Experimental and numerical investigation of nonlinear diffraction wave loads on a semi-submersible wind turbine

In a severe sea state, nonlinear wave loads can excite resonant responses of floating wind turbines either at high (structural) or low (rigid body motions) natural frequencies. In the present work, a computational fluid dynamics (CFD) model and an engineering model based on potential-flow theory with Morisontype drag are developed to investigate nonlinear wave loads on a stationary, rigid semi-submersible wind turbine under regular and irregular waves. The numerical results are validated against experimental measurements. A trimmed floater is modelled to examine the change in nonlinear wave loads due to the mean pitch angle which occurs during operation of a floating wind turbine. Furthermore, wave loads on each column are investigated numerically. Compared to the experimental measurements, the CFD model gives better estimations than the engineering model for the first, second and third order wave diffraction loads. The engineering model based on the firstand second-order potential-flow theory has large discrepancies in the phase of high order wave diffraction loads and underpredicts the amplitude of low-frequency wave loads. In the CFD simulations for the studied wave period (12.1 s), the second and third harmonic surge forces on the starboard columns are significantly larger than those on the upstream column, while first harmonic results are consistent with potential flow. The trim angle (5 ) results in an increasing surge force and pitch moment but a decreasing heave force. © 2021 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Introduction
Recently, there has been a huge increase in the use of wind turbines for generating electricity. To access a larger wind resource and reduce visual and acoustic pollution, a growing number of offshore wind turbines (OWTs) has been installed in recent years. In shallow and intermediate water depths, bottom-fixed OWTs are employed. However, in deep water, the costs of bottom-fixed foundations rise sharply, so a wide variety of floating wind turbine (FWT) concepts have been proposed, such as spar, semisubmersible and tension leg platforms (TLP). With increasing water depth, FWTs may be exposed to harsh environments and steep waves which induce highly nonlinear wave loads on the floater of FWTs. The high-frequency loads can cause springing and ringing, while the low-frequency loads can lead to the resonance in surge, sway and yaw of a moored platform. Mercier et al. [1] showed the importance of high order wave loads on TLPs through experiments. Coulling et al. [2] also used experiments and numerical tools to stress the importance of second-order difference-frequency wave forces in capturing the global response of a semi-submersible FWT. As the responses of FWTs are largely affected by nonlinear wave loads, validated modelling tools should be developed to predict these loads more accurately while keeping the computational efficiency at a reasonable level.
Most of the investigations of nonlinear hydrodynamic loads on different types of wind turbines are based on low order potential flow models [3e7]. This limits hydrodynamic modeling to linear or weakly nonlinear models and may be less reliable for analysis of FWTs in extreme condition. Some studies use high order commercial or in-house CFD codes to investigate wave loads and motion responses on various types of FWTs, as summarized in Table 1.
Although the offshore oil and gas industry has demonstrated the long-term survivability of semi-submersible platforms and can provide some guidelines for design, significant differences exist. With a smaller payload and no permanent residences for personnel, FWTs are significantly smaller than oil and gas platforms [19]. Semi-submersible FWTs are usually made of three or four vertical columns connected by cross braces which affect the flow regimes around columns. In addition, heave plates are often attached to the bottom of the columns to increase the added mass and damping. An additional lift force on heave plates due to vortex shedding may also appear [10]. Another difference between FWTs and oil and gas platforms is a mean pitch angle for the wind platform during operation due to the aerodynamic thrust force.
To better understand the nonlinear diffraction wave loads on a semi-submersible FWT, a CFD model and a potential flow theory model are used to simulate the interaction of a semi-submersible FWT with regular and irregular waves. The numerical results are compared with experimental data. Higher order sum-frequency wave loads are examined for a regular wave condition; an irregular wave state is chosen to investigate the wave loads around natural periods of rigid body motions. Because a fixed platform is considered here, we cannot directly compare responses around low frequencies. However, some qualitative conclusions are also drawn from the results.
A trimmed condition under regular wave is also modelled to examine the change of nonlinear wave loads due to the trim induced by the mean aerodynamic thrust force. In addition, wave loads on each column are extracted to better understand the effects of the multimember arrangement of the semisubmersible. Section 2 briefly describes the experiment for comparison, while Section 3 describes the CFD and potential flow theory models. In Section 4, the wave loads on the upright and trimmed floater under regular wave condition are analyzed. Then, the wave loads due to irregular waves are examined. Finally, conclusions are presented in Section 5.

