A time accurate calculation procedure for flows with a free surface using a modified artificial compressibility formulation

摘要

A time accurate, three-dimensional, finite volume, implicit scheme has been proposed for the calculations of flows with a free surface on unsteady Eulerian grids, which uses a modified form of the artificial compressibility formulation. The modification comes from the addition of the gravitational potential to the pressure term appearing in the continuity equation of the original formulation. The main advantage of this approach comes from the fact that the analysis of the inviscid flux Jacobians in terms of their eigenvalues, eigenvectors, etc., turns out to be of the same form as that of the original formulation without the body force. Also the present formulation allows one to represent the gravitational potential as a third-order accurate term by the use of Roe variables in an implicit manner, which is not possible otherwise. Thus, the formal accuracy of the original scheme is preserved. The gravitational potential appears explicitly only in the free surface boundary condition in the present formulation, which means minimal code modification to a code based on the original artificial compressibility method. The boundary conditions used at the free surface as well as at the radiating boundary differ from other formulations and are also discussed. An unsteady test case of a wave channel has been chosen to demonstrate the capabilities of the method, and comparisons are made with available experimental results. The results clearly demonstrate the time accuracy as well as the feasibility of the current method for solving flows with a free surface.