Stabilised level set methods for two-phase flow simulation

This work is dedicated to the numerical simulation of two-phase incompressible flows with a velocity field imposed or computed from a Navier-Stokes solver. The main objective is to accurately track the interface between the two fluid phases of a domain that is subject to an imposed or computed velocity field. An application of this type of problem is easily found in free surface flows or flow through pipes with a mixture of liquid and gas bubbles or a mixture of two immiscible liquids such as water and oil. We determine the position of the interface between the two phases by the level set method that consists of solving a transport equation to which we add a stabilisation term that globally reduces the error in the interface position during the simulation. These newly proposed stabilised finite element methods solve the interface transport problem using the approach of the level set method. For some cases, the formulation could produce an accurate solution of the interface position without requiring a reinitialisation procedure. However, for complex flow cases, the proposed stabilised methods used with the geometric reinitialisation method allow the elimination of oscillations. In the first part of our study, we obtain the proposed formulation by adding a term that depends on the residual value of the Eikonal equation to the SUPG variational formulation of the equation of the level set. These methods are evaluated numerically for well-known flow problems and are compared with a modified variant of the penalty method of Li et al. (Li et al., 2005). The proposed stabilised finite element methods are implemented with a temporal approximation via the semi-implicit Crank-Nicolson scheme and with a second-order spatial approximation of triangles with 6 nodes. Our techniques are promising, robust, accurate and simple to implement for a two-phase flow with one or many complex interfaces. In the second part of our research, the stabilised level set method is coupled with the mass conservation method because the level set method does not naturally preserve the mass. In this second part of our research, the stabilised variational formulation enforces the level set function to stay as close as possible to the signed distance function, while the conservation of mass is a correction step that imposes a mass (or area) balance between the different phases of the domain. The eXtended Finite Element Method (XFEM) is used to take into account the discontinuities within an element crossed by the interface and is applied to solve the Navier-Stokes equations for two-phase flow. The numerical methods are evaluated on a number of test cases such as the free surface of a liquid sloshing in a tank, dam breaks without or with an obstacle, and the problem of Rayleigh-Taylor instability