Review of experiment
In the present work, experimental results for the wave loads on the floater are from Phase I of the Offshore Code Comparison Collaboration, Continued, with Correlation, and unCertainty (OC6) project [20]. The model test was performed in the concept basin of the Maritime Research Institute Netherland (MARIN). The geometry of the floater corresponds to the DeepCwind semi-submersible floater [21] at 1:50 scale. The right-handed coordinate system used in this research originates at the center of the main column at the still water line, with positive x being in the direction of propagating waves, and z being up. The tower of the wind turbine was removed and the floater was attached to the carriage through a frame, as shown in Fig. 1. Wave loads on the stationary model were measured using a six-component gauge connecting the floater and the frame. The results presented in this paper are calculated based on the shown coordinate system. All data and results are given at full scale, except when explicitly mentioned. The uncertainty in the experiment is about 2% for the measured force and 5% for the measured moments [20].
One regular wave with height H ¼ 7.1 m and period T ¼ 12.1 s, and an irregular wave with significant wave height H s ¼ 7.1 m and peak period T p ¼ 12.1 s were selected for this research. The irregular wave was generated based on the JONSWAP spectrum with the peak enhancement factor equal to 3.3.

Computational model
The same global coordinate system as in the experiment (Fig. 1) is implemented in the computational model.

CFD model
The multiphase interFoam solver of OpenFOAM [22] is a fully nonlinear Navier-Stokes/VOF solver. Extending the interFoam solver with the wave generation and absorption toolbox, wave-s2Foam, developed by Jacobsen et al. [23], generates the waveFoam solver. Furthermore, a fully nonlinear potential flow solver, Oceanwave3D [24], is coupled with the waveFoam solver to minimize numerical diffusion and reduce the computational time for the irregular wave case [25].
where u i ði ¼ x; y; zÞ are the fluid velocity in Cartesian coordinates, r is the fluid density, p * is pressure in excess of the hydrostatic pressure, F b is an external body force including gravity and m eff is the effective dynamic viscosity. For laminar flow, m eff is equal to the laminar dynamic viscosity. For turbulent flow, it should include turbulent dynamic viscosity rn t . The local density r and the effective dynamic viscosity m eff are defined by the volume fraction a which is bounded between 0 (air) and 1 (water). The turbulent dynamic viscosity rn t is neglected in laminar flow model.
The air-water interface is tracked by the volume of fluid VOF method [26] in OpenFoam. The volume fraction is advanced in time once the velocity is known, following scalar advection equation: The smearing of the interface is vastly reduced by the introduction of the artificial velocity u r . It is only active in the vicinity of the interface where 0 < a < 1, see Berberovic et al. [27] for details. To ensure the boundedness of solution, a multi-dimensional flux limited scheme (MULES) is applied. To identify the free surface elevation, the volume fractionais integrated along a vertical line around the air-water interface.

Turbulence modelling
The effects of turbulence are incorporated in the governing equations by using different transport equations to calculate the turbulent kinematic viscosity n t . The k À u SST turbulence model has shown good results for simulating two-phase flow and predicting wave elevation [28,29] and is applied in this paper. The incompressible k À u SST model for a single fluid is a two-equation model and is given in OpenFoam as: where k is the turbulent kinetic energy, P k is the production term of k, n is kinematic viscosity, n t is the turbulent kinematic viscosity, u is the specific dissipation rate, as described in detail by Menter et al. [30]. In the present work, to avoid significant wave damping induced by increasing turbulent viscosity [31,32], a modified waveFoam solver is built to explicitly consider the variable density in the k À u SST model [32]. Table 2 shows the Keulegan-Carpenter (KC) and Reynolds numbers at model scale, assuming the wave is fully linear. The inflow velocity is the maximum horizontal water particle velocity. The KC number is around 1 for most of components. Hence, weak vortices are expected in the experiment and CFD simulations. Although the Reynolds numbers are on the order of 10 4 -10 5 , previous studies considering the same geometry suggest that the turbulence model is not essential for the CFD simulations due to the low KC number of upper column and heave plate [33]. For regular wave cases, simulations using both laminar (L.F) and turbulent (T.F) flow are carried out for comparison.

