Analysis of the incompressible Navier-Stokes equations with a quasi free-surface condition

Numerical solution of free-surface flows with a free-surface that can be represented by a height-function, is of great practical importance. Dedicated methods have been developed for the efficient solution of steady free-surface potential flow. These methods solve a sequence of sub-problems, corresponding to the flow equations subject to a quasi free-surface condition. For steady free-surface Navier-Stokes flow, such dedicated methods do not exist. In the present report we propose an extension to Navier-Stokes flow of an iterative method which has been applied successfully to steady free-surface potential flow. We then examine the sub-problem corresponding to the incompressible Navier-Stokes equations subject to the quasi free-surface condition. We consider perturbations of a uniform, horizontal flow of finite depth and we show that the initial boundary value problem associated with the incompressible Navier-Stokes equations and the quasi free-surface condition allows stable wave solutions that exhibit a behavior that is typical of surface gravity waves. This indicates that the iterative method proposed is indeed suitable for solving steady free-surface Navier-Stokes flow. Implementation of the iterative method and numerical experiments are treated in a forthcoming report.