Proceedings of the Institution of Civil Engineers -

Engineering and Computational Mechanics

ISSN 1755-0777 | E-ISSN 1755-0785
Volume 173 Issue 3, September, 2020, pp. 100-118
Themed issue on the CCP-WSI Blind Test Series 2 - part I
 
Open access content Subscribed content Free content Trial content

Full Text

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 two-phase incompressible Navier–Stokes (NS) solver OpenFOAM/InterDyMFoam with the quasi Lagrangian–Eulerian finite element method (QALE-FEM) based on the fully non-linear potential theory (FNPT) using the domain-decomposition 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 QALE-FEM 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 (CCP-WSI) Blind Test Series 2. In the numerical simulation, the incident wave is generated in the FNPT domain using a self-correction 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 non-linear potential theory (FNPT), where the fluid is assumed to be incompressible, irrotational and inviscid, and the single- or multi-phase 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 small-scale turbulence/viscous effects in the near field. For the non-breaking 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. Engsig-Karup 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 (CCP-WSI) 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 high-fidelity methods; the quasi Lagrangian-Eulerian finite element method (QALE-FEM, 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 high-fidelity 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, second-order wave theory, Stokes wave theory, stream functions and high-order potential theories (e.g. OceanWave3D, Engsig-Karup 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 large-scale wave propagations and the advantages of the NS models in resolving small-scale 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 function-splitting – for example, velocity-decomposition (Edmund et al., 2013), space-splitting/domain-decomposition (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 time-splitting 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 CCP-WSI 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 QALE-FEM with openFOAM (Li et al., 2018; Yan et al., 2019a, 2019b, 2020) and a one-way 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 second-order 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 above-mentioned 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 CCP-WSI 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 wave-induced 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.

The hybrid model, qaleFOAM, combines QALE-FEM and OpenFOAM/InterDyMFoam (Jasak, 2009) using the domain decomposition strategy. The details of the qaleFOAM have been given by Li et al. (2018) but a summary is given herein for completeness. Figure 1 illustrates the coupling of the FNPT and NS solvers that are coupled through a coupling boundary, Γc. The FNPT domain (ΩFNPT) starts from an upstream location far away from the structures, where a wavemaker is used to generate the incoming wave. The length of the FNPT domain should be sufficient to cover the inlet of the NS domain (ΩNS). In this paper, one-way coupling is adopted and, therefore, the solution of ΩFNPT is only used to provide an accurate wave condition at Γc. This means that the diffraction and radiation caused by the structures do not need to be reproduced in ΩFNPT and thus the structure is omitted from ΩFNPT. The right end of ΩFNPT is an absorption boundary and the self-adaptive wave absorber (Yan et al., 2016) is employed. The absorption efficiency of the absorber is approximately 95% in terms of wave energy for the case considered in this paper and is at a similar level for a wide range of non-linear regular and irregular waves, as demonstrated by Yan et al. (2016). Similar to all other techniques, perfect absorption is impossible and the reflection exists from the right end of ΩFNPT no matter how small it is. Such reflection can influence the structural responses when it approaches the structure site. To minimise the effect, the length of ΩFNPT needs to be sufficiently long such that the required duration of the results is obtained before the reflection wave reaches the structure site. In ΩFNPT, the QALE-FEM is used to solve the governing equations and its high robustness in modelling non-linear waves up to wave breaking (Yan and Ma, 2010b) assures a good overall robustness of the qaleFOAM, even though a long ΩFNPT may be implemented to ensure a tolerable error caused by the reflection from the end of ΩFNPT during the simulation. ΩNS is bounded by the coupling boundaries Γc at its left end and two sides in the longitude direction (dashed line in Figure 1), seabed ΓB, a pressure inlet/outlet boundary on the top ΓTOP, where the total pressure is taken as the atmospheric pressure, and the right end boundary ΓO. In ΩNS, the multiphase solver interDyMFoam, based on the finite volume method with volume of fluid technique for identifying the fluid phases, is used. In the coupling boundary Γc, the velocity and pressure for the NS solver are fed by the QALE-FEM using the following equation
u(x,y,z)=ϕ(x,y,z)zη(1Rz)ϕ(x,y,η)+Rzuw(x,y,z)z>η
1
p(x,y,z)=ρwϕtρwϕ22ρwgzzη0z>η
2
where
ρw
is the density of water;
ϕ
is the velocity potential;
η
is the free surface elevation;
u
is the velocity vector and p is the pressure. It is noted that the FNPT is a single-phase model describing only the water flow. In Equation 1, the velocity of the flow above the free surface (i.e.
z>η
the air phase) is specified by a weighted summation of the corresponding water velocity on the free surface (
ϕ(x,y,η)
) and the wind velocity,
uw(x,y,z)
, where
Rz
is a ramp function ranging from 0 to 1, to ensure a smooth transition of the fluid velocity from the water phase to the air phase.
Rz=1eβ(zzt)/lz
when the volume fraction
α
at a surface cell on Γc is smaller than 0.01, otherwise,
Rz=0
, where
β
is an exponential coefficient,
lz
is the size of the transition zone and
zt
is the vertical coordinate corresponding to the upper boundary of the surface cell where
α>0.01
. In this paper,
uw(x,y,z)=0
,
β
 = 5 and
lz
equals the vertical cell size near the free surface at Γc, which are appropriate according to the preliminary test. The volume fraction at a surface cell on Γc is defined as the ratio of the wetted surface area against the total area of the cell after the free surface at Γc is determined by
η
. Detailed numerical formulation may be found in Yan and Ma (2010a) and Jacobsen et al. (2011).
figure parent remove