Numerical domain
To reduce the computational time, CFD simulations with a stationary semi-submersible floater were carried out at model scale (1:50). In this section, all data are given at model scale.
The width (3.72 m) and water depth (3.6 m) of the experimental wave flume are modelled. The numerical wave tank (36 m) is shorter than the experimental wave flume, but long enough to dissipate the reflected waves. Fig. 2 shows good agreement in the overlayed wave elevation at the center of floater for three cycles separated in time: before reflections can reach the center of the floater, after reflections from the outlet are expected, and after reflections from both inlet and outlet are expected. The height of  the air regime is set to 1 m. Fig. 3 shows a plan view of the numerical wave tank with floater.
After spatial discretization, the size of a cell in all directions is 0.12 m. Thereafter, for the air-water interface, the mesh was refined. A mesh convergence study for free surface elevation and calculated wave loads was carried out. In Table 3, the wave height at origin (center of floater) is compared to the specified wave height at inlet (H ¼ 0.142 m) and the wave loads are compared to those with the finest mesh size (Level 4) for three different spatial discretizations. The 3-3 level of refinement, resulting in a mesh size of 0.015 m (zH/10), is applied for the air-water interface and surface of columns. Around the cross braces, which have a smaller diameter, the 4-4 level is applied, leading to cross brace mesh size 0.0075 m (zH/ 20). This local refinement allows for a high-resolution interface while keeping the total number of computational cells relatively low. For the turbulent flow model, 25 cell layers adjacent to the floater surface are generated. The thickness of the first layer is 2.0 mm and its expansion ratio is 1.2.

Boundary conditions
In the Navier-Stokes/VOF solver, boundary conditions are imposed on all surfaces in the numerical domain. The general denomination of boundary surfaces is given in Fig. 4.
The velocity and the volume fraction boundary conditions are given by stream function wave theory at the inlet and a constant current with zero velocity at the outlet. The floater surfaces and tank bottom are modelled as fixed walls. For the laminar flow model, a slip condition is applied, which means the effect of viscosity and turbulence generation are neglected. For the turbulent flow model, wall functions are applied for k (1e-5 m 2 /s 2 ) and u (1.0 1/s). A continuous wall function based on Spadling's law [34] is implemented for the turbulent viscosity. Furthermore, a Dirichlet boundary condition is set for the velocity while a Neumann condition is used for pressure and volume fraction on the fixed walls. A slip condition is applied to the front and back walls. At the atmosphere boundary, the total pressure is set to zero and an atmospheric boundary condition is applied for the velocity and volume fraction. This means that air and water are allowed to leave the numerical domain, while only air is allowed to flow back in.
The Waves2Foam toolbox [23] implements relaxation zones (blue part in Fig. 4) to avoid wave reflection from the outlet boundary (II) and to prevent internally reflected waves (I). Rectangular relaxation zones are defined in this work, as in Bruinsma's work [35].

Temporal discretization
To ensure numerical stability, the temporal discretization is determined based on the Courant-Friedrichs-Lewy condition. Different fixed time steps were chosen to carry out a time step convergence study. In Table 4, the errors are relative to the specified wave height at inlet (0.142 m, model scale) or the numerical results with the smallest time step. To limit computational effort while obtaining sufficient accuracy, a fixed time step (0.001414 s, model scale) is used in the rest of this work.
The Richardson extrapolation with the standard power-law error estimator [36] is used to obtain the wave loads at the limit of infinite temporal and spatial resolution. The resulting apparent order of convergence is 2. The discretization uncertainty is defined as the estimated discretization error multiplied to a suitable safety factor (1.25 following the recommendation from Eça and Hoekstra [37]). The total uncertainty is given by combining the temporal and spatial discretization uncertainties in the root-sum-of-squares fashion, shown in the last column of Table 6.

