# Numerical simulation of two-phase flows with complex interfaces

Zhang, Yuanjun; Reusken, Arnold (Thesis advisor); Müller, Siegfried (Thesis advisor)

*Aachen / Publikationsserver der RWTH Aachen University (2015)* [Dissertation / PhD Thesis]

Abstract

In this thesis numerical simulations of two-phase flows with complex interfaces are presented. Three classes of complex interfaces are considered, namely flows with Marangoni effects, flows with viscous interfaces and flows with insoluble surface active agents (surfactants). We restrict to immiscible incompressible two-phase flow systems. A sharp interface model, which consists of two-phase Navier-Stokes equations and interfacial conditions, is used to describe the flow. At the fluid-fluid interface the surface stress tensor is defined, which models surface forces. Three types of surface stress tensors are considered, namely the constant surface tension force, the variable surface tension force and the viscous interface according to the Boussinesq-Scriven law. The surfactant transport is modeled by a convection-diffusion equation on the interface. Constitutive relations, e.g. the linear relation or the Langmuir model, relate the surfactant concentration to the surface tension coefficient. The DROPS package, which is developed at the Chair for Numerical Mathematics at RWTH Aachen University, is used to perform numerical simulations of three dimensional two-phase flow problems. The package provides a framework for such problems, and includes a level set method for capturing the unknown interface, a pair of P2-XFEM finite elements for discretizing the two-phase Navier-Stokes equations, a trace finite element method for the surfactant transport equation, a Gauss-Seidel type decoupling scheme for handling the coupled systems, fast iterative solvers etc. The main contributions of this thesis are the following. Numerical methods are developed to treat two-phase flows with complex interfaces. These methods can be categorized into two groups: the numerical treatment of surface stress tensors and the numerical treatment of the nonlinear coupling between fluid dynamics and interface dynamics. By introducing the surface stress tensor and applying the partial integration of the surface force functional, different classes of complex interfaces can be treated with a unified approach. The direct calculation of the mean curvature of the interface, which involves second derivatives, is avoided. Instead we concentrate on the discretization of the projection operator at the interface. The viscous surface force terms, which depend on the velocity unknowns, are discretized and treated implicitly in the momentum equation. The two-phase Navier-Stokes equations and the surfactant equation, coupled through the surface tension coefficient, are solved with a Gauss-Seidel type decoupling scheme. The aforementioned numerical methods are implemented in the DROPS package. A systematic methodology is applied to validate the numerical solver for the three classes of flows with complex interfaces. We construct benchmark problems, in which theoretical predictions exist and can be considered valid, and perform numerical experiments of these model problems. Numerical results are compared with theoretical predictions. Very good agreements have been achieved. We also discuss certain properties of the numerical methods, e.g. the convergence rate of the decoupling schemes and the linear algebra aspects of the trace finite element method etc. At last, a more complex problem, namely the breakup of droplet in a simple shear flow, is numerically investigated. Theoretical analysis is not known for this problem. We compare numerical results with a recent numerical simulation study.

### Identifier

- URN: urn:nbn:de:hbz:82-rwth-2015-055026
- REPORT NUMBER: RWTH-2015-05502