Level set methods for prediction of two-phase incompressible flow

Multi-phase flows have been deeply studied in the last two decades. Such an interest rose from the fact that the number of the possible applications in the engineering field are uncountable. Some examples are the chemical reactions, it is likely that they involve more than one phase of reactants or products at a time. In this framework the numerical simulation of those phenomena acquires an huge importance for predicting the physics of a system and better understanding the small scale evolution of the flow. The only viable way to accomplish this task is to adopt the DNS methodology for resolving the Navier-Stokes equations. Even though in the present day we do not dispose enough speed and memory from the current supercomputers to take advantage from such an approach for engineering applications, the DNS will remain the only numerical method that can provide new physical insights at all scales of turbulence in flows involving complex physical phenomena. This thesis concerns about the study of the interface evolution between two different phases via the implementation of the Level Set Method (LSM). The Level Set method is an Eulerian method that belongs to the front capturing techniques such as the Volume of Fluid method (VOF). According to these methods the interface is reconstructed from suitable field scalar variables, in the case of the LSM from a distance function. Another category of methods for computing multifluid flows is the Front Tracking (FT). In this case a separate front marks the interface while a fixed grid is adopted for the fluid in every single phase. We preferred to adopt the Level Set method above the others techniques since it is simple to handle, it manages easily breaks-up and merges of the interface, and it is a good candidate for the exploiting of high-order spatial discretization schemes. The main drawback of the Level Set method is the lack of mass conservation. The task of the current thesis is to compare different techniques to abolish, or at least diminish, this problem. The work was accomplished with the essential contribution of the Mechanical and Aerospace Department (MAE) of the University of California - Irvine. The group possesses a DNS code for the resolution of the Navier-Stokes equations written in Fortran 90, with objects and features from Fortran 2003. We took advantage from this code, which represented the starting point of the current project. A numerical model was developed using the environment provided by Octave 3.8.1, whose flexibility and simplicity is the main strength for a dynamic encoding and testing phase. The finite difference discretization was used to reduce the differential equations to algebraic equations. We adopted a structured Cartesian grid and the domain is in two dimensions. Besides the traditional method for the calculation of the interface evolution we implemented and compared three more methods. The Level Set method theory is deepened in the first part of the thesis. Here the attention is focused on the main features of the method and the techniques to advect and reinitialize the scalar function. The theory also provides us a geometrical toolbox for the calculation of useful properties. The numerical methods for the discretization of the differential equations were presented. Four Level Set methods were implemented: the first one is the traditional advection method of the Level Set function, the second introduces a modification in the reinitialization step to avoid the shifting of the surface, the third method embeds in the advection equation a source term in order to avoid the reinitialization step and the last one is a mass-conservative method. The verification of the proposed methods was provided. Thus, we compared the methods in term of mass conservation and considering their ability to maintain the exact shape from the analytic solution. Finally the proposed methods are applied to a incompressible flow case test, that is a falling droplet in air. The Navier-Stokes equations were resolved with the projection method, the surface tensions were treated as source terms localized within the finite thickness of the interface.