Potential flow theory model
In addition to the CFD model, a numerical model based on potential flow theory has been built using the software SIMA (SIMO-RIFLEX) [38,39], developed by SINTEF Ocean and widely used in the analyses of numerous types of offshore platforms and wind turbines.
In present simulations, only the stationary semi-submersible floater is modelled. First order potential flow forces, such as the wave excitation force, and second-order sum-frequency and difference-frequency wave force transfer functions (QTFs) are estimated based on a potential flow solution (WAMIT [40]) and subsequently input to SIMA. In addition, viscous effects are considered by applying drag forces from Morison's equation to the columns and cross braces. A constant drag coefficient in the normal direction (0.774, based on towing tests [20]) is applied for each part of the floater. The axial drag force on the heave plate is calculated based on Equation (9). D is the diameter of the heave plate. U is the water particle velocity along the axial direction. Here, we use C da ¼ 2:48 based on previous comparisons of a similar engineering tool with experimental data from the DeepCwind test campaign [41].
In the SIMA simulations, different force contributions are investigated, as summarized in Table 5. For simulations labelled 'SIMA', the experimentally measured wave elevation is used as input, while simulations indicated as 'Linear' use a fully linear regular wave (same period and height as experimental wave). When the experimentally measured wave elevation is used as input, all components are treated as linear, and all components are assumed to travel in the positive x-direction.

Results
In this section, nonlinear wave diffraction loads on a semisubmersible FWT are studied in detail, and all the results are shown at full scale. Considering symmetry, nonlinear wave loads on the starboard column are presented for only one of the trailing columns (starboard, see Fig. 1).

Regular wave
In the examination of regular wave results, wave amplitudes and loads are extracted from five out of 20 steady-state wave cycles. First, second and third harmonic components are obtained by filtering the signals to keep components around 1u, 2u and 3u, where u is the wave frequency. The amplitude is the average value over the five wave cycles and the phase represents the initial phase. The uncertainty in the measured wave elevation is estimated as 0.03 m full scale [42].
Numerically calculated waves at the center of floater are compared against the experimental results in Fig. 6, which shows a good agreement except for a fully linear regular wave (SIMA þ Linear). The small difference between SIMA and

