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# Introduction

A brief introduction on the available numerical methods for free surface flows will be given. For more comprehensive reviews the reader is referred to [1] and [2]. The numerical computation of free surfaces flows can be divided in two different approaches. One method uses an Eulerian grid, which is fixed in time. The other method use a boundary fitted grid which deforms in time, and follows the free surface of the flow. The first method is denoted interface capturing and the second interface tracking method in [1]. There are advantages and drawbacks of both methods. The major advantage of the interface capturing method is the capability of treating breaking waves. While the major disadvantage is the loss of accuracy in the determination of the location of the free surface. This method can be divided in to subgroups, Marker and Cell methods (MAC), and Volume of Fluid (VOF) methods. MAC methods use virtual massless particles, markers, convected with the flow. The location of the free surface is found by interpolating between the markers which are closest to the free surface boundary. The method was introduced in [3], later the method has been applied for breaking waves problems, [4].

In the VOF method a new scalar is introduced in the discretized governing equations. The scalar describes the fraction of a cell filled with fluid. The value of the scalar is zero if the cell empty of fluid, and the value is one if the cell is totally filled with fluid. Cells located in the free surface will be partly filled with fluid, thus the scalar will have a value between zero and one. From this information it is possible to obtain an approximation of the location of the free surface. Examples of the use of the VOF method can be found in [5].

However, the most used method for problems related to ship hydrodynamics, such as computation of waves generated by hulls and hydrofoils, are the interface tracking method, which use a boundary fitted grid that follows the boundary of the free surface. The main reason that this is the most popular method for these problems is that it determines the location of the free surface with higher accuracy compared to the interface capturing methods. This method can, however, not threat problems which includes waves that are breaking. The method has been used widely in the recent years in computing both viscous and non-viscous, as well as stationary and time dependent flows. Examples of stationary non-viscous computations can be found in [6]. Viscous 2D computations has been performed by Hino, [7], who computed the same test case as in the present method. Unsteady flows are computed in [8] and [9], the former studied the acceleration of a Wigley hull, whereas the later studied sinusoidal heaving of the same hull.

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