Figure 1. Schematic diagram of the domain decomposition and the coupling approach of the qaleFOAM (ΩFNPT does not include the floating structure)

It is noted that Equation 2 can be used to specify the pressure at Γc of ΩNS, acting as a pressure boundary condition. However, applying both Equations 1 and 2 under velocity and pressure boundary conditions at Γc results in a scenario that the velocity–pressure relation at such boundary follows Bernoull's equation and thus the NS equation is not satisfied, possibly yielding unsmoothed NS solutions near Γc. In the qaleFOAM, two techniques have been employed to overcome the problem. The first one is to use Equation 1 to specify the velocity boundary condition and to impose the fixed flux pressure condition, available in OpenFOAM, as the pressure boundary condition. The second approach is to implement a transitional zone near Γc (Figure 1), similar to the relaxation zone suggested by Jacobsen et al. (2011). In the transitional zone, the NS solution f (velocity and pressure) is corrected by
fQALEw+fNS(1w)
, where subscripts QALE and NS denote QALE-FEM solution and NS solution, respectively; w is the weighting function, which is 1 on Γc and 0 on the other boundary of the transitional zone and the exponential function following Jacobsen et al. (2011) is employed. This does not only ensure a smooth transition of the solutions within the transitional zone, but also absorb the reflection/radiation waves from the structures. The length of the transitional zone is determined based on a preliminary test, which suggests that a length of 1–2 characteristic wavelength is sufficient (Li et al., 2018).
The wave in the qaleFOAM is generated by the QALE-FEM in ΩFNPT using a second-order wavemaker theory (Schaffer, 1996) and propagates towards ΩNS through the coupling boundary Γc. Due to the fact that neither the shape nor the motion of the wavemaker is provided in the blind test, to reproduce the wave conditions identical to that in the laboratory, a self-correction technique (Ma et al., 2015) is employed in this study. A summary of this technique is given here for completeness. The initial amplitudes and phases of the wave components specifying the motion of the wavemaker are given by
ai0=2S(ωi)Δω
and
φi0=kixfωitf
, i = 1, 2…N, where xf and tf are the expected focusing location and time, respectively. The target spectrum S*(ω) and phase φ* are obtained by applying fast-Fourier transform to the measured surface elevation η*(t, xr) at a specific gauge location xr in the experiment. Then iterations are carried out in the following procedures: (i) at the nth iteration, the wavemaker motion is specified by using ain and φin, based on the second-order wavemaker theory (Schäffer, 1996), and the surface elevation ηn(t, xr) is recorded; (ii) the amplitude and the phase of each component are corrected by
ain+1=ainS(ωi)/S(ωi)
,
φin+1=φin+φm(ωi)φmn(ωi)
, where the subscript m denotes the average phase within the range [ωi − Δω/2, ωi + Δω/2]; (iii) the error between η*(t, xr) and ηn(t, xr) is calculated by using the formula, Err = max[(η* ηn)2/η*2]. If Err is sufficiently small, the iteration stops; otherwise, n = n + 1, go to step (i). Although this approach seems to calibrate the wave at a specific location, numerical investigations have indicated that this technology leads to a satisfactory agreement between the numerical wave elevation and the experimental data at other locations (Ma et al., 2015; Yan et al., 2020).
In the right end of the NS domain, ΓO, a full absorption of the reflected wave from this boundary or a free transfer of the incoming wave is expected. In our previous paper (Li et al., 2018), this boundary was treated in the same way as the left end. The numerical investigation by Li et al. (2018) has demonstrated the effectiveness of this approach for a satisfactory absorption of the reflected waves. However, in this paper, the improved passive wave absorber (Yan et al., 2020; Yan et al., 2020) is employed. A fixed flux pressure condition is imposed on the boundary applying such an absorber; the fluid velocity above the free surface (air phase) imposes a zero-gradient condition, whereas the fluid velocity below the free surface (water phase) is given by
Uh(t)=ω~(t)cosh(k~(t)(z+d))sinh(k~(t)d)η~(t)nh
3
Uzz=0
4
where
Uh
and
Uz
are the horizontal and vertical velocity components, respectively;
ω~
,
k~
and
η~
are instantaneous wave frequency, wave number and wave elevation recorded at the location of the absorber and
nh
is the normal direction of the absorber surface. Once
η~
is recorded,
ω~
can be obtained using the extended Kalman filter (EKF) and
k~
can be determined using the linear wave dispersion. The effectiveness of the improved passive wave absorber has been demonstrated in Yan et al. (2020) and readers are referred to these references for further details. For the boundary on the floating body surface, the moving-wall velocity boundary condition and a zero-gradient pressure condition are imposed.
In the qaleFOAM, the NS equation, continuity equation and the transport equation for the volume fraction are solved in the arbitrary Lagrangian-Eulerian forms in order to use the dynamic mesh technique. After the governing equations are solved, the force and moment on the floating body can be evaluated. The following six-degree-of-freedom (6DoF) motion equation is solved in a body-fixed coordinate system (Obxbybzb, as shown in Figure 1), where the origin Ob locates at the centre of the gravity of the floating body, following Yan and Ma (2007) and Ma and Yan (2009)
MU˙c=F
5
IΩ˙+Ω×IΩ=N
6
dSdt=Uc
7
Bdθdt=Ω
8
where F and N are the external forces and moments acting on the floating body in the body-fixed coordinate system;
Uc
and
U˙c
are translational velocity and acceleration at its gravitational centre (rotational centre);
Ω
and
Ω˙
are its angular velocity and acceleration;
θ(α,β,γ)
are the Euler angles and S is the translational displacement. In Equations 5 and 6,
[M]
and
[I]
are the mass and inertia-moment matrices, respectively.
B
in Equation 8 is the transformation matrix formed by Euler angles and is defined as
B=cosβcosγsinγ0cosβsinγcosγ0sinβ01
It is easy to deduce that
Ω×IΩ=0
and
[B]
is a unit matrix for the cases with 3DoF – that is, surge, heave and pitch. After the translational and rotational motions of the floating body are obtained by Equations 5–8, the OpenFOAM mesh will be updated using the dynamic mesh technique.

