turbulence model is modified accordingly to account for the effects of a wavy interface. In this case, the concept of interfacial roughness is used to model the wavy nature of the interface.
Numerical results are compared with analytical solutions for laminar flow and experimental data for turbulent flow. It is also demonstrated that the current numerical method offers more flexibility in simulating stratified pipe flow problems with complex shaped interfaces, including three-phase flow, than seen in any previous approach.