This section presents the main experimental results and discusses the strategies adopted to reproduce them in the numerical model. Firstly, the effects of wind forces on surge, heave and pitch natural periods obtained from decay tests are investigated and comments on the hull mean position and attitude are drawn. Response in waves are presented in
Section 3.2, aiming to study the concomitant effect of not aligned wave and wind. For the present study, the angle between waves and wind is 30°, as shown in
Figure 12.
3.1. Decay Tests
Decay tests were carried out to assess natural periods and damping levels.
Table 3 brings the experimental results for natural periods for surge, heave and pitch motions. Three wind velocities were considered—no wind,
m/s and
m/s. In addition, two different initial inclination angles of the floater were taken into account: even keel and +5° trim (hereafter, the latter condition will be denoted by “with trim adjustment (TA)”). The TA was adopted in the tests to reduce the mean tilt angles for
m/s and
m/s wind conditions, thus emulating the effects of a ballast compensation.
Table 4 presents the mean surge and mean trim angles for both conditions, with and without TA.
The wind thrust force induces significant changes in the surge and pitch natural periods obtained from the decay tests. In fact, similar results are found in the literature [
19,
28,
29]. However, the origins of surge and pitch shifting in period are different. Due to the mean thrust load, the FOWT platform drifts to a mean point far from the trivial unloaded position. As discussed by [
27,
30], the new equilibrium position changes the mooring system stiffness, mainly for horizontal motions, as they are the most affected by mooring restoring forces.
Figure 13 brings the stiffness-offset relation, considering different tilt angles of the floater, for surge, pitch and heave motions only (possible couplings included). Notice that the plots in
Figure 13 represent the coefficients of the 3 × 3 symmetric mooring system stiffness matrix ([
27]). As a consequence of this change, the natural periods also change.
Table 5 presents the comparison between the natural periods estimated from the decay tests and those predicted by the numerical model. Mass/inertia and hydrostatic stiffness matrices were computed using Edtools
® with added mass matrix obtained from WAMIT
®.
The natural periods numerically obtained present a fair agreement with experimental data for surge and heave motions, but the same cannot be said regarding the pitch motion. For surge motion, the WAMIT
® model with the adjusted mooring stiffness (from 62 kN/m to 56 kN/m) predicts the physical model response quite well. In
Figure 13, it is notable how the mean offset and attitude of the system change the stiffness coefficients, with the offset, being the principal reason for period changing. In turn, changes in the mooring stiffness or phase shifts of the rotor loads do not seem to explain the reduction observed for the pitch period and further investigation on this issue is needed.
Finally,
Table 6 brings the values of the linear and quadratic damping coefficients derived from the free decay measurements together with the values of the linearized damping coefficients. The linear and linearized dampings are expressed as a percentage of the critical damping for each dof, while the quadratic damping coefficients are presented as a percentage of the mass/inertia of the respective dof. The estimation of the damping levels followed the procedures presented in [
31]. The thrust force notably increases the surge and pitch damping ratios. This effect is even more pronounced in the latter. Regarding the quadratic terms, the damping coefficient
(i.e., proportional to the square of the velocity) is the most affected by the increase of wind velocity. For the linearized damping model, the result indicates that the linearized damping ratio also grows with the wind velocity for surge and pitch, but remains practically unchanged for heave motion. It is important to emphasize that the results present similar trends concerning the ones presented by [
19].
3.2. Responses in Waves
For assessing the concomitant effects of wind and waves, Response Amplitude Operator (RAOs) were derived from the experimental white-noise type wave test.
Figure 14 depicts the heave and pitch RAOs, for all the three wind conditions tested: no wind,
m/s and
m/s. Only the cases without TA are shown at this first moment.
The pitch RAO is observed to substantially change with the wind thrust, similarly to what could be inferred from decay tests. It is also remarkable that the pitch response for the conditions with wind in
Figure 14b present an additional peak at T ≈ 16 s (i.e., near natural period in heave) when compared to the no wind case, pointing to a relevant coupling effect between heave and pitch. Furthermore, a significant increase in the resonant response can be readily identified.
