
advantage of this method is that it can treat surfaces of any shape and that it can avoid any logical problems associated with the surface intersection. But because F has the nature of a step function, its derivatives must be computed in a special way, and higher-order-accurate difference schemes can not be directly applied. Never theless the applicability of the VOF method is so wide, and because the method needs only one variable as sociated with the surface in each cell, it has also an economical advantage. Miyata et al." developed the TUMMAC-VI method and demonstrated simulations of 3D breaking wave around vertical cylinder piercing the free-surface. In the TUMMAC-VI method, the marker-density technique, which is a generalized or simplified version of the VOF method, is introduced. In the method, marker-density Pm is assumed to be continuous through the computational region including the interface and no special treatment for evaluating its derivatives is employed. Because of the numerical diffusion, the transition region of pm becomes wider and wider through the computation, but when flows have inflow boundary where definite distri bution of the p,n is given, sufficient accuracy is achieved using a fine grid system".
Recently Park et al.4) improved the body boundary condition of the TUMMAC-VI method, and extended the applicability of marker-density technique to the flow about arbitrary three dimensional bodies. But flows around ship or surface piercing struts include not only wave making phenomena but also complex viscous flow field, and to simulate these viscous phenomena, like boundary layer separation or vortex shedding, the use of body fitted coordinates which can resolve boundary layer around three dimensional body is inevitable.
The principal objective of this study is to extend the marker-density technique to three dimensional body fitted curvilinear coordinates, and thus to establish a powerful computation method that can simulate complex-flows around three dimensional body of arbi trary shape including breaking waves. A brief explana tion of the new computational method is given in sec tion 2, and then accuracy validation is made for flow (1)
For the incompressible viscous flow, the Navier-Stokes equation can be written into the following conservative form,
The stress tensor T is defined as (4) where I is the unit matrix and c is the normalized pressure without hydrostatic component defined as: (5) The continuity equation for the incompressible flow is as follows. (6) 2. 3 Solution algorithm In the WISDAM V method, the MAC-type algorithm is employed and all the terms except for the pressure term are evaluated explicitly and only the pressure term is treated implicitly.
Say, the velocity field of the (n+1)-th time level is, 
Here, V is the Laplacian operator defined as (10) where J is the Jacobian and Si is the area vector. Detailed definitions of these are found in references 5) and 6). gij is the contravariant metric tensor defined as follows. (11) Discretizing Eq. (9) using 2nd order central differencing we have 19 points differencial expression, and it is solved iteratively by the SOR method.
4 Free-surface conditions Kinematic condition
The free-surface conditions consist of the kinematic condition, which is the continuity equation on the surface, and the dynamic condition, which stands for the stress equilibrium on the surface. Several numerical represen tations of the kinematic condition are available, includ ing the marker particle method or the height function method. As the objective of this study is to establish a computation method that can treat complicated free surface including breaking waves in the framework of curvilinear coordinates, the marker-density method developed by Miyata et al.2) is employed. The marker density method is a generalized interpretation of the VOF method1), and its advantage is that it can cope with surfaces of arbitrary geometry efficiently.
The marker-density Pm is defined as follows. 
And above equation is discretized using the area vector and Jacobian in the same manner as momentum equa tion.
In this study, the Navier-Stokes equations are solved only within the water layer, but Eq. (13) is solved in the whole computational domain using velocity extrapola tion method described below.
3rd order upwind differencing is used for the spacial differencing of Eq. (13), and 2nd order Adams Bashforth method is used for the time differencing.