Upright condition
The surge forces on the upright floater are presented in Fig. 7. SIMA5 is not shown because the axial drag force on the heave plate (z direction) does not contribute to the surge force (x direction) on the upright floater. The amplitude and phase for the total and first harmonic surge force are predicted very well (within 2.5%) by all numerical simulations. For higher order surge forces, CFD results show a similar phase as the experiment. The simulations with laminar flow overpredict the second harmonic surge force and underpredict the third harmonic, while CFD results with turbulent flow are closer to the experimental measurements. Compared to the laminar flow, the turbulent flow has a lower pressure drag contribution and a higher friction drag contribution. In SIMA, the viscous effects are considered through the water particle velocity. Because the mass loads, proportional to water particle acceleration, dominate for the considered structure and wave, the phase difference between water particle acceleration and velocity leads to a phase-lead in the SIMA 4 surge force compared to the experiment. Furthermore, SIMA 4 underpredicts the third harmonic surge force, and there is a phase difference between SIMA and experiment for higher order results, which can be explained by decomposing the higher order surge forces. Even though the experimental wave is treated as a linear wave in SIMA, the measurements contain higher order components compared to the fully linear regular wave. A comparison of SIMA 1e4 and the 'Linear' results shows that the nonlinearity in the wave elevation dominates the higher order surge forces. Additionally, the sum-frequency QTF and integration of the drag force to the undisturbed linear free surface contribute to the second harmonic surge force. In the SIMA 2e4 models, the combination of highfrequency wave components and the sum-frequency QTF lead to a third harmonic surge force. The nonlinear drag force also contributes to the third harmonic surge force. The 'Linear' results show good phase agreement compared to the experiment. For the SIMA 1e4 results, the treatment of nonlinear components of the incoming wave leads to phase discrepancy. The applied constant normal drag coefficient in SIMA may also contribute to amplitude differences in the higher order surge forces.
The heave forces on the upright floater are compared in Fig. 8. SIMA 1e4 and CFD with laminar flow underestimate the amplitude by 15%, while CFD with turbulent flow agrees well with the experiment (within 5.8%). Turbulent flow increases the first and third harmonic heave force but reduces the second harmonic. Flow separation around the edge occurs due to the geometric singularity and leads to a larger vertical drag force. However, the RANS model used in this paper may not simulate this effect correctly, and overpredicts the third harmonic heave force by over 50%. Like the surge force, the CFD model has a similar phase as the experiments, while SIMA 1e4 deviates for all harmonics.
When axial drag forces on the heave plates are not included (SIMA 1e4), most of the heave excitation comes from potential flow theory, and the first and third harmonic heave forces are underpredicted, while the second harmonic is overpredicted. Additionally, the normal drag force on the cross braces slightly increases the third harmonic heave force (SIMA 2 vs SIMA 3). From the comparisons between SIMA 1e4 and the 'Linear' results, nonlinear incoming wave also contributes to the higher order heave forces. Furthermore, the phase of the higher order heave force also deviates from the experiments in the 'Linear' results.
The axial drag force (SIMA 5 and Linear 5) increases the first and third harmonic heave force, while the integration of the drag force to the linear free surface also increases the second harmonic. SIMA 5 predicts the phase better than SIMA 1e4, but still differs from the experimental results in the higher order results. This suggests that potential flow theory and Morison-type drag with a tuned axial drag coefficient can predict the amplitude of the heave force very well, without really capturing the physics, as evidenced by the difference in the phase. Fig. 9 compares the numerically estimated pitch moment against the measured result. All numerical models underpredict the first harmonic amplitudes (from 1.6% for SIMA 5e7.8% for CFD with turbulent flow) but have similar phase as the experimental data. The discrepancy in amplitude may be caused by the underprediction of the heave forces. In the CFD simulation, the second harmonic pitch moment performs similarly as the surge and heave forces, but for the third harmonic, the model with turbulent flow underestimates the amplitude by 28%. The SIMA 1e4 simulations underpredict higher order amplitudes and differ in phase from the experimental results. The phase difference also exists when the axial drag force on the heave plate is included (SIMA 5).
For the second harmonic pitch moment, a comparison of the SIMA and Linear 1e4 models shows that nonlinearity in the incoming wave is important (linear potential flow theory dominates). Furthermore, the sum-frequency QTF reduces the amplitude (SIMA 2 vs. SIMA 1). The normal drag force has no obvious effect on the second harmonic pitch moment.
For the third harmonic, the moment induced by the sumfrequency QTF is more important than other contributions for SIMA 1e4. Meanwhile, by comparing SIMA 3e4 and 'Linear' results, the third order contributions from normal drag forces due to highfrequency wave components reduce the amplitude. The axial drag force (SIMA 5 and Linear 5) mainly contributes to the third harmonic pitch moment and has no significant effect on the first or second harmonic. In conclusion, the first and second harmonic pitch moments in SIMA are mainly from potential flow theory and only the axial drag forces on the heave plates have a significant effect on the third harmonic.
The differences between measured wave loads in the experiment and calculated wave loads in the CFD with turbulence model are compared with the estimated uncertainty in Table 6. As shown, the differences for the total wave loads are within the range of sum of uncertainties in experiment and numerical results. Both the differences and uncertainties in higher order components may be larger, however.
The experiments do not measure surge forces on individual columns, but this comparison can be carried out numerically, shown in Fig. 10 and Fig. 11. Here, the sum-frequency QTF is not included in SIMA. Therefore, wave nonlinearity is important for the first and second harmonic surge forces, while the drag force is dominant in third harmonic in these results.
For the first harmonic surge force, SIMA and CFD are in better agreement for the upper column (within 2.4%) than for the heave plate (within 10.2%). In the CFD model, the intersections between heave plates and cross braces are modelled, which reduces the heave plate flank area where the pressures are integrated by 1.33%. The SIMA model considers the heave plates as closed cylinders. As with the total force, due to the constant drag coefficient in SIMA, there is a large difference in the amplitude and phase of higher harmonic surge forces between SIMA and CFD.
For the CFD results, except for the higher order surge force on the upper column of the upstream column (Fx2), the surge forces have similar phase for both laminar and turbulent flow in each harmonic. According to the KC number in Table 2 and results from other researchers [43], a pair of weak vortices is formed in the wake of upper column of upstream column which laminar flow model cannot simulate. In Fig. 10, the flattened peak value of the surge force on the main column (Fx1) is due to the geometry: the top of main column has larger diameter which leads to a ledge (see left part of Fig. 1). This ledge leads to the large difference in the third harmonic surge force of main column between laminar and turbulent flow model. This phenomenon is not modelled in SIMA.
For both the upstream and starboard column, although the diameter of the heave plate is twice that of the upper column, the surge force on the upper column is larger because the water particle acceleration decreases as one moves downward. To better understand these results, the surge forces on each column under fully linear regular waves with different wave periods and the same wave steepness were estimated using the SIMA 4 model, shown in Fig. 12. Based on the previous analysis (Fig. 7), the main difference between the total and first order amplitude (right part of Fig. 12) is due to the third harmonic drag force. The total surge forces on the upper column (Fx2 and Fx4) are also larger than the forces on the heave plate (Fx3 and Fx5), and the discrepancy increases for shorter waves. Comparisons among the higher order surge forces lead to the same conclusions, but the discrepancy depends on frequency.
Comparing the forces on components of the upstream column and starboard column in the CFD simulations (Fig. 11), the amplitude of higher order surge forces on the starboard column is larger than those on the upstream column. The same can be observed in

