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%0 Thesis %A Gross, Sven %T Numerical methods for three-dimensional incompressible two-phase flow problems %C Aachen %I Publikationsserver der RWTH Aachen University %M RWTH-CONV-112777 %P XIII, 216 S. : Ill., graph. Darst. %D 2008 %Z Zusammenfassung in engl. und dt. Sprache %Z Aachen, Techn. Hochsch., Diss., 2008 %X In this thesis a numerical approach for the simulation of three-dimensional incompressible two-phase flows is presented. It is based on a level set method for capturing the interface. The mathematical model consists of the incompressible Navier-Stokes equations and an advection equation for the level set function. The effect of surface tension is modeled by a singular force term located at the interface. For the spatial discretization we use finite elements on a nested hierarchy of tetrahedral grids. An adaptive multilevel refinement algorithm allows for local refinement and coarsening of the grid hierarchy. By partial integration of the Laplace-Beltrami operator the weak formulation of the surface tension force term can be stated in such a way that second derivatives induced by the curvature can be avoided. It is shown that a standard Laplace-Beltrami discretization on a piecewise planar approximation of the interface only yields an order of 1/2 w.r.t. the H1 norm, and on the other hand that by a slight modification this order can be increased up to a value of at least 1. The pressure distribution is continuous in both phases, respectively, but has a jump across the interface due to surface tension. The approximation of such functions in standard finite element spaces yields poor results with an order of 1/2 w.r.t. the L2 norm. The introduction of an extended finite element (XFEM) space provides second order approximations. For this purpose a standard finite element space is augmented by additional basis functions incorporating a jump at the interface. For the time discretization a one-step theta-scheme is applied which leads to a coupled system of level set and Navier-Stokes equations. The coupling can be treated by a Picard iteration. By applying a linearized variant of the theta-scheme the equations can be decoupled. The nonlinearity of the Navier-Stokes equations is handled by a fixed point approach. The arising Oseen problems are solved by an inexact Uzawa method or by Krylov subspace methods, where problem-adapted preconditioners are applied which account for the jump of the material properties between both phases. For the reparametrization of the level set function a Fast Marching method is used. The methods have been implemented in the software package DROPS. The structure of the code and basic design concepts are briefly discussed. We also consider parallelization aspects, as the consumption of memory resources and computational time are typically huge for complex problems such as two-phase flows. The correct implementation and the accuracy of several numerical components is analyzed by means of some test cases. Finally, examples originating from droplet and falling film applications are considered. These two-phase systems play an important role in chemical engineering processes and are some of the major research topics in the collaborative research center SFB 540 at the RWTH Aachen University. Some numerical results for simulations of levitated droplets, rising bubbles and a falling film are presented. %K Numerische Strömungssimulation (SWD) %K Oberflächenspannung (SWD) %K Inkompressible Strömung (SWD) %K Adaptives Gitter (SWD) %K Mehrphasenströmung (SWD) %K Dreidimensionales Modell (SWD) %K Finite-Elemente-Methode (SWD) %F PUB:(DE-HGF)11 %9 Dissertation / PhD Thesis %U https://publications.rwth-aachen.de/record/50223