The evolution of the free interface between two incompressible viscous fluid flows occurs in many physical problems. The numerical simulation of these free surfaces presents great difficulties in scientific computing, especially when the Reynolds number is very large.
In this thesis, a new numerical approach is introduced to simulate the two-dimensional viscous incompressible fluid flows with free interface. A mapped coordinate system is used to transform the curved geometry into a rectangular region so that a standard grid can be applied to the region. The Navier-Stocks equations using the stream function and vorticity are solved together with the interface conditions between two fluids. The second-order backward difference formula is used for time evolution. An implicit/explicit method is used to avoid iterative procedure when handling the nonlinear terms. A periodic boundary condition is assumed in the horizontal direction and spectral methods are applied. The equations are then written as a first order ODE system in the vertical direction. In order to achieve a stable numerical method for calculations with huge Reynolds number as well as a high accuracy, a forth-order boundary value problem solver is introduced to solve this ODE system. In order to get the initial conditions for the simulation, linear stability analysis with the interface is performed within the mapped geometry. The non-linear parts of the system are input into the simulation gradually in order to get a smooth transformation from the linear initial condition to the non-linear simulation.
As an important application of this new method, the viscous effects on the Benjamin-Feir Instability for the deep water wave are studied. The simulation results show that the viscosity has a strong impact on the development of the Benjamin-Feir instability. For waves of small wavelength where viscous effects are mere pronounced, the Benjamin-Feir instability will be suppressed by the viscous damping after a certain amount of time.