This paper presents a numerical investigation on the interaction between focused waves and wave energy converter (WEC) models using a hybrid solver, qaleFOAM, which couples a twophase incompressible Navier–Stokes (NS) solver OpenFOAM/InterDyMFoam with the quasi Lagrangian–Eulerian finite element method (QALEFEM) based on the fully nonlinear potential theory (FNPT) using the domaindecomposition approach. In the qaleFOAM, the NS solver deals with a small region near the structures (NS domain), where the viscous effect may be significant; the QALEFEM covers the remaining computational domain (FNPT domain); an overlap (transitional) zone is applied between two domains. The WEC models, mooring system and the wave conditions are given by the Collaborative Computational Project in Wave–Structure Interaction (CCPWSI) Blind Test Series 2. In the numerical simulation, the incident wave is generated in the FNPT domain using a selfcorrection wavemaker and propagates into the NS domain through the coupling boundaries and attached transitional zones. An improved passive wave absorber is imposed at the outlet of the NS domain for wave absorption. The practical performance of the qaleFOAM is demonstrated by comparing its prediction with the experimental data, including the wave elevation, motion responses (surge, heave and pitch) and mooring load.
Reliable prediction of structural responses in waves plays an essential role in the design, deployment and operation of the offshore and marine structures, such as the wave energy converters (WECs). For survivability of the structure, additional attention needs to be paid to its behaviour under realistic extreme wave conditions. Such extreme wave conditions are often represented in physical and numerical wave tanks by a focused wave group – for example, the new wave theory (Tromans et al., 1991). Consequently, modelling the wave–structure interaction (WSI) in focused waves has attracted interest from both academia and industry.
To model WSIs, numerous numerical models and software have been developed based on a wide range of theoretical models, including the fully nonlinear potential theory (FNPT), where the fluid is assumed to be incompressible, irrotational and inviscid, and the single or multiphase Navier–Stokes (NS) models with or without turbulence modelling. The performances of these models rely on the effectiveness of generating incident waves in the far field, modelling the wave propagation, simulating structural responses and resolving smallscale turbulence/viscous effects in the near field. For the nonbreaking extreme waves, it is widely accepted that the FNPT model can satisfactorily reproduce the wave conditions and model their propagation in a large computational domain (e.g. EngsigKarup et al., 2016; Grilli et al., 2001; Ma and Yan, 2006; Ma et al., 2001, 2015; Ning et al., 2008, 2009; Stansby, 2013; Wang et al., 2018). For simulating structural responses, the FNPT model can also deliver a promising accuracy if the structure is relatively large compared with the wavelength (Bai and Eatock Taylor, 2006; Celebi et al., 1998; Hu et al., 2020; Kashiwagi, 2000; Ma and Yan, 2009; Tanizawa and Minami, 2001; Wu and Hu, 2004; Yan and Ma, 2007), due to insignificant viscous effects involved in such problems. This was further confirmed by the final report of the first Collaborative Computational Project in wave structure interaction (CCPWSI) blind test held in ISOPE 2018 (Ransley et al., 2019), where cases with a fixed FPSO (Floating Production Storage and Offloading) subjected to extreme wave conditions were numerically simulated using various numerical models. The blind test minimises the possibility of numerical calibrations or tuning of the numerical models due to the fact that the experimental data is released after the numerical predictions are submitted, and, therefore, its results largely reflect the reliabilities of the numerical models in daily practices when the experimental data are not available. Ransley et al. (2019) concluded that FNPT methods have performed equally similar to the highfidelity methods; the quasi LagrangianEulerian finite element method (QALEFEM, Ma and Yan, 2006, 2009; Ma et al., 2015; Yan and Ma, 2007), is at least 1.5 orders of magnitude faster than the quickest NS code and has comparable predictive capability in these cases (Ransley et al., 2019), where the viscous and the turbulent effects are insignificant (Yan et al., 2019b).
However, if the size of the structure is relatively small compared with the characteristic wavelength – for example, within the range of the application of the Morison's equation (usually the size of the structure is smaller than 0.2 characteristic wavelength), the viscous effects become important. The viscous effects may also be significant when the motion of the structure is significant (e.g. Hu et al., 2020; Yan and Ma, 2007) and/or the fluid is sloshing in a confined zone (e.g. Yan et al., 2019a). For such problems, the NS models may be necessary and the potential theory is not suitable, unless an appropriate artificial viscosity is applied (e.g. Yan and Ma, 2007). The artificial viscosity is often numerically calibrated using available experimental data or reliable highfidelity numerical results. This obviously causes inconvenience and uncertainty into the numerical practices when the experimental data is often not available. Compared with the FNPT models, the NS model is more time consuming, as evidenced by Ransley et al. (2019), not only due to its higher degree of complexity of the governing