On the Solution of Nonlinear Free-Surface Problems by Surface- Singularity Techniques

This report describes an attempt to solve the two-dimensional problem of a body performing steady translation near a free surface. The aim was to solve the problem in its full generality without any linearizations or small- perturbation assumptions. The method of solution is based on iterative use of the well-known surface singularity technique and thus, if successful, the method could be generalized to three dimensions. At each stage of the procedure the free-surface shape is assumed known and the singularity strength adjusted to satisfy the constant-pressure boundary condition. In general, the normal velocity on the free surface is not zero. The free-surface shape is then altered by some algorithm, and the procedure is iterated to obtain zero normal velocity. The key to approach is the iterative algorithm. Various algorithms were tested by applying them to the problem of a submerged point vortex and comparing the free-surface shape obtained in convergent cases with published results. After considerable experimentation, a procedure was devised that gives very good results as long as the wave heights are not too large. For large wave heights, convergence could not be obtained, and some as yet undiscovered change in the method is needed.