Efficient Conservative Numerical methods for fluid flow problems with Interfaces

The surface fluid flows coupled with porous media flows have many applications in science and engineering. The mathematical models are the coupled partial differential equations with interface conditions. It is of importance to develop efficient numerical methods for modeling and analyzing these interface problems. In the dissertation, we first investigate the long wave solutions for the surface flows on inclined porous media. The flows are derived by the Navier-Stokes equations governing the clear flows in the surface fluids and the nonlinear Forchheimers flow equations are for the porous media flows in substrates. We derive out a corresponding Orr-Sommerfeld problem by linearizing the infinitesimal perturbations in the system of coupled equations for analyzing the long wave solutions of surface flows. Numerical analysis is further developed by using Chebyshev collocation method to its eigenvalue problems to analyze critical condition and stable regions of long wave solutions. Secondly, a new mass preserving solution-flux scheme is proposed for solving parabolic multilayer interface problems. The domain is divided into staggered meshes for layers. At regular grid points in each subdomain, the solution-flux scheme is proposed to approximate the equation. The important feature of the work is that at the irregular grid points, the novel corrected approximate fluxes from two sides of the interface are proposed by combining with the interface jump conditions, which ensure the developed solution-flux scheme mass conservative while keeping accuracy. We prove theoretically that our scheme satisfies mass conservation in the discrete form over the whole domain for the interface equations. Numerical experiments show mass conservation and convergence orders of our scheme and the application of the heat propagation in the multilayer media. Thirdly, an efficient conservative pressure velocity scheme for coupled free and porous media flow system is proposed. The Stokes equations are applied in the free flow domain and Darcy's law is used to model the porous media flow. The system couples the two flows of an incompressible fluid at fluid-porous media interface via an appropriate set of interface conditions including the Beavers-Joseph Saffman condition. Based on the efficient treatments of the interface conditions, a time-splitting conservative pressure-velocity scheme within a staggered grid framework is developed to solve primitive variables in the free and porous media flow system. We prove theoretically that our scheme satisfies mass conservation in discrete form over the whole domain. Numerical simulations are carried out for two model problems, two real model problems and a realistic problem. Numerical results show the convergence, mass conservation and excellent performance of the proposed scheme.