Experimental setup

The experimental campaign was performed on a three-dimensional small physical model of a floating offshore wind turbine operating in deep-water conditions, with a water depth equal to 6 m, and subjected to incident regular and irregular wave loading. The tested configuration reproduces, at a Froude scale of 1:40, the OC3-Hywind (Jonkman, Reference Jonkman2010) spar-buoy concept. The floating platform consists of two main cylindrical sections connected through a conical transition element. The upper cylinder has a height of 0.4 m and an outer diameter of 162.5 mm, whereas the lower cylinder is 2.6 m high with an outer diameter of 235 mm. The two portions are connected by a conical section with a length of 0.2 m. At the base of the structure, a removable ballast element with a height of 0.1 m is installed and filled with lead in order to achieve a total draft of 3 m and position the CoM of the floating body 2.25 m below the still water level (SWL).

The floating support structure is equipped with the reference US National Renewable Energy Laboratory (NREL) 5 MW wind turbine (Jonkman et al., Reference Jonkman, Butterfield, Musial and Scott2009), which is widely employed in both experimental campaigns and numerical investigations involving floating offshore wind systems. The adoption of this reference turbine ensures consistency with previous studies and provides a reliable benchmark for aero-hydro-servo-elastic analyses. The mooring system consists of three mooring lines symmetrically distributed around the platform with an angular spacing of $ 120{}^{\circ} $ , keeping the incident wave direction perpendicular to the turbine rotor plane. With respect to the wave propagation direction ( $ x $ -axes, Figure 1), the Mooring Line 1 is aligned with the incoming wave direction, whereas the Mooring Lines 2 and 3 are inclined by $ 60{}^{\circ} $ . Each mooring line is equipped with a dedicated load cell in order to measure the tensions transmitted to the floating platform during the experimental campaign.

Figure 1.

Side and plan views of the investigated platform, showing the three CoM configurations and the mooring-line arrangement.

Figure 1 Long description

Figure 1 shows side and plan views of the investigated spar-buoy platform. Three centre-of-mass configurations are indicated along the vertical axis. The plan view shows three mooring lines arranged symmetrically at 120 degrees around the platform, with line 1 aligned with the wave direction and lines 2 and 3 inclined symmetrically.

The experimental measurements provide a detailed characterisation of the dynamic behaviour of the system under wave action. In particular, the platform response was analysed in terms of translational motions along the three principal directions ( $ x $ , $ y $ , $ z $ , Figure 1), namely surge, sway and heave, respectively, together with the rotational motions around the corresponding axes, namely roll, pitch and yaw. These measurements constitute the reference dataset employed to validate the numerical model and to assess its capability to reproduce the 6DOF response of the floating platform.

Numerical framework

The numerical framework is based on the solution of the Reynolds-Averaged Navier–Stokes equations, coupled with a Volume of Fluid (VOF) formulation to describe the evolution of the free surface between two immiscible phases, namely water and air. This modelling strategy allows the main features of the wave-structure interaction problem to be captured, including the propagation of the incident wave field, the motion of the air-water interface and the hydrodynamic loads acting on the floating body. The governing equations consist of the mass conservation equation and the momentum conservation equation, expressed as Equations (1) and (2):

(1) $$ \nabla \cdot \mathbf{u}=0, $$

(2) $$ \frac{\partial \rho \mathbf{u}}{\partial t}+\nabla \cdot \left(\rho \mathbf{u}\mathbf{u}\right)=-\nabla p+\nabla \cdot \left({\mu}_{\mathrm{eff}}\nabla \mathbf{u}\right)+\rho \mathbf{g}+{\mathbf{f}}_{\sigma }, $$

where $ \rho $ is the fluid density, $ \mathbf{u} $ is the velocity vector, $ p $ is the pressure, $ {\mu}_{\mathrm{eff}} $ is the effective viscosity, $ \mathbf{g} $ is the gravitational acceleration, $ t $ is the time and $ {\mathbf{f}}_{\sigma } $ represents the surface tension contribution. In OpenFOAM, this term is evaluated from the local curvature of the air–water interface, together with the prescribed surface tension coefficient. For the characteristic length scales considered in the present study, surface tension effects are expected to be small compared with gravitational effects and, therefore, to have a negligible influence on the global platform response, although the term is retained in the governing equations. The air–water interface is tracked through the VOF with the transport equation of the volume fraction, Equation (3):

