A dual unknown location on the volumic grid associated to the Rhie & Chow interpolation technique is used for the construction of pressure equation. The three components of velocity are located on the nodes of the grid, pressure at the centre of elementary volumes and free surface elevation at the centre of free surface interfaces.

Transport equations are written on the nodes of the mesh, the pressure equation is solved at the centre of elementary volumes and normal dynamic free surface condition at the centre of the free surface interfaces. The two tangential dynamic conditions and the kinematic condition form the set of velocity boundary conditions on the free surface.

The fully linear system obtained by second order finite difference schemes for the velocity components, the pressure and the free surface unknowns is solved at each iteration using a multigrid method with three levels of grid. A generalised Rhie & Chow technique is used to ensure the invertibility of the pressure block.

Numerical results concerning the free surface, the velocity and the pressure field around a Series 60 CB=0.60 (*Rn*=4.5.10^{6}, *Fn*=0.316) show a good agreement with experiments. The problem of singularity of kinematic condition on the hull is well solved and we can calculate the formation of unsteady meniscus near the wall in the whole boundary layer.

Efficiency of k-ε model, in spite of free surface conditions, waves, pressure and velocity fields are presented here, for the steady state and also during the unsteady phase.

Navier-Stokes-Reynolds equations are written under a convective form for a three-dimensionnal turbulent flow in a Newtonian incompressible fluid. The 3 components of velocity *(u*^{i}*),* pressure *(p)* including the gravitational effects (*ρgx*^{3}) and turbulent kinetic energy (2/3*ρk*) are the dependant unknowns. Independent unknowns are the 3 directions of curvilinear co-ordinates *(ξ*^{i}*)* and the time *(t), (x*^{i}*)* is the Cartesian basis and *Ua* the forward velocity, the curvilinear system is chosen to simplify boundary conditions on the hull and on the free surface. *ξ*^{2}=0 et *ξ*^{3}=0 are the equation of wetted part of the hull and of the free surface respectively at each time.

A partial four-dimensionnal transformation of the Cartesian space moving with time in a curvilinear computation space is then applied. The metric of this transformation uses covariant basis *(a*_{i}*)* and contravariant basis *(a*^{i}*),* contravariant metric tensor *(g*^{ij}*),* control grid functions *(f*^{i}*)* and deformation velocities of the computational domain . Transport equations in the frame moving with the hull are written:

(1)

and the continuity equation:

(2)

A classical k-ε model is given for completely developed turbulent flow and does not allow to describe parietal flow where the turbulent viscosity is negligible versus molecular viscosity. The Jones and Launder' s model [7] allows to integrate transport equations up to the wall. It gives the damping function, describing attenuation of turbulence, as a function of turbulent Reynolds number In the curvilinear space *(ξ*^{i}*,t)* the two transport equations for *k* and are:

(3)

(4)