H. Li and E.E. Bachynski
Renewable Energy 171 (2021) 709e727 the first and third order surge forces in SIMA simulations (Fig. 11) while the opposite is seen in the second order surge force. The results under different wave periods (Fig. 12) show that the surge force comparison between the starboard and upstream column depends on frequency.

Trimmed condition
Considering the existence of a mean pitch angle for the wind turbine during operation, nonlinear wave diffraction loads on a 5 trimmed floater are studied in this section. The floater is rotated 5 clockwise around Point A shown in Fig. 1. No corresponding experimental data are available. Only results from CFD with turbulent flow and the SIMA model are shown in this section. Furthermore, we focus on the integrated wave loads on the complete floater. Surge force (along x axis), heave force (along z axis) and pitch moment (around y axis) are calculated in the same global coordinate system as the upright condition which is shown in Fig. 1. The amplitudes of load responses divided by the xth (x ¼ 1,2,3) power of the displaced submerged volume V (х10 4 m 3 ) are also compared to eliminate the effect of changed submerged volume, which is denoted as 'Amplitude/V x ' in Figs. 13e15.
The surge forces under upright and trimmed conditions are presented in Fig. 13. The first harmonic surge force under trimmed condition increases (4.8% for CFD and 2.4% for SIMA) compared to the upright condition. After rotating the floater, a larger portion of the columns are submerged, but the water particle acceleration at the same position of columns decreases. Hence, considering the changed volume, the first harmonic surge force under trimmed condition decreases (5.2% for CFD and 4.7% for SIMA). The higher order results generally decrease, especially when considering the changed volume.
As shown by 'SIMA5þ5 ' in Fig. 13, the axial drag force increases the first harmonic surge force by 3.2% and decreases the third harmonic by 4.2% (SIMA5 has the same value as SIMA4 under upright condition). For the trimmed floater, the axial drag forces on the heave plates have a component along x direction. The water particle velocity along the column axis is the resultant velocity of the vertical and horizontal water particle velocity, which have a phase difference. Hence, the contribution of this additional drag force depends on the phase.
The heave forces under upright and trimmed position are compared in Fig. 14. All harmonics of the heave force decrease after rotating the floater except for a slightly increased second harmonic CFD result. Considering the changed volume, all harmonics decrease. The increased heave force on the upper column cannot compensate for the loss of heave force on the heave plate. In the SIMA 4 model without axial drag force, this decreasing heave force relates to the change of immersed geometry. By comparing 'SIMA4þ5 ' and 'SIMA5þ5 ', the axial drag force increases the heave force in each harmonic, especially at the third harmonic (around 200%). Furthermore, the axial drag force (SIMA5þ5 ) is mainly calculated based on the vertical water particle velocity, which leads to a phase shift for each harmonic heave force compared to the 'SIMA4þ5 '. Normalizing by submerged volume, there is almost no change in the SIMA5 results due to trim.
The comparisons of numerically estimated pitch moment are shown in Fig. 15. For the CFD simulations, except for the deceasing second harmonic results (15.3%), there is an increasing pitch moment (within 9.3%) on the trimmed floater. All harmonic pitch moments in SIMA 4 model increase (around 7% for the first harmonic, 60% for the second harmonic, 100% for the third harmonic) in the trimmed condition due to the changed results from linear potential flow theory. The axial drag force increases the third harmonic pitch moment (14 times larger) and decreases second harmonic pitch moment (11.2%) by comparing 'SIMA4þ5 ' and 'SIMA5þ5 '. However, the total pitch moment increases by 1%. Similar to the heave force, when dividing by submerged volume, there is almost no change in the SIMA5 results due to trim.

