Mass Conserving Simulations of Two Phase Flow

Abstract: Consider a mixture of two immiscible, incompressible fluids e.g. oil and water. Since the fluids do not mix, an interface between the two fluids will form and move in time. The motion of the two fluids can be modelled by the incompressible Navier-Stokes equations for two phase flow with surface tension together with a representation of the moving interface. The parameters in the Navier-Stokes equations will depend on the position and other properties of the interface. The interface should move with the velocity of the flow at the interface. Since the fluids are incompressible, the density of each fluid is constant. Mass conservation then implies that the volume occupied by each of the two fluids should not change with time. The object of this thesis has been to develop a new numerical method to simulate incompressible two phase flow accurately that conserves mass and volume of each fluid correctly.Numerical simulations of incompressible two phase flow with surface tension have been a challenge for many years. Several methods have been developed and used prior to the work presented in this thesis. The two most commonly used methods are volume of fluid methods and level set methods. There are advantages and disadvantages of both of the methods.In volume of fluid methods the interface is represented by a discontinuity of a globally defined function. Because of the discontinuity it is hard both to move the interface as well as to calculate properties of the interface such as curvature. Specially designed methods have to be used, and all these methods are low order accurate. Volume of fluid methods do however conserve the volumes of the two fluids correctly.In level set methods the interface is represented by the zero contour of the globally defined signed distance function. This function is smooth across the interface. Since the function is smooth, standard methods for partial differential equations can be used to advect the interface accurately. A reinitialization is however needed to make sure that the level set function remains a signed distance function. During this process the zero contour might move slightly. Because of this, the volume conservation of the method becomes poor.In this thesis we present a new level set method. The method is designed such that the volume of each fluid is conserved, at least approximately. The interface is represented by the 0.5 contour of a regularized characteristic function. As for standard level set methods, the interface is moved first by an advective step, and then reinitialized. Unlike traditional level set methods, we can formulate the reinitialization as a conservation law. Conservative methods can then be used to move and to reinitialize the level set function numerically. Since the level set function is a regularized characteristic function, we can expect good conservation of the volume bounded by the interface.The method is discretized using both finite differences and finite elements. Uniform and adaptive grids are used in both two and three space dimensions. Good convergence as well as volume conservation is observed. Theoretical studies are performed to investigate the conservation and the computational time needed for reinitialization.