equations, but also due to the fact that it requires a much finer temporal–spatial resolutions to achieve convergent results. For these reasons, the NS models are rarely applied to modelling WSIs in large spatial–temporal domains. In many applications (e.g. Hildebrandt and Sriram, 2014; Hu et al., 2014, 2017), the computational domain of the NS model is confined to a limited space near the structure (near field). This implies that one needs to accurately specify the wave field at the wave generation boundaries of the computational domain. A few tools (e.g. Hu et al., 2014; Jacobsen et al., 2011) are available for imposing the wave conditions using different wave theories – for example, the linear wave theory, secondorder wave theory, Stokes wave theory, stream functions and highorder potential theories (e.g. OceanWave3D, EngsigKarup et al., 2008). Recently, developments of hybrid models, combining the NS solver with another simplified theory, have attracted interest from researchers across the world to deal with WSIs. These hybrid models take the advantages of the simplified theories in fast modelling the largescale wave propagations and the advantages of the NS models in resolving smallscale viscous/turbulent effects, vortex shedding and flow separation, fluid compressibility and aeration. By applying the NS model to a small temporal/spatial zone – for example, near the structure or where/when breaking wave occurs, and the simplified model to the rest of the domain, they are expected to achieve highly efficient solutions without comprising the overall computational accuracy. Typical approaches including the functionsplitting – for example, velocitydecomposition (Edmund et al., 2013), spacesplitting/domaindecomposition (e.g. Colicchio et al., 2006; Fourtakas et al., 2018; Hildebrandt and Sriram, 2014; Li et al., 2018; Sriram et al., 2014; Yan and Ma, 2010a; Zhang et al., 2020) and timesplitting approaches (e.g. Wang et al., 2018) have been attempted. Systematic reviews of the development of the hybrid models can be found in Sriram et al. (2014), Li et al. (2018), Wang et al. (2018) and Zhang et al. (2020). The effectiveness of the hybrid model in improving the computational robustness has been reported by recent CCPWSI blind tests for modelling the interaction between the focused wave and the floating bodies (Ransley et al., 2020a, 2020b). It was concluded that the hybrid methods combining the FNPT with NS solvers, including the qaleFOAM combining QALEFEM with openFOAM (Li et al., 2018; Yan et al., 2019a, 2019b, 2020) and a oneway hybrid model combining the FNPT with smoothed particle hydrodynamics (SPH) (Zhang et al., 2020), demonstrate a potential improvement in the required CPU effort when compared with the most robust NS solvers involved in the test, including the one adopting the linear and secondorder wave condition in the OpenFOAM (wave2Foam, Jacobsen et al., 2011). We agree that the implementations of different numerical models – for example, the computational domain and mesh sizes, are considerably influenced by users’ experiences, since no specific domain/mesh is provided for standardisation in the abovementioned blind test. Nevertheless, the comparison by Ransley et al. (2020b) may demonstrate a better practical performance of the hybrid model for WSI problems compared with both the potential theory and NS solvers.
This paper contributes to the CCPWSI Blind Test Series 2, where the cases with two simplified WEC models subjected to focusing waves with different wave conditions are set. The details of the case configurations can be found in Ransley et al. (2020a). The sizes WEC model in this test are considerably smaller than the characteristic wavelength, implying that the associated viscous effect may be significant. Furthermore, one WEC model is a cylinder with a moonpool at its centre, where the liquid sloshing is expected to bring additional viscous damping for suppressing the waveinduced motions of the WEC model. Following Yan et al. (2020), the qaleFOAM with an improved passive wave absorber is applied to model the cases considered in the blind test. The numerical results of the motions of the WECs have been obtained before the experimental data were released. This paper mainly focuses on the comparison with the experimental data to demonstrate the practical performance of the qaleFOAM. For this purpose, all results presented in this paper are originally submitted ones but additional quantitative analysis is added.
For all cases considered by the CCPWSI blind test, the experiment was performed in the wave basin at the University of Plymouth that features 35 m long, 15.5 m wide and 3 m depth. Flap wave paddles are installed to generate threedimensional waves. The temporal variation of surface elevations at various locations is recorded by 13 wave gauges (WGs) with a sampling frequency of 128 Hz. The diagrams of the geometry of the wave basin and the distribution of the gauges can be found in Ransley et al. (2020a). Three wave conditions are used and are summarised in Table 1. Two models of pointabsorber WECs with a specific mooring system are initially placed at where WG5 is located. The geometries of these models are illustrated by Ransley et al. (2020a). The mass (m), moments of inertias (I_{xx}, I_{yy} and I_{zz}) at the centre of the mass (CoM) are summarised in Table 2, where Z_{C0M} denotes the vertical distance from the CoM to the bottom of the models. For both models, the mooring line is a linear spring with a stiffness of 67 N/m and a rest length of 2.224 m.