Irregular wave
An irregular wave state (H s ¼ 7.1 m and T p ¼ 12.1 s) is considered to investigate the wave loads near natural periods of rigid body motion. In the CFD model, as the wave maker motion in the experiment is not available, the measured irregular wave at the center of body is used to generate a time series of wave elevation at the wave maker as an input signal at the inlet of potential flow domain, following the procedure shown by Bachynski et al. [44]. The outlet boundary of the potential flow domain is defined by setting a constant current with zero velocity. The spatial discretization in the CFD domain is the same as in the regular wave case. The cell size in the potential flow domain is 10 times larger. The center of the two computational domains is co-located with the center of the floater. Due to excessive numerical damping for wave generation with a turbulence model, the CFD laminar flow model is used. The wave in SIMA is obtained by filtering the measured wave elevation at the origin with a 0.025 Hz cut-off frequency. For the SIMA 2 model, the sum-frequency QTF is replaced with the difference-frequency QTF. In this section, the power spectral density (PSD) of responses is calculated based on the time series from 1200s to 4800s and we only focus on low frequencies. In the wavefrequency region, the agreement is very good among all results, and is not shown.
Experimental and numerical time series of 20 wave cycles extracted from 1-h simulations and the wave power spectra (in log scale) for the full hour are compared in Fig. 16. Although there is a difference in the low-frequency domain, the values are quite small compared to the ones in the wave-frequency domain where good agreement occurs and have no significant effect on the obtained wave energy. Hence, to be fair, CFD can reproduce the wave elevation very well. Based on the difference between wave power spectrum at inlet and center of CFD domain, most of numerical damping in the CFD domain occurs in the low-frequency domain. The effect of the cut-off frequency (0.025 Hz) is visible in the SIMA simulations.
The numerically estimated low-frequency surge force power spectra are compared against experimental results in Fig. 17 (left). SIMA underestimates the low-frequency surge forces, and by 26.6% at surge natural frequency. CFD also underpredicts the surge forces by 16.2% at the surge natural frequency. On the other hand, CFD results have more significant wave energy (within 20%) at low frequencies.  Fig. 17 (right) presents the low-frequency surge force spectra considering different force contributions in SIMA. All of the components of the measured irregular wave are treated linearly in SIMA. The surge natural frequency is below the cut-off frequency, so there is no significant linear wave excitation (SIMA 1) at surge natural frequency. The difference-frequency QTF (SIMA 2) has a slight influence on the low-frequency surge force. The surge force at surge natural frequency is mainly from integration of the Morison drag force to the linear free surface (SIMA 4). The axial drag forces on the heave plates have no effect on the low-frequency surge force (SIMA 5 vs SIMA 4).
The heave force spectra are compared in Fig. 18 (left). SIMA underestimates the low-frequency heave forces, and by 20.5% at the heave natural frequency. The CFD model shows better predictions of heave forces, however, there is also significantly less wave energy (18%) at the heave natural frequency in the CFD model. It is important to note that the laminar flow model used in CFD simulation cannot correctly capture the flow around the heave plate. A large part of the heave force comes from linear potential flow theory (SIMA 1), see right part of Fig. 18. The difference-frequency QTF (SIMA 2) and the normal drag forces (SIMA 3 and 4) have minor effects on the heave force. The axial drag forces on the heave plates (SIMA 5) increase the heave force by 89.5% at the heave natural frequency.

