*4.6. Mesh Convergence Study*

In order to support the goodness of the numerical scheme created, a grid independence analysis has been carried out. This process proves the stability of the numerical method and ensures that solution does not change when refining the mesh. This convergence study is based on the number of cells used to discretize the wave height that is generated by the oscillations of the hull. In the wave pattern there is the energy lost by the device during the free decay and it is fundamental to capture it accurately. Moreover, prism layer properties are base size independent and computed as absolute values, thus base size has little influence in the overall mesh. The number of layers that are necessary to discretize the wave is very dependent on wave steepness; according to [5], at least 10 layers are suggested and even more for a sharp interface between fluids. Not using enough cells some information may be lost and wave transport and generation may be inaccurate, resulting in overly-damped or magnified waves. Cells count is very dependent on the number of layers of sea refinement because most of the cells come from this mesh control and from the overset zone, as confirmed by results presented in Figure 7.

In order to choose the appropriate mesh sizes CPU time is considered, as shown in Figure 8: passing from 15 cells to 20 means an increase of about 115%. So the configuration of 20 cells per wave height has been excluded because of the prohibitive computational time.

**Figure 7.** Number of total cell varying the cells per wave height.

**Figure 8.** CPU Time (*s*) for different cells per wave height.

The cells count of the overset is related with the size of the sea refinement, because cells at the interface between background and overset must have same volume, especially at sea surface. Convergence study is done on the kinematics of the free decay with respect to Figure 9, which is also the data used for the post-processing and evaluation of damping coefficients.

**Figure 9.** Different kinematics in terms of pitch angle for various cells per wave height.

The wave height with several probes has also been monitored in order to understand the impact of mesh resolution in areas far from the floater.

The coarsest mesh setups have been compared to the finest ones corresponding to 20 cells in terms of relative errors and the associated root-mean-square as shown in Figures 10 and 11.

**Figure 10.** Relative error for different meshes.

**Figure 11.** RMS of error compared to 20 cells per wave height.

Finally, 15 cells per wave height have been chosen, in order to ensure a good compromise between wave capturing and computational cost specially considering the trend of errors in Figure 10, the low related rms (Figure 11) and its computational cost in terms of time (Figure 8).

#### **5. Results**

#### *5.1. CFD Scenes*

In this section, results are presented in terms of scalar and vector scenes in order to show the validity and the reliability of the conducted CFD simulations for pitch and roll free decay motions. An example of the wave pattern generated by the pitching motion is shown in Figure 12, starting from a non-equilibrium angle equal to 10◦. The outcomes are consistent with the physical phenomenon.

The hydrostatic pressure referred to the waterline is well defined as shown in the example of a roll simulation in Figure 13.

**Figure 13.** Hydrostatic pressure referred to the waterline.

The interface is smooth and well-captured, as shown in Figure 14, where the line convolution integral (LIC) of velocity is presented. Such a visualization tends to show a texture where points along the same streamline have the same colour; Displaying LIC of velocity is more computationally efficient the actual streamline scene which the latter is very expensive in terms of computational costs. Generally this technique [35] involves convoluting a white noise image along streamlines computed from the vector field. In the resulting image, the streamlines cover the entire domain of the vector field. This technique has the advantage of being able to visualize large and detailed vector fields in a reasonable display area. Compared with simpler integration-type approaches, which entail following the flow vector at each point to produce a line, LIC produces a whole image at every step.

**Figure 14.** Line Integral Convolution of velocity in roll simulation.

Lastly, the target value of *y*+ = 1 on the wet surface of the hull, on average, has been correctly obtained, as shown in Figure 15. The scene in Figure 15 refers to an intermediate time step and proves an adequate mesh sizing within the scope of capturing a low value of *y*+ which results into an accurate interpolation by the solver in terms of wall stresses and frictional velocity.

**Figure 15.** Wall Y+ on wet surface, grey color refers to the surface in air.

#### *5.2. Post-Processing and Damping Coefficients Identification*

The kinematics of pitch and roll motion versus physical time are presented in Figures 16 and 17, respectively. In order to graphically represent and evaluate the importance of non-linearities, each free decay time trace has been normalized with respect to the initial displacement, as reported in Figures 18 and 19, respectively for pitch and roll. While the pitch normalized responses significantly overlap, suggesting linearity, roll normalized responses are diverse, showing a clear nonlinear behaviour. In particular, the larger the initial displacement, the faster the decaying rate, suggesting a higher content of non-linearity. This is hereafter quantified through the identification of the nonlinear damping coefficient *β*, as discussed in Section 3.

