4.2.2. Overset Region

Two main factors influence the overset's dimensions:


For the overset sea refinement, we used the shape displayed in Figure 6 and isotropic cells because, otherwise, when the region rotates, the anisotropic sea refinement at the water line will rotate along with the hull.

**Figure 6.** Hull mesh refinement based on maximum rotation angle.

#### *4.3. Volume of Fluid*

The Volume of Fluid (VOF) Multiphase model is an Eulerian numerical model suited to simulate flows of several immiscible fluids on numerical grids, capable of resolving the interface between the different phases [29]. The spatial distribution of each phase at a given time is defined in terms of a variable that is called the volume fraction. A method of calculating such distributions is to solve a transport equation for the phase volume fraction. The method uses the STAR-CCM+ Segregated Flow model which solves each of the momentum equations in turn. The VOF Multiphase model is the standard model to deal with marine environment and wave generation in a RANS simulation [21].

#### *4.4. Turbulence, Wall Treatment, Y+*

K-Epsilon is a two-equations model that solves transport equations for the turbulent kinetic energy and the turbulent dissipation rate in order to determine the turbulent eddy viscosity. Such model provides a good compromise between robustness, computational cost and accuracy and are well suited for external flows and when gradients at wall are not too strong [30,31]. In our case a Realizable Two layer K-Epsilon Model is used [32]. This model contains a new transport equation for the turbulent dissipation rate with respect to standard K-Epsilon. In addition, the turbulent viscosity is expressed as a function of mean flow and turbulence properties, rather than assumed to be constant as in the standard model. A two-layer wall treatment is adopted: this method solves for but prescribes and turbulent viscosity algebraically with distance from the wall in the viscosity-dominated near-wall flow regions. In this approach, the boundary layer is divided into two layers. The values specified in the near-wall layer

are blended smoothly with the values computed from solving the transport equation far from the wall. The equation for the turbulent kinetic energy is solved across the entire flow domain. The two-layer approach, which resolves correctly for both low and high *y*+, using respectively standard wall function and blended wall function according to Reichardt's law [33], fits perfectly with the free decay problem: after few oscillations the velocities are smaller and *y*+ value decreases during the numerical experiment. Based on such considerations, a target value of *y*+ = 1 has been chosen, since the aim of this work is to define damping coefficient accounting for non-linear effects, thus it is important to describe properly the boundary layer zone where viscous stress, separation or vortex shedding that occur. The value of *y*+ = 1 is obtained considering the flat-plate boundary layer theory [34], obtaining a first cell height of 4.222 · <sup>10</sup>−<sup>4</sup> m, considering a reasonable total thickness of 3 cm and using a number of 10 layers, which guarantee a smooth grow rate of the cells.

#### *4.5. Time Step*

In Star-CCM+ the multiphase VOF model requires the use of an implicit unsteady approach. A second order time-marching discretization has been chosen since the first order is numerically dissipative and the property of the wave may not be transported correctly. First order time discretization is the only model unconditionally stable, but it has a drawback: the leading term of the truncation error of convective flux resembles a diffusive flux. This numerical, or artificial, diffusion that is introduced in the solution is magnified in multidimensional problems if the flow is oblique to the grid. A very fine grid would be necessary to obtain accurate solutions. On the other hand, since a second order approach does not require such a refined mesh-discretization, it is faster and more accurate than a first-order time-discretization. Time step is limited by two different conditions:


The more severe limitation on the time step comes from the latter condition; therefore, it is convenient to change the time step according to the different initial conditions: since the oscillation period is always the same, the further away from equilibrium is the initial displacement, the smaller the time step needs to be in order to maintain the same courant number.