H. Li and E.E. Bachynski
Renewable Energy 171 (2021) 709e727 Fig. 19 (left) compares the low-frequency pitch moment spectra. Similar to the surge and heave forces, SIMA underpredicts the pitch moment. This is consistent with previous results showing five times smaller load responses at the pitch natural frequency of a mooring FWT seen in simulations using the engineering tools compared to the experimental measurements [5]. The CFD model underpredicts the low-frequency pitch moment by about 10%. Again, this underprediction may in fact be more severe than it seems, as the CFD model includes more wave energy at low frequencies.
The comparisons of low-frequency pitch moment spectra with different force contributions in SIMA are shown in Fig. 19 (right). The pitch moment from the difference-frequency QTF (SIMA 2) is significant. In addition, the normal drag forces (SIMA 3) and integrations to the linear free surface (SIMA 4) have little effect on the pitch moment at low frequencies. The axial drag forces on the heave plates (SIMA 5) increase the moment at very low frequencies, but the results remain lower than experimental measurements.

Conclusions
Nonlinear wave loads are important for designing the floater and mooring system for FWTs. In this paper, nonlinear diffraction wave loads on a semi-submersible FWT are studied using two numerical approaches based on CFD and potential flow theory as well as experimental measurements.
Based on the comparisons under a regular wave, good agreement (within 10%) between numerical models and experiment is achieved for wave-frequency loads. For higher order wave loads, CFD correctly estimates the phase, and CFD with the turbulence model generally give the best results. The lower heave force and pitch moment in the laminar flow may be due to the lack of simulation for weak flow separation occurring around heave plate. SIMA (potential flow with Morison drag) has large discrepancies in prediction of phase and amplitude of higher order wave loads. The dominant higher order components in SIMA are related to treating a nonlinear wave measurement as a linear input, which leads to phase discrepancy against CFD and experimental measurements. The normal drag force of columns and cross braces in SIMA model has significant contributions to the surge force while the axial drag forces on the heave plates increase the heave force and the pitch moment. The proper drag coefficient can decrease the amplitude discrepancy in higher order loads, but a phase difference nonetheless exists in the SIMA results. This discrepancy demonstrates the limitation of potential flow theory with Morison-type drag for estimating high order diffraction wave loads.
Regarding the low-frequency wave loads on the fixed structure under an irregular wave, both CFD and SIMA underpredict the loads, but the discrepancy for CFD results is smaller. In SIMA, all of the components of the measured irregular wave are treated linearly. The difference-frequency QTF has significant contributions to the pitch moment. Normal drag forces on the columns and cross braces lead to an increased surge force at surge natural frequency. In addition, the axial drag forces on the heave plates largely increase the heave force at heave and pitch natural frequencies, but only contribute to the pitch moment at very low frequencies. Due to excessive numerical damping of the wave using a turbulence model in CFD, only laminar flow was considered in the irregular waves. Further developments of hybrid methods with CFD with a turbulence model and potential flow theory models are needed.      In order to better understand the effects of the multimember arrangement of the semisubmersible, surge forces on each column are extracted. Higher order surge forces on the starboard column are larger than those on the upstream column in the CFD simulations under this regular wave (12.1 s), but this conclusion is frequency-dependent (based on potential flow simulations at additional frequencies). Furthermore, the total surge force on the upper column is larger than the force on the heave plate, and this discrepancy increases for shorter waves.
The effects of 5 mean pitch angle on the nonlinear wave loads are also investigated in regular waves, using the same global coordinate system as in the upright condition. In the trimmed condition, the surge force and pitch moment increase while the heave force decreases. Furthermore, this 5 mean pitch angle reduces the higher order surge and heave forces while increasing the higher order pitch moment. In SIMA, the axial drag forces on the heave plates increase the third harmonic heave force and pitch moment, but decrease the third harmonic surge force, which is due to the phase difference between vertical and horizontal water particle velocity. Considering the effect of changed volume, the surge and heave forces decrease, and the pitch moment slightly increases under trimmed position.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.