The right-hand side of Eqn. (5) is equal to zero by the continuity equation.
The algorithm of the present scheme is shown is about the same as that of the MAC method. The pressure Φ and velocity (u,v,w) are defined at the same grid points of regular mesh. The marker particles putted on the free surface of water are moyed only the z direction and fixed toward x and y directions. At first, E, F and G of Eqn. (3) are calculated. After the given H,F and G are substituted in the Poisson equation, the pressure Φ is solved iteratively by the SOR method. The velocity (u.v,w) at the (n+l)-th time step is calculated by Eqn.(3).
Assuming the free surface by H(x,v,z,t)=z - η (x,y,t), the free surface elevation η (t,x,y) is calculated by the linear interpolation from velocity of moving marker particles on the free surface. As material derivative of the equation of the free surface with respect to time t is equal to zero, the following equation is obtained from ∂ H/∂t＝ 0 as follows,
This calculated process are repeated in time marching procedure till we obtain the steady solution. The calculation starts from zero velocity at initial non-dimensional time
T＝0.0, and the accelerations of velocity are continued to T=0.1.
The body-fitted curvilinear coordinates system (ξ, η ,ζ) is introduced. The fundamental equations obtained on the physical coordinates systems are transformed to the computational coordinates. The relation of between the physical coordinates and the computational ones is
ξ＝ξ（x,y,z,t), η＝η（x,y,z,t),ζ＝ζ（x,y,z,t),τ＝ｔ. （7）
Keeping the numerical computation stable, the convection terms are evaluated by the third-order upstream difference and the fourth-order central differences are used for the spatial differentials.
4. NUMERICAL CALCULATION
A computed flat plate has the unity non-dimensional wetted chord length in the still water, and B/L is 0.44 (B is the plate breadth). The steady attack angle is 5.0° as same as the conditions of wave experiments as shown in chapter 2. Fig.5 shows the computational grid system and the body-fitted coordinate is generated by a geometrically method. The numbers of the grid are 127 × 41× 31 in x,y and z directions, respectively. The computational domain is -1.228≦ x≦ 4.80, -1.248≦ y≦ 0.0, -0.377≦ z≦ maximum wave height. As the arrangement of grid point gives large effects on the accuracy of calculation, they are clustered near the leading edge, the trailing edge, the port side edge, the bottom of the planing plate and the free surface of water, because the flows near there are very complicated. The minimum spacing of the grid in the non-dimensional physical coordinates is 0.0071 constantly in x and y directions, and 0.001 in z direction. The maximum spacing is 0.05 in x, 0.11 in y and 0.03 in z directions. The time increment Δt is 0.0003, which is limited by the CFL condition.
Boundary conditions in the computational domain are in the followings. On the bottom boundary, the uniform flow and the