Case ID  A_{n}: m  f_{p}: Hz  h: m  H_{s}: m  kA 

1BT2  0.25  0.3578  3.0  0.274  0.128778 
2BT2  0.25  0.4  3.0  0.274  0.160972 
3BT2  0.25  0.4382  3.0  0.274  0.193167 

Model  m: kg  Z_{C0M}: m  I_{xx}: kg m^{2}  I_{yy}: kg m^{2}  I_{zz}: kg m^{2} 

1  43.674  0.191  1.620  1.620  1.143 
2  61.459  0.152  3.560  3.560  3.298 
For all wave conditions, the corresponding emptytank simulations are carried out to examine whether the target waves are generated properly. The wave is generated using the selfcorrection wavemaker in the left end of Ω_{FNPT} aiming to reproduce the same time history of the wave elevations recorded at WG5. In the empty tank test, Ω_{FNPT} starts from the wavemaker and the length of Ω_{FNPT} is 50 m that is longer than the physical wave tank. As indicated above, this is to minimise the error caused by the reflection from the right end of Ω_{FNPT}, where a selfadaptive wave absorber is imposed that produces approximately 95% absorption efficiency. Ω_{NS} starts at x = 11.55 m, between WG1 and WG2. Generally speaking, the length of Ω_{NS} should be sufficient to accommodate the transitional zone, whose thickness is 1.5 m in the front side and 0.5 m near the size boundaries of Ω_{NS}, according to the preliminary investigations. To investigate the absorption efficiency of the improved passive wave absorber applied at the right end of Ω_{NS}, Ω_{NS} ends at x = 17.55 m, where WG8 is placed. Using such a configuration, the gauge data at WG8 can be used as a reference to qualify the absorption efficiency. The height and width of Ω_{NS} are 6 and 3 m, respectively. For all cases, the laminar model is employed when specifying the turbulence properties.
The comparisons of the wave elevations at different WGs between the results of qaleFOAM and the experimental data are shown in Figures 2–4, where the results of qaleFOAM with two different mesh resolutions are plotted. As observed, two sets of the results of qaleFOAM are almost identical to each other, demonstrating a satisfactory convergence of the qaleFOAM in the emptytank test. More importantly, the results of qaleFOAM agree well with the corresponding experimental data. This conforms a satisfactory reproduction of the target waves at WG5 by the selfcorrection wavemaker technique, even though the tank geometry and the wavemaker used in the qaleFOAM are different from the experiment.
Although the agreements between the results of qaleFOAM and the experimental data at WG5 have proven that the present passive wave absorber applied at Γ_{O} can effectively prevent the wave reflected at Γ_{O} from influencing the wave condition at WG5 during the required duration of the simulation (the blind test requires the submission of the time history ranges from 35.3 to 50.3 s), a further analysis has been carried out to quantitatively evaluate the absorption efficiency. As stated by Yan et al. (2016), the theoretical approach based on the linear regular wave theory may not be applicable to highly nonlinear focusing waves considered in this paper, the absorption efficiency is estimated through the relative difference between the numerical results adopting the absorber and the reference data that do not include reflection wave – for example, the corresponding results obtained using a longer tank. One may agree that the wave elevation in Ω_{FNPT} by the QALEFEM can be regarded as the reference data, since Ω_{FNPT} is sufficiently long and the reflection wave from the end of Ω_{FNPT} does not reach WG8 at t = 50.3 s. Figures 5 and 6 compare the wave elevations recorded at WG5 and WG8, respectively. As observed from Figure 5, the results of qaleFOAM are very close to the corresponding QALEFEM results. The relative differences between them during t = 35.3 to 50.3 s are all within 2% for three cases (yielding an absorption efficiency of 98%). Nevertheless, at WG8 (Figure 6), the QALEFEM results agree with the experimental data, whereas the qaleFOAM with the wave absorber results in a slightly different results from others due to the reflection from Γ_{O}. The relative difference between the QALEFEM results and the results of qaleFOAM are 2, 4 and 6% (yielding absorption efficiencies of 98, 96 and 94%) for cases 1BT2, 2BT2 and 3BT2, respectively. This is consistent with what Yan et al. (2020) concluded.
The results shown in Figures 2–6 are obtained in a wave tank without WEC models. For the cases with WECs, mesh convergent tests are also carried out. For each WEC model, four sets of computational mesh are generated using the snappyHexMesh tool and adopted in the convergent test. The horizontal (d_{sh}) and vertical grid sizes (d_{sv}), the total number of grids, N_{t} and the number of grids on the structure surface, N_{s}, are summarised in Table 3. In order to capture the nonlinear WSI and also smallscale viscous effects – for example, boundary layer separation, near the structure, the mesh near a confined zone surrounding the WEC model with a radius of 0.5 m is refined. One example of the mesh near the WEC is illustrated in Figure 7.