(3) $$ \frac{\partial \alpha }{\partial t}+\nabla \cdot \left(\alpha \mathbf{u}\right)+\nabla \cdot \left(\alpha \left(1-\alpha \right){\mathbf{u}}_r\right)=0, $$

where $ \alpha $ denotes the water volume fraction and $ {\mathbf{u}}_r $ is an artificial compression velocity, active only in the interface region, which counteracts numerical smearing. Values of $ \alpha =1 $ and $ \alpha =0 $ identify cells fully occupied by water and air, respectively, while intermediate values define the free surface region.

The governing equations are discretised with the finite volume method on a collocated, unstructured grid. Time integration is performed with a first-order implicit Euler scheme, with an adjustable time step limited by the maximum Courant number, set equal to 0.5. The adopted threshold lies below the stability limit of the explicit MULES algorithm and is consistent with the recommendations of Larsen et al. (Reference Larsen, Fuhrman and Roenby2019) for wave propagation in interFoam. Convective terms in the momentum equation are discretised with a second-order linear-upwind scheme with a cell-limited gradient, which prevents spurious velocity overshoots in the most distorted cells. The turbulence equations were solved with a first-order upwind scheme, the volume fraction with the compressive van Leer scheme and diffusive terms with limited non-orthogonal correction. Pressure–velocity coupling is solved with the PIMPLE algorithm, using four outer correctors per time step with the grid updated at each iteration and an under-relaxation factor of 0.5 on the rigid-body acceleration, to prevent added-mass instabilities.

Turbulence is modelled with the $ k $ - $ \omega $ SST closure, combined with the formal stabilisation proposed by Larsen and Fuhrman (Reference Larsen and Fuhrman2018). Standard two-equation closures are unconditionally unstable in the nearly potential flow region beneath non-breaking waves, where the turbulent kinetic energy and the eddy viscosity grow exponentially, resulting in an excessive damping of the propagating waves (Devolder et al., Reference Devolder, Rauwoens and Troch2017; Larsen and Fuhrman, Reference Larsen and Fuhrman2018). They showed that, for high-steepness regular waves with a wave steepness of ~0.07, comparable to the upper range considered in the present study, the standard $ k $ - $ \omega $ SST model can lead to a substantial reduction in wave height along the numerical flume, between 40% and 50%, while the modified formulation largely preserves the incident wave amplitude. The stabilised formulation is therefore particularly relevant for the present simulations, in which the incident waves propagate over two wavelengths before reaching the platform, since excessive numerical dissipation during propagation could modify the wave height and consequently affect the hydrodynamic loading acting on the structure.

In the first stage, the motion of the floating platform is handled using a mesh morphing strategy, following the approach adopted in previous OpenFOAM-based studies on floating-body dynamics (Chen and Hall, Reference Chen and Hall2022; Palm and Eskilsson, Reference Palm and Eskilsson2022). In this method, the computational grid is not regenerated at each time step, but is, instead, continuously deformed according to the rigid-body motion of the structure. The 6DOF displacements of the platform are used as inputs for the mesh deformation procedure, while the magnitude of the mesh motion gradually decreases with the distance from the floating body. This allows the cells located close to the structure to follow its motion, while the outer regions of the domain remain almost unaffected. The mesh deformation of the model is performed using a spherical linear interpolation procedure, which provides a smooth transition of the displacement field and helps to preserve the quality of the computational grid during the simulation (Shoemake, Reference Shoemake1985). In this approach, the mesh motion is controlled according to the distance from the moving body. Mesh points located within a prescribed inner distance from the platform follow the full rigid-body motion, whereas points located beyond an outer distance remain fixed. In the intermediate region, the imposed translation and rotation are progressively reduced using a smooth distance-based weighting function. This approach preserves the mesh topology without requiring remeshing and limits local cell distortion as long as the body motion remains moderate. However, as the platform motion increased, the mesh deformation became progressively more severe, and under the highest wave condition, excessive cell distortion eventually led to numerical instability, motivating the use of an alternative strategy for the most demanding wave condition. For the extreme sea state, where the platform motions exceed the deformation that the mesh morphing algorithm can accommodate without excessive grid distortion, the body motion is instead handled using the overset mesh method, natively available in OpenFOAM^®. In this approach, the computational domain is composed of two independently generated, overlapping meshes: a background mesh, spanning the full computational domain and fixed in space, and an inner mesh, containing the floating platform, which moves rigidly with the body. At each time step, the flow field is interpolated between the two meshes in the overlap region using a cell-volume-weighted scheme, which reconstructs each interpolated cell value from a volume-weighted average of the donor cells in the other mesh. Unlike mesh morphing, this method avoids cell deformation during the simulation, thereby preserving the quality of the individual meshes provided that an adequate overlap region is maintained.

