Validation of a potential flow-based numerical model for engineering design of Floating Offshore Wind Turbine foundations

Designers apply numerical models for scrutinizing the complex response of Floating Offshore Wind Turbine (FOWT) foundations to environmental conditions at sea. This paper presents a numerical model for predicting the wave-induced motions of FOWT foundations. The model combines two solvers, one for generating the forcing wave field and the other for computing the foundation response based on Potential Flow Theory. With its robustness and relatively low computational demand, the proposed model aims to assist the initial exploratory stages of the FOWT design process. A validation of the proposed model against a laboratory experimental campaign is presented. The performance of the model is investigated, with highlighting the role of second-order hydrodynamic effects such as viscous damping and slow-drift.


Introduction
The Floating Offshore Wind Turbine (FOWT) industry has been growing considerably in the last decade.Several FOWTs are expected in the coming years (e.g.Hywind Tampen, 94.6 MW), projecting the total installed capacity to approximately 260 GW by 2050 [1].Nevertheless, FOWT-associated costs are still high, due to immaturity of technology and supply chain; as an example, Floating Wind has currently a Levelized Cost Of Energy (LCOE) four times higher than bottom-fixed [1].Amongst all sources of expenses, the foundation is one of the elements with the highest potential for massive cost reductions [1].Beside advancing on e.g.standardization and manufacturing, de-risking the foundation design process will thus be key to drive down the costs of the whole FOWT technology.
Predicting the complex response of the foundation to waves, wind, and currents is an essential step of the design process.For this scope, designers rely on numerical models, typically in combination with physical model tests.Therefore, the development of robust, reliable, and accurate models will help designers better tackle the response analyses and their uncertainties, hence reduce conservatism that is one of the reasons for the high foundation-related costs.
This study focuses on a numerical model for predicting the hydrodynamic response of FOWT foundations to waves.The model combines two solvers, one for propagating the incident (forcing) wave field and the other for computing the induced motions of the foundation.The response solver is based on the (linear) Potential Flow Theory, which ensures low computational demand and robustness.These characteristics are convenient in the first stages of the design process, when numerous simulations are performed for scrutinizing the foundation response in different loading scenarios.Nevertheless, potential flow-based models offer mid-fidelity capabilities [2], because of their inherent limitations with respect to nonlinear wave forcing and viscous drag.However, these limitations are commonly accepted in the preliminary and comparative design analyses.A list of mid-fidelity tools commonly used for FOWT response modelling is given in [2].

The numerical model
The proposed model combines the wave propagation solver MIKE 3 Wave Model FM (MIKE3-WFM) and the potential flow solver MIKE 21 Mooring Analysis (MIKE21-MA).MIKE3-WFM is developed by DHI for wave transformation studies in coastal and offshore environments.MIKE21-MA is developed by DHI for moored vessel response studies primarily, but the engine handles the hydrodynamic response of any floating object.The scientific background of each solver is briefly described in the next subsections, with more emphasis on MIKE21-MA.