The reasons for the heave-pitch coupling were first experimentally investigated. For this, the two trim conditions presented in
Table 4 are considered: with and without TA.
Figure 15 brings pitch RAO for wind velocities
m/s and
m/s. Mean trim angles for both velocities are shown in the legend.
By comparing the results, it is possible to realize that the effect induced by ballast changing is remarkable, especially in
Figure 15a, as the adjusted trim for
m/s leads to a pitch response that is almost the same observed for the no wind condition. From
Figure 15b, it is possible to conclude that the heave-pitch coupling increases significantly with the mean trim angles.
In order to better understand the behavior of the wave test, a progressive approach for the corrections of the numerical model will be presented next. Firstly, the platform tilt is investigated. Then, hydro and aerodynamic related damping coefficients are proposed. The following paragraphs present the results of this sequence of changes when taking into account the worst wind case in terms of trim, that is, m/s.
The effect of the trim angle was investigated by means of the construction of two numerical meshes: one in even keel condition and another with the proper trim
.
Figure 16 brings the comparison between the numerical predictions with both meshes and the experimental heave and pitch RAOs. Marked curves and full lines represent experimental and numerical results, respectively. Both, no wind (in black) and
m/s (in red) are presented. Note the that the only effect of the wind on the numerical model up to this point is the heeled mesh. It is also important to mention that the proper mooring system stiffness (i.e., taking into account the mean offset and attitude of the platform) was considered for the WAMIT
® model.
From
Figure 16b, the mean potential effects arising from the hull tilt can be concluded to indeed be the main responsible for the appearance of the additional peak near the heave natural frequency. In practice, this effect can be mitigated by using an active control of the floater ballast, or at least in part if a static mean tilt is adopted as a design measure in view of the prevalent wind directions. However, if higher tilt angles of the FOWT are tolerated, these results indicate that additional caution should be taken in the numerical modelling to avoid inaccuracies in the motion predictions.
Still from
Figure 16b, it is remarkable that simply modeling the structure with proper trim is not sufficient to recover the motion amplitudes for periods higher than 18 s. In fact, one may readily notice that the canceling frequency and peaks from the numerical model are not yet in good agreement with the experiments. Thus, a second step for the numerical modeling was to introduce a proper adjustment of external damping coefficients, both from aero and hydrodynamic sources, based on the test data and the wind turbine response curve. Special care was taken for the off-diagonal terms, responsible for coupling effects. Surge and pitch motions are well known to be often strongly coupled for FOWT structures, but the experiments indicate that external heave-pitch damping coefficients are also relevant. As mentioned before, this coupling is more significant the larger the trim angle is. Then, both surge-pitch and heave-pitch damping coefficients seem to be necessary to reproduce the cancellation point and peaks observed in the experiment, as illustrated by
Figure 17.
The main hypothesis for the heave-pitch external damping term is that it is due to hydrodynamic viscous drag, which is not computed by a potential flow tool such as WAMIT
®. By modeling the structure with Morison’s elements and linearizing the quadratic drag force with the statistical linearization approach discussed in [
32], a preliminary quantification of this damping coefficient was made, apparently with good results. However, this procedure still requires a thorough validation and thus these results will not be used here. Finally, following the formulation presented in
Section 2.6, a proper calibration of the aerodynamic damping coefficients was made. For that, the computed value of
kN/(m/s) was considered based on the rotor dimensions and the mean wind speed of 11.6 m/s; Then, with the hub height
m, the damping coupling terms for surge-pitch were defined as
kN s and
kNm s. In turn, the heave-pitch coupling was empirically defined as
kN s.
Figure 18 brings the final calibration of the numerical pitch RAOs, now taking into account these external damping coefficients. Even though some discrepancies still persist for wave periods above 20 s, the predicted amplitudes are now much closer than the ones obtained with the initial version of the numerical model whose calibration was made exclusively based on no wind conditions.
In summary, the analysis above indicates that the region close to wave periods of 15 s is mostly affected by the floater tilt angle, whereas for periods above 20 s up to the pitch natural period the aerodynamic rotor coupling is more important. The response in this region also suffers a strong influence of the quadratic viscous damping terms and, therefore, different adjustments would be needed for different wave heights.