Open boundary conditions for the Navier-Stokes equation

The purpose of this report is to address in a general sense the question of open boundary conditions of in and outflow type for the Navier-Stokes equation and to some extent relate that to the boundary conditions used for the Euler equations. Both the continuous and semi-discrete problem is analysed using the energy-method and the Laplace-transform technique. The energy method is used to derive well-posed boundary conditions for the continuous problem. For the semi-discrete problem, the energy method is used to prove that by using the well-posed boundary conditions for the continuous problem and adding a suitable numerical boundary condition well-posedness is preserved. By employing the Laplace-transform technique, the spectra for different types of boundary conditions are obtained. The spectra are analysed and, in particular, it is shown how the choice of boundary conditions strongly affects the convergence to steady-state.