The wave propagation solver MIKE3-WFM
MIKE3-WFM solves the three-dimensional incompressible Reynolds-averaged Navier-Stokes equations, after splitting the total pressure into a non-hydrostatic and a hydrostatic component and operating a vertical sigma-coordinate transformation.A height function is applied for the advection of the free surface.The spatial discretization of the governing equations is performed in conserved form using a cell-centered Finite Volume method.The interface convective fluxes are reconstructed using the approximate Riemann solver in [3], which enables robust and stable simulations of e.g.breaking waves.The vertical convective and diffusive terms are discretized in time using an implicit scheme, while a second-order explicit Runge-Kutta scheme is applied for the other terms of the governing equations.The non-hydrostatic pressure is treated with the fractional step approach developed by [4].More information on the solution algorithm is given in [5]; a collection of validation cases is found in [6].
2.2.The potential flow-based solver MIKE21-MA MIKE21-MA solves the equations of motions for the 6 Degree-Of-Freedom (DOF) of the floating rigid body.The governing equations are written in time-domain according to the impulse response function approach of Cummins [7], that is Equation 1 relates the motions of the objects (x k ) with the inertia forces (M jk and a jk ), the damping due to radiated waves (K jk ), the hydrostatic restoring forces (C jk ), and the external excitation forces, either linear due to incident waves (F j,D ) or nonlinear (F j,nl ) due to moorings, winds, currents, viscous damping, and second-order drift forces.In order to reduce the computational demand required by the direct solution of Eq. 1, MIKE21-MA applies the hybrid frequency-time domain method [8].Under the assumptions of this approach, the governing equations can be written in frequency-domain as A numerical solution of Eq. 2 is obtained with an inbuilt solver (MIKE21-Frequency Response Calculator), which applies the Boundary Element Method on the body surface discretized with quadrangular panels.The radiation-diffraction analysis returns A jk (ω) and B jk (ω).The coefficients M jk and C jk are also derived at this stage from mass and geometry input.The terms a jk (t) and K jk (t), used in Eq. 1, are calculated through the relations [8] A In particular, an inverse Fourier Transform on Eq. 3b yields With the K jk coefficients known from Eq. 4, a jk is calculated from Eq. 3a with ω → ∞.
The nonlinear external forces, i.e.F j,nl in Eq. 1, are calculated individually.In particular, second-order wave drift forces are computed for surge, sway, and yaw using the Newman approximation [9].Mooring line forces for catenary systems are obtained with a quasi-static approach.Viscous damping forces (F j,visc ) are calculated as where the constant (B 0 ), linear (B 1 ), quadratic (B 2 ), and cubic (B 3 ) coefficients are provided by the user via 6x6 matrices.

Coupling between MIKE3-WFM and MIKE21-MA
The coupling between the two solvers works such that the incident wave field is first reproduced with MIKE3-WFM (without including the floating structure); then, the floater response is computed with MIKE21-MA using the modelled waves as a forcing.The coupling lies thus on the term F j,D (t) in Eq. 1.Following the mentioned hybrid frequency-time approach and the Haskind relations [10], this term is first calculated in frequency-domain as where Φ j is the radiation potential, p I is the first order dynamic pressure, and Φ In is the incident wave potential (the subscript n denotes the operation x • ∇).The radiation potential Φ j is an output of the radiation-diffraction analysis.The terms p I and Φ In are derived from surface elevation and wave kinematics modelled by MIKE3-WFM.The procedure explained in [11] is applied for such derivation.With knowing F j,D (ω) in frequency-domain, an inverse Fourier Transform gives F j,D (t) in time-domain.

Experimental campaign
The validation study was carried out against a physical model test campaign on the FOWT foundation TetraSpar, developed by Stiesdal Offshore.The campaign was conducted in the deep-water basin of DHI in 2017, by Stiesdal Offshore, Technical University of Denmark (DTU), and DHI.The basin was 30 m wide and 20 m long (Fig. 1).Still water level was at 3 m from the bottom.Waves were generated with sixty wave paddles and absorbed with a perforated parabolic plate (Fig. 2).Eleven wave gauges were installed in the basin; gauge no. 9 (WG9), positioned next to the FOWT, was used in this study as a reference.A wind generator, located behind the wave maker, provided the wind forcing for the tests.The 1:60 FOWT was placed at 5 m from the wave maker.A 10 MW turbine (prototype), developed by DTU, was mounted on TetraSpar (Fig. 3).The structure comprised a floater and a counterweight (Fig. 4), connected through six chains.A Qualisys MotionTrack system and 6DOF-accelerometers were installed to measure the motions.A catenary mooring system was used; three chains, spread with angles of 120°, anchored the side tanks to the bottom of the basin (Fig. 2).The test program comprised moored decay, wave-only, and combined wind-wave tests.Regular and irregular realizations of five sea states were generated.Table 1 lists the experimental tests presented in this validation study.Cases with combined waves and wind were not included.It is noted that the decay and the wave-only tests were conducted with the turbine mounted on the foundation, but idled. .Model validation -wave forcing modelling with MIKE3-WFM The wave fields of all wave-only cases listed in Table 1 were modelled.The same irregular sea state realizations tested experimentally were reproduced, with a duration of 3 hours in full-scale.After an initial sensitivity analysis, the same setup was applied for all cases and only the input wave conditions were changed.
The computational domain encompassed the entire laboratory basin.Relaxation zones for wave generation and absorption were included, with a width of 5 m and 6 m respectively (Fig. 5, left).The FOWT, not included in the simulation, was ideally placed at (X,Y) = (0,0), while WG9 was at (X,Y) = (0,2).The mesh consisted of quadrilateral elements.As long-crested waves were simulated, the horizontal grid was one-element wide along the spanwise direction.The cell size along the streamwise direction was 0.02 m until and past the floater position, and it progressively increased to 0.50 m towards the end of the domain (Fig. 5, right).The vertical discretisation was achieved with 18 layers with a variable thickness that gradually decreased towards the free surface (Fig. 5, right).Figure 6 shows the results for all simulated cases in comparison with the experimental measurements.Power density spectra and exceedance probability distributions were obtained from the measured and modelled surface elevation at WG9.In general, a good agreement is recognized.However, the wave energy was underestimated around the peak frequency in SS03 and SS05, whereas it was overestimated in SS64 and SS11.It is noted that the energy content of SS11 was slightly underestimated at the higher frequencies, indicating that the adopted mesh did not fully capture the shorter waves for that sea state.The exceedance plots show that the extreme wave events were generally well captured; some larger discrepancies are seen for SS64 and SS11.

