Numerical methods for mass transport equations in two-phase incompressible flows

摘要

In this thesis, we presented numerical methods for discretizing and solving the mass transport problem in two-phase flows. The level set method is used for capturing the time-dependent interface. The motion of the fluid is described by the two-phase Navier-Stokes equations. For the spatial discretization of these equations we use the known methods in the literature, namely the emph{improved} Laplace-Beltrami discretization for the surface force and the extended finite element (XFEM) for the pressure approximation. The combination of these methods delivers optimal error bounds when the surface tension coefficient is constant. For the general case with a variable surface tension coefficient, we introduce a new discretization of the localized surface force term. The solution of the mass transport equation must satisfy certain interface conditions, which imply that in general both the concentration and its derivatives are discontinuous across the interface. A simple transformation is often used in the literature to eliminate the discontinuity of the solution, which, however, results in a suboptimal approximation error bound O(h^{1/2}) in the L^2 norm for the finite element discretization. We use the Nitsche-XFEM method to handle the Henry condition and obtain an optimal error estimate O(h^2) in the L^2-norm for the spatial discretization in the case of a emph{stationary} interface. The semi-discretization resulting from the Nitsche-XFEM method is combined with the standard theta-scheme and an optimal time discretization error bound is also obtained. This method can also be applied for problem with moving interface but a full error analysis is not available. Finally, we performed numerical simulations of the coupled two-phase Navier-Stokes and mass transport equations for rising droplet problems for both cases of constant and concentration-dependent surface tension coefficients. For the latter case, different phenomena were observed, such as the occurrence of the so-called stagnant cap in the velocity field and a significant change in the droplet rising velocity. Due to the absence of a stabilization method for the discretization of the mass transport problem, we restrict ourselves to the case of medium diffusivity instead of the physically correct (much smaller) diffusivity. Effects of the initial concentration and the size of the convection (relative to the diffusion) on the droplet rising velocity and the droplet concentration at steady state are investigated.