The computational domain is defined so as to reproduce the main geometrical characteristics of the reduced-scale experimental facility. For the mesh morphing simulations, it consists of a total length of 23 m in the $ x $ direction ( $ -8 $ m to $ 15 $ m), 8 m in the $ y $ direction and 14 m in the $ z $ direction ( $ -7 $ m to $ 7 $ m), with the floating structure positioned close to the domain centre. A reference mesh is initially defined on the basis of the characteristic dimensions of the computational domain, the geometry of the floating structure and the spatial resolution required to describe the wave field. The background mesh is initially defined based on common meshing practices adopted in previous OpenFOAM studies of wave-floating-body interaction (Windt et al., Reference Windt, Davidson, Chandar, Faedo and Ringwood2020; Chen and Hall, Reference Chen and Hall2022). It is generated using characteristic cell dimensions of $ 0.25 $ m, $ 0.50 $ m and, within a graded vertical distribution, $ 0.10 $ m near the free surface, coarsening towards the top of the domain. In the region surrounding the floating structure, the grid is progressively refined using the native OpenFOAM utility snappyHexMesh, so that the cell dimensions progressively decrease towards the body. An additional refinement region is introduced around the free surface, extending vertically from $ z=-1.6 $ m to $ z=0.35 $ m, in order to provide an adequate representation of the wave kinematics and reduce numerical dissipation during wave propagation. The adequacy of this reference discretisation is assessed through the grid-convergence analysis described in Section “Numerical validation,” and the resulting mesh consists of ~0.98 million cells.

Following the parametric strategy adopted by Khan et al. (Reference Khan, Barajas, Gaeta, López Lara and Archetti2024), different inertial configurations of the floating system are investigated by varying the vertical position of the CoM. In particular, three configurations are analysed, in which the overall CoM of the turbine-platform system is placed at depths of 1.92, 1.82 and 2.02 m below the SWL, corresponding respectively to the dimensionless depth $ {z}_{\mathrm{CoM}}/{z}_{\mathrm{floater}} $ of 0.640, 0.607 and 0.673, as displayed in Figure 1. The first configuration corresponds to the reference case, while the other two are introduced to assess the sensitivity of the platform response to changes in mass distribution. The resulting platform motions and mooring-line tensions are then compared in order to evaluate the influence of the CoM position on the global dynamic response of the system and on the loads transmitted to the mooring system.

The dynamic response of the floating structure is analysed under regular, unidirectional wave conditions, reproducing the experimental sea states reported by Russo et al. (Reference Russo, Contestabile, Bardazzi, Leone, Iglesias, Tomasicchio and Vicinanza2021). Two wave tests are considered: WT1 with $ H=0.13 $ m and WT2 with $ H=0.25 $ m. For all tests, the wave period was kept constant at $ T=1.6 $ s and the simulated physical time was equal to 60 s.

In addition to the displacement analysis, the tensions induced in the three mooring lines are also assessed. For this purpose, the lumped-mass-based MoorDyn mooring model is adopted (Hall and Goupee, Reference Hall and Goupee2015). Each mooring line is discretised into a series of elastic segments connected by lumped-mass nodes. The model accounts for the axial stiffness and internal damping of the lines, submerged weight, hydrodynamic drag and added-mass effects, as well as seabed interaction when applicable. The motion of each node is governed by the following relation:

(4) $$ {\mathbf{M}}_i{\ddot{\mathbf{r}}}_i=\sum {\mathbf{F}}_i, $$

where $ {\mathbf{M}}_i $ is the effective mass matrix of the $ i $ -th node, including the added-mass contribution, $ {\overset{"}{\mathbf{r}}}_i $ is the node acceleration and $ \sum {\mathbf{F}}_i $ represents the total applied force, including line tension, internal damping, weight and buoyancy, hydrodynamic drag and seabed interaction. During the coupled simulations, the motion of the platform fairlead points is transferred to MoorDyn, while the resulting mooring forces return to the six-degree-of-freedom solver.