Model  Mesh  d_{sh}: m  d_{sv}: m  N_{t}: M  N_{s} 

1  Finest  0.04  0.015  1.550  10 348 
1  Fine  0.05  0.0175  0.956  7512 
1  Medium  0.06  0.02  0.613  5346 
1  Coarse  0.08  0.02  0.358  3882 
2  Finest  0.04  0.015  1.549  20 128 
2  Fine  0.05  0.0175  0.937  13 920 
2  Medium  0.06  0.02  0.612  9840 
2  Coarse  0.08  0.02  0.376  7244 
Figures 8 and 9 compare the motions of and the mooring force of the WEC models 1 and 2, respectively, subjected to the wave condition 3BT2 that has the highest wave steepness (kA = 0.193127) within all wave conditions defined by the blind test. It is observed that the present results are insensitive to the mesh resolutions for all mesh; especially the results with medium mesh satisfactorily agree with the corresponding results with finer mesh. Relative errors of the results of qaleFOAM with different mesh sizes are quantitatively analysed. Results are summarised in Table 4, where the relative errors of the numerical results with medium mesh in terms of both the peak value (E_{p}) and the root mean square (RMS) error using the time histories during t = 35.3 to t = 50.3 s are listed. Similar to Brown et al. (2020), the results of finest mesh are regarded as the reference values for the analysis. Considering the fact that the maximum relative error shown in Table 4 is the RMS error of 7.5% for model 1 subjected to wave 2BT2, one may agree that the medium mesh is sufficient to achieve convergent predictions in the WEC motions and the mooring force, although a similar numerical uncertainty analysis by Brown et al. (2020) is not presented.

Error: %  Model 1 1BT2  Model 1 2BT2  Model 1 3BT2  Model 2 1BT2  Model 2 2BT2  Model 3 3BT2 

