surface and body boundary conditions have then been introduced in the model, with applications related to radiation and diffraction problems for submerged bodies, using a fully Lagrangian description of the free surface (Ferrant 1994). A further step in the development of the nonlinear model was achieved with the treatment of surface-piercing bodies, involving bi-cubic spline interpolations at the free surface and extrapolation techniques at the waterline (Ferrant 1995).
We describe significant results of two versions of the code differing in the formulation of the free surface motion. The first version is based on a semi-Lagrangian formulation with an explicit treatment of the incident wave through a stream function model. This model is primarily dedicated to radiation and diffraction problems in waves of moderate steepness, without overturning. Two different applications of this version are presented. First we describe the nonlinear simulation of the diffraction of long waves by a surface-piercing vertical cylinder. Higher order loads and free surface motions are computed and compared with results from the third order frequency domain model of Malenica & Molin (1995). Thenafter, the simulation of the nonlinear free motion of a floating circular dock in a regular wave is presented.
The second version of the code uses a fully Lagrangian description of the free surface and material boundaries, which potentially allows the simulation of the generation of steep three dimensional waves up to overturning, as well as their interaction with material boundaries. It is applied here to the simulation of large amplitude standing waves in a three dimensional tank. This Lagrangian version is under development and we hope to be able to present results on the generation of steep three dimensional in a wave tank in a very near future.
The fluid is heavy, inviscid and incompressible. The problem is started either form rest or from prescribed initial conditions so that the flow remains irrotational in the fluid domain D. The constant atmospheric pressure at the free surface Sf is taken as the pressure of reference. The fluid velocity derives from a scalar potential satisfying Laplace equation in the fluid domain:
(2) Δ (M,t)=0
An initial boundary value problem for is obtained by applying suitable boundary conditions on the surfaces limiting the fluid domain: the free surface Sf, the body surface Sb, and the external surface Se which may include a bottom at finite but not necessarily constant depth.
On the body and bottom surfaces, Neumann conditions are applied:
where n is the unit normal vector exterior to the fluid and Vb is the local velocity of the body surface, relative to the fixed coordinate system Oxyz with z pointing upwards and z=0 on the calm water level. In the case of infinite depth, the bottom condition is replaced by:
(5) --> 0 for z -->–∞
Computations presented in this paper concern problems with a flat bottom at constant depth. The bottom condition is thus not explicitly satisfied, but is accounted for by an additional symmetry. The general case of a non uniform bottom may be treated without difficulty at the cost of additional unknowns.
On the free surface both kinematic and dynamic conditions must be satisfied. The kinematic condition states that the mass flux through the free surface is zero, and writes, in Lagrangian form:
Surface tension being ignored, the dynamic