Numerical Computations for a Nonlinear Free Surface Problem in Shallow Water

Abstract

A finite-element method is developed to simulate a numerical towing tank, in the scope of potential theory. The exact nonlinear free-surface flow problem formulated in an initial/boundary value problem is replaced by an equivalent weak formulation and then discretized by the finite-element method. Emphasis is made on the present simulation to include the dry bottom occurring at the downstream of a transom stern model stretched from the free surface to the tank bottom in supercritical speeds. The dry bottom has been observed in an earlier study of Bai et al. But it was not clear whether the phenomenon was physical or simply due to numerical instability. In the present paper, we have introduced a number of numerical schemes to improve the numerical stability and to simulate the dry bottom in more robust manner. Also made is a series of experiments to validate the numerical results and the existence of the dry bottom at the downstream. Numerical simulations and towing-tank tests are made for wedge-shaped ships with different stern shape, beam-draft ratios and the Froude numbers. For the model with transom stern, both the computed results and the experimental observations show the generation of the dry bottom behind the transom stern. We dedicate this article to our teacher, Prof. John V. Wehausen.