Model validation -motion response simulations with MIKE21-MA
The response analyses were carried out under the assumption that the whole floating system, comprising of counterweight, floater, tower, and idled turbine, responded as a rigid body in the experimental campaign.This assumption was supported by the measured tensions of the chains connecting the floater to the counterweight, which did not show the chains to be slack under the waves.The whole floating system will be thus referred to as "the floater" in the remainder of this paper.

Precursor Radiation-Diffraction analysis in frequency-domain
The response simulations in time-domain were preceded by the frequency-based radiationdiffraction analysis on the discretized floater in Fig. 7.It is reiterated that no external forcing (waves, mooring) was applied at this stage.As a symmetry plan existed, the geometry was built for half of the floater, in order to reduce the computational demand.The floater surface was discretized with 7,342 quadrangular panels with an average size of 0.018 m.A quality check of the results was done at this stage on the (displacement) Response Amplitude Operators (RAOs) of the floater.As an example, Fig. 8 displays the RAOs for 0°-incident wave direction (waves from astern).The RAOs followed the expected trend generally.The amplitudes approached 0 for very short waves; the heave amplitude approached 1 for very long waves.At frequencies near 0, the surge phase was close to +90°, the heave phase to 0°, and the pitch phase to -90°.

Damping calibration in time-domain
The modelled damping of the floater was calibrated to include missing second-order viscous effects (pressure drag), which were conjectured at the heave plates as well as past the (slender) cylindrical structural elements.The tuning parameters were the B 1 and B 2 coefficients in Eq. 2.
2. An initial estimate of those coefficients was found with the method of Faltinsen [12] applied on the available experimental decay tests.The coefficient values were then refined with iterative trial and error simulations of the decay motions in time-domain, until the comparison with measurements did not show further improvement.Figure 9 shows the results of the modelled surge, have, and pitch decay tests.Results of the other DOFs are omitted for the sake of brevity.Response in time domain and damping  .The crucial role of the damping calibration is clearly recognized.The agreement with the experimental data was poor without calibration, whereas it improved substantially when the calibrated damping coefficients were applied.Good results were achieved overall.Surge test showed larger discrepancies; this was attributed to the adopted quasi-static approach for the catenary mooring modelling.Figure 9: Experimental and modelled surge, heave, and pitch motions.Numerical results are without and with damping calibration.Damping ratio is not shown for surge due to the short duration of the experimental time series