**Figure 16.** Pitch kinematic.

**Figure 18.** Normalized Pitch kinematic.

**Figure 19.** Normalized Roll kinematic.

The different behaviour of pitch and roll in terms of damping is clearly presented in kinematics figures, and the roll motion results less damped than pitch. This behaviour is justified considering the radiation curves related to the hull of this work, shown in Figure 20 where the pitch motion has a more significant value with respect to roll damping, on a point of detail greater than roll of about two orders of magnitude. When oscillations become small and non-linearities less important, roll needs more time to reduce the amplitude of motion. Anyway, roll is strongly damped at the beginning, when oscillations are big because the quadratic part, i.e., proportional to the square of velocity, is predominant in roll viscous damping while the linear part is almost negligible.

**Figure 20.** Radiation curves.

Each pair of consecutive peaks (considering both maxima and absolute values of minima) defines a *αeq* and a *δmean*,*i*, according to Equations (10) and (11), respectively. Such parameters are shown in Figures 21 and 22 for pitch and roll responses, respectively.

**Figure 22.** All roll peaks data.

As regards peaks data in Figures 21 and 22, the wide spread in pitch data at low angles is justified by oscillations around the equilibrium point. Such low angles measurements are negligible as far as the identification of viscous damping coefficient is concerned and thus will be neglected.

About the roll motion the previous spread is absent because it is overall less damped; In addition two trends are clear (Figure 22), one until about 2◦, the other one afterwards. This behaviour is possibly related to the linearization assumptions done in order to define and obtaine *αeq*. Therefore, the identification regards only the left side of the plot, where points are well aligned and so the linearization appears to be more valid.

Taking into account the Equation (10) and also referring to the plots in Figures 23 and 24, the more non-linear behaviour of roll is confirmed. Indeed, considering Equation (10), if *α* represents the intercept and *β* the slope, it is clear the different pattern both in pitch and roll motion. Pitch kinematics are more damped than the roll one, but the first has a linear behaviour instead of the second one which results less damped, but mainly quadratic. With regards to roll data in Figure 24, points corresponding to *δ*<sup>0</sup> = 10◦ differ from the 5◦ and 7◦ because the first case involves much larger angles and consequently a more non-linear behaviour.

Numerical results are summarized in Table 1 in terms of coefficient of determination *R*<sup>2</sup> of the linear fit, the period of oscillations *T* and finally *α* and *β* respectively corresponding to intercept and slope of the linear regression, or in a more physical meaning, the linear and quadratic damping terms.

The computation of values of *α* and *β* throughout the logarithmic decrement here proposed can be considered as a starting step for solving Equation (6). Taking into account the previous outcomes, the equation is solved with a Runge-Kutte scheme (RK) of 4th-order accuracy with an adaptative time step [36].

Furthermore, this technique has been enriched considering a parametrization of 100 values *α* and *β* centred on the outcomes computed through the linear regression and shown in Table 1. For each value, the goodness of fit has been computed with the cost function Normalized Mean Square Error:

$$\text{fit} = 1 - \frac{||\mathbf{x}\_{ref} - \mathbf{x}||^2}{||\mathbf{x}\_{ref} - \mathbf{mean}(\mathbf{x}\_{ref})||^2} \tag{19}$$

where *xref* and *x* represents respectively the reference time history (i.e., the one obtained from CFD) and the one computed through RK-solver, the goodness of fit equals to 1 means a perfect match of results. By way of example and for sake of brevity, outcomes are presented in a visual way through plot in Figures 25 and 26, considering pitch and roll free decay with an initial condition of 10◦, showing CFD results, Equation (6) solved with values in Table 1 and finally with *α* and *β* corresponding to the maximum of goodness of fit (GoF).

**Figure 23.** Cut pitch data and outcomes.

**Figure 24.** Cut roll data and outcomes.

**Table 1.** Numerical results for pitch and roll motion post-processing.


**Figure 25.** Pitch results.

For clarity's sake, zoomed portions of plot are presented in Figures 27 and 28 as far as the first oscillations where large amplitude occur and Figures 29 and 30 as regards the final tail.

**Figure 27.** Large oscillations in pitch.

**Figure 29.** Tail pitch results.

**Figure 30.** Tail roll results.

A different behaviour of the optimization of *α* and *β* is observed as regards pitch and roll. Concerning the first, the maximization of GoF tends to a best fit of larger oscillations as shown in Figure 27 and a poorest fit of the final tail of the decay (Figure 29); The opposite behaviour is evident concerning the roll motion, where the largest amplitudes are not perfectly fitted by the optimal values of *α* and *β* (Figure 28), even if the fit is nevertheless better than the guess value of Table 1. The explanation of such trend is analogous to the considerations about the cloud of data in Figures 21 and 22. Results are summarized in Tables 2 and 3, respectively for pitch and roll motion.


**Table 2.** Outcomes for pitch free decay motion.

\* Low value due to the small amout of points after the angle cut. † The solution of differential equation can not be performed considering all the initial conditions.


**Table 3.** Outcomes for roll free decay motion.

where