The LEONARDO supercomputer hosted by CINECA was employed for the simulations. An adjustable time-step strategy was adopted, with the maximum Courant number set to $ {Co}_{\mathrm{max}}=0.5 $ , selected as a compromise between numerical stability, temporal resolution and computational cost. The computational cost varies depending on the simulated wave conditions and the resulting platform motions; however, the dominant factor affecting the computational time is the adopted mesh-handling strategy. The mesh morphing simulations required ~5 h per test, using 192 MPI processes, while the overset simulations required ~18 h per test, reflecting the additional cost of the overset interpolation at each time step.

Numerical validation

The numerical model is validated by comparing the numerical results with the reference experimental data (Russo et al., Reference Russo, Contestabile, Bardazzi, Leone, Iglesias, Tomasicchio and Vicinanza2021). The comparison focuses primarily on the platform response in the six degrees of freedom in order to verify that the numerical model correctly reproduces the main physical mechanisms and dynamic behaviour observed in the experiments.

Prior to validation, the numerical model is subjected to a preliminary verification to assess its numerical performance and limit artificial numerical phenomena. This verification includes a grid-convergence study and an empty-tank wave-propagation test, focusing respectively on mesh sensitivity and wave-propagation accuracy.

A grid-sensitivity study is performed starting from the reference mesh described in Section “Numerical framework.” Three systematically refined grids containing ~0.44, 0.98 and 2.31 million cells are therefore considered, referred to as the coarse, intermediate and fine grids, respectively. A refinement ratio of ~ $ r=1.3 $ is adopted between consecutive grids in each spatial direction. The convergence analysis is performed considering the platform surge and pitch responses, together with the upstream mooring-line tension. The results obtained with the coarse and intermediate grids are compared with those provided by the fine grid. The relative difference is evaluated as

(5) $$ {\delta}_{\phi }=\left|\frac{\phi_i-{\phi}_f}{\phi_f}\right|\times 100, $$

where $ {\phi}_i $ represents the quantity obtained using either the coarse or intermediate grid, while $ {\phi}_f $ is the corresponding fine-grid value. Considering the peak values of the analysed quantities, the intermediate grid shows a considerably closer agreement with the fine-grid solution than the coarse grid for the platform motions. For surge, the relative difference with respect to the fine grid decreases from ~ $ 46.1\% $ for the coarse grid to $ 6.6\% $ for the intermediate grid. Similarly, for pitch, the difference decreased from ~ $ 13.6\% $ to $ 1.6\% $ . The upstream mooring-line tension exhibits a lower sensitivity to grid refinement, with relative differences of ~ $ 4.6\% $ and $ 4.2\% $ for the coarse and intermediate grids, respectively. Most of the improvement is obtained already between the coarse and the intermediate grid, whereas further refinement yields comparatively diminishing returns. Although the fine grid remains the most accurate discretisation in principle, its use for the full simulation campaign would only be justified if it provided a comparable accuracy gain. The computational times associated with the coarse, intermediate and fine grids are ~2.6, 5.0 and 12.4 h, respectively. Given that the intermediate grid already agrees with the fine-grid solution within a few percent for all three quantities, and that its accuracy is independently confirmed against experimental measurements, the intermediate grid is retained for all subsequent simulations as the more efficient choice, with ~0.98 million cells for the mesh morphing simulations. Figure 2 shows the resulting background meshes for the mesh morphing (a) and overset (b) approaches, together with a detail of the refined region surrounding the platform, common to both meshing techniques (c).

Figure 2.

Computational meshes adopted in the present study: (a) mesh morphing background mesh, lateral view; (b) overset background mesh, lateral view; (c) detail of the mesh in the vicinity of the platform, representative of both meshing techniques.

