A semi‐implicit numerical method for the free‐surface Navier–Stokes equations

Abstract

SUMMARYSemi‐implicit methods are known for being the basis of simple, efficient, accurate, and stable numerical algorithms for simulating a large variety of geophysical free‐surface flows. Geophysical flows are typically characterized by having a small vertical scale as compared with their horizontal extents. Hence, the hydrostatic approximation often applies, and the free surface can be conveniently represented by a single‐valued function of the horizontal coordinates. In the present investigation, semi‐implicit methods are extended to complex free‐surface flows that are governed by the full incompressible Navier–Stokes equations and are delimited by solid boundaries and arbitrarily shaped free‐surfaces. The primary dependent variables are the velocity components and the pressure. Finite difference equations for momentum, and a finite volume discretization for continuity, are derived in such a fashion that, after simple manipulation, the resulting pressure equation yields a well‐posed piecewise linear system from which both the pressure and the fluid volume within each computational cell are naturally derived. This system is efficiently solved by a nested Newton type iterative scheme, and the resulting fluid volumes are assured to be nonnegative and bounded from above by the available cell volumes. The time step size is not restricted by stability conditions dictated by surface wave speed, but can be freely chosen just to achieve the desired accuracy. Several examples illustrate the model applicability to a large range of complex free‐surface flows and demonstrate the effectiveness of the proposed algorithm. Copyright © 2013 John Wiley & Sons, Ltd.