hydrostatic pressure are used, because the bottom of the computational domain is considered sufficiently deep and no free surface effects. On the forward flow, the uniform flow velocity and the hydrostatic pressure are given as u= 1.0, v＝w＝0.0 and Φ＝0, and no wave height. On the far rear flow boundary, the velocity and the pressure are extrapolated from the inside. On the central plane boundary, the symmetrical condition is implemented for the velocity and the pressure. On the lateral plane, the velocity and the pressure are extrapolated from the inside. On the free surface, the followings are assumed that the surface tension is neglected, that the atmospheric pressure is used and that the velocity is extrapolated from the lower grid points. On the body boundary, the no-slip condition is implemented as u＝v＝w＝0, and the Neumann boundary condition for the pressure is given by substituting the no-slip condition into Hqn. (3).
In the initial condition at non-dimensional time T=0.0. the free surface elevation near the plate is not zero and the pressure becomes z/Fn though the velocity is zero in the entire domain. Generally, the trailing edge of a planing plate is immersed in still water. After the fluid motion started, the free surface at the rear of the trailing edge goes down gradually and last, it is smoothly connected to the trailing edge.
The process is so complicate that the computing algonthm and the body-fitted curvilinear coordinates can not be constructed following to every instant. To overcome this, the initial free surface is assumed to be the same as wave generating plate and to be on a line smoothly connected to the trailing edge of the plate. The same assumption is adopted to the initial free surface in the both sides of the plate.
In spite of these unrealistic initial conditions and grid generations, it is confirmed that the computations do not diverge and the computed free surface pattern becomes realistic after one thousand computing time steps.
The computations are carried out in Re=104, Fn=1.29. Reynolds number is restricted in low value, because the turbulence model is not included. The computation is tried to a non-dimensional time T=4.5 and there, the flow becomes steady.
5. RESULTS
The calculated waves are shown in perspective in Fig.6. The wave height is magnified three times in the figure. When the foxy becomes steady, the wave features near the side edges of the plate become qualitatively similar to the experimental ones.
The stem cross waves and the triangle waves are developing, however, the stern divergent waves are not appeared. The reason is considered to be in lower Reynolds number The spray is also not present because of the strong non-linearity of phenomena. It is considered that the finer grid and improved computations are necessary to the accurate simulation.
The developing wave contour maps at Fn=1 .29 are shown in Fig.7. Solid lines show positive values of the free surface elevation and dotted lines show negative values. The wave configuration is symmetrical to the x-axis, and triangle waves and stern cross waves can be seen clearly in Fig.7.
Fig.6 Perspective view of wave pattern at Fn= 1.29
