Efficient Numerical Solution of Steady Free-Surface Navier–Stokes Flow

Numerical solution of flows that are partially bounded by a freely moving boundary is of great importance in practical applications such as ship hydrodynamics. The usual method for solving steady viscous free-surface flow subject to gravitation is alternating time integration of the kinematic condition, and the Navier–Stokes equations subject to the dynamic conditions, until steady state is reached. This paper shows that this time integration approach is often inefficient. It proposes an efficient iterative method for solving the steady free-surface flow problem. The new method relies on a different but equivalent formulation of the free-surface flow problem, involving a so-called quasi free-surface condition. The convergence behavior of the new method is shown to be asymptotically mesh-width independent. Numerical results are presented for two-dimensional flow over an obstacle in a channel. The results confirm the mesh-width independence of the convergence behavior, and comparison of the numerical results with measurements shows good agreement.