Wave-induced response simulations in time-domain
Response simulations were performed with applying the forcings in Section 4. The mooring system was modelled with a quasi-static catenary approach.Computed motions were postprocessed as power density spectra and exceedance probability distributions (based on peak-topeak values).
Results for SS11 are presented in Fig. 10.Only surge, heave, and pitch are depicted; sway, roll, and yaw were not significant as waves were aligned with the symmetry plan of the floater.Looking at the exceedance distributions, it is seen that heave was well predicted, while both surge and pitch were slightly underestimated at the tail.The spectral analysis disclosed that, similarly for each DOF, the model captured mainly the response to first-order wave loading, as the majority of the modelled energy was located at around the peak frequency of the incident wave spectrum (vertical black dashed line).In fact, the energy was overestimated at these frequencies for heave and pitch.This is partially motivated by the overpredicted incoming wave energy for SS11 (Fig. 6).Another reason is that the model did not fully reproduce the lowfrequency (sub-harmonic) components of heave and pitch, which mainly originated from second-(and higher-) order wave forcing effects (slow-drift).This circumstance was expected, because the slow-drift contributions were not accounted for these two DOFs.Instead, the low-frequency surge was obtained, as the Newman approximation was applied for this DOF; nevertheless, the energy content was overestimated.This suggests that the Newman approximation was not optimal in this case; however, it is believed that the adopted quasi-static approach and the neglected hydrodynamic effects in the catenary modelling played an important role as well.The effects of the second-order forces were more evident for the other sea states.As an example, it can be seen in SS06 (Fig. 11) that the low-frequency response was more prominent than the first-order wave response (vertical black dashed line).This led to larger underpredictions for heave and pitch, whereas surge was substantially overestimated.
In order to assess the performance of the model within its primary validity range, i.e. response to first-order wave loading, numerical and experimental results were further compared with filtering out the mentioned sub-harmonic motions.In particular, components at frequencies lower than 0.28 Hz were excluded.This threshold was slightly larger than the highest natural floater frequency (Fig. 4) and, moreover, at the lower bound of SS64 frequency range, which had the longest waves.As an example of results based on the filtered time series, Fig. 11 displays the exceedance probability distributions marked with star symbols.With applying the filter, it is clear how better the model agreed with the experimental data, indicating the important role of the missing second order-wave loads in the model predictions.where R σ is the ratio of the standard deviation (σ) of the numerical and experimental time series respectively, and R 1% is the ratio of the numerical and experimental motion (m) corresponding to 1%-probability of occurrence respectively.The closer the value of these two metrics to one, the better the agreement between numerical and experimental results.
Figure 12 shows an overview of R σ and R 1% for all simulated sea states.In sea states 05, 06, 64, and 11, the larger discrepancies were ± 25% approximately; results for SS03, i.e., the sea state with the smallest waves, suffered from large underprediction of surge and pitch, even after filtering out the slow-drift.This was attributed to inaccuracies in the wave kinematics modelling for this sea state.

Conclusions and Future Work
This study presented a validation of a numerical model for predicting the response of FOWT foundation to waves.The model combined a solver for wave propagation (DHI's MIKE3-WFM) and a potential flow-based solver (DHI's MIKE21-MA) for computing the 6DOF-response of the foundation.The validation study was based on water surface elevation and 6DOF-response measurements from a physical model test campaign conducted on the FOWT foundation TetraSpar, developed by Stiesdal Offshore.First, the experimental wave field was modelled with MIKE3-WFM, without the foundation.Comparison of power spectra and wave exceedance probability distributions showed the good results of the wave modelling in all different simulated  The motions due to second-order wave loads were not fully captured instead, which generally led to underestimation of the motions, for surge especially.Future work will concern a deeper investigation of the second-order hydrodynamics, including a more advanced catenary mooring modelling.Furthermore, the wind forcing will be included, which will allow to validate and apply the proposed model in combined wind-wave design scenarios.

Figure 1 :Figure 2 :
Figure 1: Top-view of the test setup

Figure 3 :Figure 4 :
Figure 3: TetraSpar foundation by Stiesdal Offshore with the 10MW turbine by DTU during the experimental campaign at DHI

Figure 5 :
Figure 5: Computational domain used for the wave modelling with MIKE3-WFM.Left: horizontal extent including wave generation/absorption zones.Right: vertical layers with variable thickness and transition between fine and coarser grid

Figure 7 : 1 Freq
Figure 7: Discretised floater used for the response modelling with MIKE21-MA

Figure 10 :
Figure 10: Experimental and modelled floater motions for SS11 (test T0178).Top: power density spectra; the vertical dashed line marks the wave spectrum peak frequency of SS11.Bottom: exceedance probability distributions of peak-to-peak motions

Figure 11 :
Figure 11: Experimental and modelled floater motions for SS06 (test T0185).Top: power density spectra; the vertical dashed line marks the wave spectrum peak frequency of SS06.Bottom: exceedance probability distributions for raw (•) and filtered ( * ) peak-to-peak motions

Figure 12 :
Figure 12: Overview of model performance in the present validation study.Two measures, R σ and R 1% , are used to quantify the discrepancies with the experimental measurements

Table 1 :
Experimental tests used for the present validation study