The initial condition of pm is given so that the value of pm is exactly the same as the fractional volume of the fluid within a cell, and so the transition region of pm is very narrow, typically one cell width. But because of the numerical diffusion, pm is diffused in the course of time marching unless very find grid spacing is used, thus surface location is obscured and the accuracy of this method is inferior to the other methods like the marker particle method or the height function method. Never theless it is guaranteed that the marker-density method can treat the surface of arbitrary geometry in a very simple way. In addition, because the flow field has an inflow boundary in this study, where definite distribu tion of pm is given, the diffusion of pm has the upper limit, and thus location of the surface is represented with certain degree of accuracy. Dynamic conditions For the problem of ship wave, the surface tension and viscous stresses on the free-surface play only negligible role, and then the following conditions are imposed on the free-surface positioned using the method discribed above. (14) ( 15) where x3 is the height of the free-surface from the still free-surface and n is taken normal to the free-surface. 
where, ufs is extrapolated velocity and, h, is the dis tance from the point of ufs to the free-surface. Eq. (16) means that the velocity of a B-point is extrapolated mainly from the direction nearest to the free-surface. In the case of the right side of Fig. 1 , the value of ufs is extrapolated from the 3 points namely u1, u2 and u3. But because 1, is the smallest, the value of u, is domi nant.
On the left side of Fig. 1 , the scheme of velocity field extension is shown.
The velocity of E-point is extrapolated from the adjacent points which already have their value, and when more than 2 qualified points exist, simple averaging is used.
This procedure is continued until all the points in the computational domain are filled up.
5 Other boundary conditions
No slip condition is imposed on the body surface. All the velocity components are set at zero on the hull surface and the pressure is extrapolated with zero normal gradient.
The marker-density is also extrapolated with zero normal gradient on the body surface, that is, the free-surface is assumed to have right angle to the body surface. This is not always the case for ship wave, but because the nearest grid points are located so close to the hull that this causes little prob lem.
At the inflow boundary the uniform flow velocity is given and the still free-surface and thus standard pres sure is kept here throughout the computation. At the outflow boundary the zero-normal-gradient condition is used for all the flow variables including the marker density. On the centerline boundary, symmetric condi tions are imposed.
Since the 0-type grid system is employed for the transverse sections of the computational domain, the Comparison is made of the pressure distribution on the hull surface in Fig. 4 , again the previous calculation shows better agreement with the measured one except at the both ends of the ship.
Wave contours in Fig. 5 shows good agreement with the measured result, which is not shown here, around the bow and stern, where relatively fine grid spacing is allotted. But in the far field the accuracy is far from satisfactory, and this means that the marker-density method requires very fine grid spacing.
In Fig. 6 , longitudinal variation of the marker-density distribution is shown. In the region from the inflow boundary to the fore end of the hull, the distribution of the marker-density remains definite but as it is convect ed through the computational domain, its transition region becomes wider. In the present case, however , the width of the transition region remained within 3 cell. Fig. 11 , and the comparison of the bow wave contour is made in Fig. 12 Shown in Fig. 13 is the time evolution of the bow breaking wave around the point when the flow accelera tion is finished at T=0.4.
Even after the uniform Table 2 The principal dimensions of models In Figs. 14 and 15, the comparison is made of the photograph of the experiment and the perspective view of the calculated bow wave of the container carrier model LC442 running at the Froude number 0.4. And the calculated wave contour map is shown in Fig. 16 . In contrast with the bulker model M55, the breaking wave front is nearly the straight line and emanating from the hull surface, that is refered as oblique wave here.
After Baba"' found that there is a momentum deficient wake caused by the bow wave breaking in 1969, the wave breaking has been explained and discus. sed in relation with the nonlinear wave called free surface shock wave and the vortical motion under the free-surface. The characteristic vortex under the breaker is sometimes called necklace vortex11). In Fig.  17 iso-surfaces of the magnitude of vorticity J W are shown. As seen in Fig. 17(A) , for the case of the Wigley's hull, almost no significant vorticity is generat ed by the wave except for the very narrow region just in front of the bow. In contrast, the case of the normal breaking wave namely for the bulker model in Fig. 17  (B) , and the case of the oblique breaking wave in Fig.  17(C) , certain amount of vorticity is generated just under the breaking wave crest respectively. In Fig. 18 , distribution of I wj at three longitudinal sections are shown. At the wave front strong deficit of the velocity is caused by the breaking motion, and the mixing layer of the momentum is formed under the free-surface. Baba10) found the momentum deficient wake by this breaking motion far behind the ship. but in this calcula tion, this shear layer is numerically dissipated before the midship.
Concluding remarks
The 

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