A consistent boundary element method for free surface hydrodynamic calculations

Abstract

SUMMARYFree surface phenomena are described by equations that exhibit two types of non-linearities. The first is inherent to the equations themselves and the second is caused by the application of boundary conditions at a free surface at an unknown location. Numerical calculations usually do not specifically recognize the second non-linearity, nor treat it in a fashion consistent with the more obvious non-linearities in the boundary conditions. A consistent formulation is introduced in the present paper. The field equation is integrated and the free surface boundary conditions are applied on the unknown geometry by means of appropriate series expansions. The consistent formulation introduces improvements in accuracy and computing speed. The method is demonstrated on several hydrodynamic free surface problems and an error analysis is included.

An implicit boundary element solution with consistent linearization for free surface flows and non‐linear fluid–structure interaction of floating bodies

“…In this example, very similar to the one considered by Medina et al [27] the free surface motion under the action of the gravitational forces is followed in a rectangular tank. Initially, the free surface has a triangular shape (Figure 2) and the boundary conditions are given by null ux across the walls of the tank.…”

“…It should be noted that the formulations are either equivalent or similar to the formulations from literature for certain sub-cases. For example, the boundary element equations are equivalent to the ones employed, among others, by Longuet-Higgins and Cokelet [17], Lin and Yue [7], Nakayama [39], Medina et al [27], Haack et al [26] etc. The treatment of the time derivative of the potential is similar to the one applied by Grilli [24] or developed by Kang and Gong [38] for one moving body in an inÿnite uid.…”

“…This non-linearity has been for the ÿrst time exposed by Ligget [28] and used by Medina et al [27] to formulate a semi-implicit free surface formulation. Donescu and Virgin [42] demonstrated the use of a fully implicit method in the context of a consistent linearization for the solution of non-linear free surface problems.…”

“…Another approach, employed by Abe and Kamio [40] is based on using weighted residuals for the non-linear boundary conditions on the free surface. A signiÿcantly di erent approach, adopted by Medina et al [27], consists of explicitly considering that the boundary element equations depend on the free surface position. This key element has been used in the linearization of the integral equations and the boundary conditions with respect to the potentials, uxes and free surface position deÿned by displacements on preferred directions of motion chosen at the previous time step.…”

“…Implicit or semi-implicit procedures have mainly consisted of segregated approaches used in FIDAP [44] and by Haack et al [26] in which iterations between the computation of the free surface position and the solution of the linear boundary element equations for potentials and uxes are used at each time step until convergence. A di erent approach adopted by Medina et al [27] (based on Liggett [28]) explicitly considers that the boundary element equations depend on the free surface position and uses linearization on preferred directions determined at the previous time step.…”