Figure 2 Long description

Figure 2 compares the computational meshes adopted for the two mesh-motion strategies. Panel (a) shows the mesh-morphing background mesh in lateral view. Panel (b) shows the overset background mesh in lateral view. Panel (c) shows the locally refined mesh around the platform, representative of both approaches, with increased resolution near the body and free surface.

Second, an empty-tank wave-propagation test, with no structure present, is carried out to verify the accurate propagation of the incident wave field and to quantify artificial numerical damping, which could otherwise influence the predicted six-degree-of-freedom motions of the structure. Regular waves (H = 0.13 m, T = 1.6 s) are considered within the scope. In such tests, the free surface resolution, as detailed in Section “Numerical framework,” is assessed, namely corresponding to ~5 cells per wave height in the vertical direction and 64 cells per wavelength in the horizontal direction. Although a vertical resolution of ~10 cells per wave height is commonly adopted in numerical wave tank simulations (e.g., Jacobsen et al., Reference Jacobsen, van, Capel and Borsboom2018; Chen et al., Reference Chen, Warmink, van Gent and Hulscher2021; Imbò et al., Reference Imbò, Scaravaglione, Saponieri, Lauria and Barajas2025), the selected resolution is retained to limit the computational cost, and its adequacy was assessed through the dedicated wave propagation test. With the free-surface refinement region extending from 1.6 m below to 0.35 m above the free surface, as adopted in the final mesh, the total discrepancy between the generated wave height, measured 7 m from the generation boundary, and the propagated wave height, measured two wavelengths further downstream at 15 m, was below $ 5\% $ , confirming that the selected resolution is adequate to reproduce wave propagation with limited numerical attenuation.

Building on the verified spatial discretisation and wave propagation accuracy described above, the numerical model is finally validated against the free-decay and regular-wave measurements reported by Russo et al. (Reference Russo, Contestabile, Bardazzi, Leone, Iglesias, Tomasicchio and Vicinanza2021). The natural frequencies from the free-decay tests are obtained from the oscillation period of the structure, while the response under regular waves is characterised using the standard deviation (STD) evaluated over the periodic steady-state window. The difference between the numerical model and the experimental measurements is quantitatively evaluated through the relative percentage error (RPE), $ ({\phi}_{\mathrm{CFD}}-{\phi}_{\mathrm{exp}})/{\phi}_{\mathrm{exp}}\times 100 $ . As shown in Table 1, the numerical model predicts natural frequencies of 0.196 Hz in heave and 0.190 Hz in pitch, compared with experimental values of 0.210 Hz and 0.189 Hz, respectively, corresponding to a RPE of $ 6.5\% $ and $ 0.5\% $ , confrming that the model correctly reproduces the restoring stiffness and inertial properties of the platform. Under regular waves, the predicted surge response is in very close agreement with the experimental STD ( $ -2.0\% $ ), while the pitch response is underestimated by $ 16.3\% $ and the mooring line tension is overestimated by $ 10.7\% $ . These discrepancies are consistent with the small absolute magnitude of the measured quantities (STD of $ 0.40{}^{\circ} $ and $ 0.196 $ N, respectively), for which even minor differences translate into larger relative errors, and remain within the typical uncertainty of physical model tests at this scale. Overall, the validation confirms that the numerical model reproduces both the free-decay dynamics and the regular-wave response of the platform with an adequate accuracy for the parametric analysis presented in the following sections.

Table 1.

Validation of the numerical model against the experimental campaign of Russo et al. (Reference Russo, Contestabile, Bardazzi, Leone, Iglesias, Tomasicchio and Vicinanza2021)

Table 1. Long description

Table 1 compares experimental and CFD results. In free decay, the heave natural frequency is 0.210 Hz experimentally and 0.196 Hz numerically, with an RPE of −6.5%; pitch is 0.189 and 0.190 Hz, with an RPE of +0.5%. For regular waves with H = 0.13 m and T = 1.6 s, surge STD is 0.020 m experimentally and 0.0196 m numerically, with an RPE of −2.0%; pitch STD is 0.40 degrees and 0.335 degrees, with an RPE of −16.3%; mooring-line 1 tension STD is 0.196 N and 0.217 N, with an RPE of +10.7%.