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Twenty-First Symposium on NAVAL HYDRODYNAMICS
were satisfactory. We have extended this method with, among other things, a frequency independent absorbing boundary condition .
In this paper we apply the method to a LNG carrier at service speed. The forward speed of the commercial tanker considered up to now is very low, the maximum Froude number is 0.018, i.e. 2 knots, while the usual speed of a 125,000m3 LNG carrier is about 20 knots (i.e. Froude number is 0.2). This fact causes some problems in our algorithm. We study increasing speed and the effect on our absorbing boundary condition. To remove the instabilities on the free-surface due to increasing forward speed, we introduce upwind discretization. Both cases were done in the two-and tree-dimensional algorithm.
In the first section we give the main idea of the Prins' algorithm and our extension to a frequency independent boundary condition. In the second section we study increasing speed and the the effect on our algorithm. In the third section results are presented for a 125,000m3 LNG carrier, at deep water. We will calculate the drift force or added resistance not only for low forward speed, but also the added resistance for higher forward speed has been studied. In order to check the method, we compare our results with measurements of Wichers  and with strip theory results [1, 2]. The last section we give the conclusions and ideas for further research.
The time-domain algorithm given below is based on the one given by Prins .
The physical fluid domain is an infinite (or large) domain. The computational domain cannot be infinite, so we have to introduce artificial boundaries and proper boundary conditions. In the literature several methods have been proposed to absorb free surface waves. On the basis of a literature search, Prins decided to use an extension of the Sommerfeld radiation condition for two families of waves. The disadvantage of this Sommerfeld condition is that it is dependent on the wave frequency, so it cannot handle non-harmonic waves, and on the forward velocity.
Keller and Givoli  introduce a semi-discrete DtN-method, using an artificial boundary, dividing the original domain into a computational and a residual domain (the interior and exterior). In our method we use a three-dimensional boundary condition independent of the wave frequency, using the idea of the Givoli's method with Prins' algorithm. In the interior domain we use the same mathematical model as Prins  use but we do not implement a Sommerfeld radiation condition on the artificial boundary.
The interior problem
We consider a vessel sailing with an uniform velocity U in the negative x-direction, or an uniform current with velocity U is directed in the positive x-direction. Regular waves are travelling in the water-surface in a direction which makes an angel β with the positive x-direction, see figure 1. The coordinate system is chosen such that the undisturbed free surface coincides with the plane z=0 and the centre of the gravity of the hull is on the z-axis, with z pointing upwards. The hull is free to move in all directions and to rotate around the main axes.
FIGURE 1: The geometry
We assume the following restrictions: there is no viscosity, the fluid is incompressible and homogeneous, and the flow is irrotational. We introduce the velocity potential Φ, which has to satisfy the Laplace equation
By using the dynamic and kinematic conditions and splitting the potential into a steady and an unsteady part, like
we get the linearized free-surface condition on the undisturbed free surface
with subscripts denoting the partial derivative. In contrast with the two-dimensional algorithm, we do not include terms of Including them would cause us to calculate higher derivatives of the unsteady potential at the free surface. This would increase the computation time, because the need of the very fine mesh.