E_{p} (surge)  0.17  2.53  1.69  0.49  0.84  1.83 
E_{p} (heave)  0.81  1.05  0.69  0.30  0.22  0.18 
E_{p} (pitch)  0.27  0.39  0.71  0.44  1.12  2.48 
E_{p} (force)  0.32  0.44  0.27  0.15  0.08  0.10 
RMS (surge)  5.22  7.47  5.36  1.60  1.12  1.75 
RMS (heave)  1.06  1.90  1.91  0.79  0.97  0.49 
RMS (pitch)  2.08  4.97  4.11  3.53  3.38  3.60 
RMS (force)  0.18  0.28  0.26  0.14  0.15  0.16 
By using the medium mesh, the motions of the WECs subjected to three wave conditions are numerically simulated and analysed in this section. For demonstration, Figure 10 illustrates the free surface profiles near the WEC models at three instants around the focusing time – that is, t = 45, t = 45.5 and t = 46 s, for the cases with wave condition 2BT2. As expected, the presence of the WECs does not seem to disturb the surrounding wave field, conforming to the typical feature of slender bodies (the sizes of the WECs considerably smaller than the characteristic wavelength).
More importantly, the comparisons between the results of qaleFOAM and the corresponding experimental data shown in Figures 11 and 12 largely reflect the practical performance of the qaleFOAM in modelling the motions of the WECs in extreme waves. For three motion modes and the mooring loads, the qaleFOAM seems to satisfactorily capture the peak values. The corresponding errors are summarised in Table 5. Except for the pitch motion, the relative errors in surge, heave and mooring load are all below 2%. However, the relative error in peak pitch angle is 16.4 and 8.51% for models 1 and 2 subjected to wave 2BT2. Not only the peak pitch angle, the spectra shown in Figures 12(e) and 12(f) and the corresponding time histories shown in Figure 11 have also revealed an unsatisfactory prediction by the qaleFOAM. Similar phenomena are observed in other cases with different wave conditions. The corresponding motion responses and the mooring loads are shown in Figures 13 and 14 and the quantitative errors of the peak values are summarised in Table 5. In fact, numerical results by other numerical methods in Ransley et al. (2020a, 2020b) behave similarly in terms of predicting pitch motion. A recent analysis of sensitivity by Windt et al. (2020) has shown that the pitch motion is sensitive to the centre of rotation and the moment of inertia. Ransley et al. (2020a, 2020b) did not provide the freedecay test for pitch motion and, therefore, it is difficult to quantify whether the error in pitch motion is due to incorrect measure of the centre of rotation and the moment of inertia.

Error: %  Model 1 1BT2  Model 1 2BT2  Model 1 3BT2  Model 2 1BT2  Model 2 2BT2  Model 3 3BT2 

E_{p} (surge)  6.94  1.36  7.52  6.20  1.14  7.39 
E_{p} (heave)  9.77  0.54  3.53  3.41  1.38  1.31 
E_{p} (pitch)  44.0  16.4  19.9  6.38  8.51  6.75 
E_{p} (force)  4.72  1.59  0.86  3.45  0.87  2.00 
In this paper, the qaleFOAM is used to numerically simulate the cases defined by the CCPWSI Blind Test 2 (Ransley et al., 2020a). All wave conditions summarised in Table 1 have been considered. The effectiveness of the qaleFOAM in modelling focused wave group is assessed by comparing the wave elevations in the empty tank tests, where the WEC models are not placed. The results confirm a promising accuracy of the qaleFOAM in modelling highly nonlinear water waves. In addition, the convergence test has demonstrated a good convergence property in terms of predicting the motions of the WECs and the associated mooring forces. The comparisons of the motion responses of the WECs between the present numerical results and the experimental data demonstrate a satisfactory accuracy of the qaleFOAM for modelling the highly nonlinear WSI problems addressed in this paper.
It is further noted that the CPU time spent in cases 1BT3, 2BT3 and 3BT3 to achieve convergent results during t = 35.3 and t = 50.3 s are, respectively 12 h using an 8processor MPI parallel computing in a workstation with Intel Xeon E52680, 2.4 GHz, 32 GB RAM. This demonstrates a satisfactory robustness of the present qaleFOAM.
Acknowledgements
The authors gratefully acknowledge financial support from EPSRC projects (EP/M022382, EP/N006569 and EP/N008863) and UKIERIDST project (DSTUKIERI2016170029).
d  water depth 
instantaneous wave number  
normal direction of the absorber surface  
p  pressure 
horizontal velocity component  
vertical velocity component  
fluid velocity  
w  weighting function ranging from 0 to 1 
free surface elevation  
recorded wave elevation at the wave absorber  
density of the water  
velocity potential  
instantaneous wave frequency 