For all cases considered by the CCP-WSI 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 three-dimensional 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 point-absorber 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 (Ixx, Iyy and Izz) at the centre of the mass (CoM) are summarised in Table 2, where ZC0M 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.

Table

Table 1. Wave condition

Table 1. Wave condition

Case ID An: m fp: Hz h: m Hs: 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
Table

Table 2. Mass and moment of inertia

Table 2. Mass and moment of inertia

Model m: kg ZC0M: m Ixx: kg m2 Iyy: kg m2 Izz: kg m2
1 43.674 0.191 1.620 1.620 1.143
2 61.459 0.152 3.560 3.560 3.298
3.1 Wave generation and absorption

For all wave conditions, the corresponding empty-tank simulations are carried out to examine whether the target waves are generated properly. The wave is generated using the self-correction wavemaker in the left end of ΩFNPT aiming to reproduce the same time history of the wave elevations recorded at WG5. In the empty tank test, ΩFNPT starts from the wavemaker and the length of ΩFNPT 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 ΩFNPT, where a self-adaptive wave absorber is imposed that produces approximately 95% absorption efficiency. ΩNS starts at x = 11.55 m, between WG1 and WG2. Generally speaking, the length of ΩNS 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 ΩNS, according to the preliminary investigations. To investigate the absorption efficiency of the improved passive wave absorber applied at the right end of ΩNS, ΩNS 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 ΩNS 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 empty-tank 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 self-correction wavemaker technique, even though the tank geometry and the wavemaker used in the qaleFOAM are different from the experiment.

figure parent remove

Figure 2. Comparison of the wave elevation recorded at different locations (case 1BT2, empty tank test, dsv = 0.0175 m). (a) WG1, (b) WG3, (c) WG5, (d) WG8

figure parent remove

Figure 3. Comparison of the wave elevation recorded at different locations (case 2BT2, empty tank test, dsv = 0.0175 m). (a) WG1, (b) WG3, (c) WG5, (d) WG8

figure parent remove

Figure 4. Comparison of the wave elevation recorded at different locations (case 3BT2, empty tank test, dsv = 0.0175 m). (a) WG1, (b) WG3, (c) WG5, (d) WG8

Although the agreements between the results of qaleFOAM and the experimental data at WG5 have proven that the present passive wave absorber applied at ΓO can effectively prevent the wave reflected at ΓO 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 non-linear 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 ΩFNPT by the QALE-FEM can be regarded as the reference data, since ΩFNPT is sufficiently long and the reflection wave from the end of ΩFNPT 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 QALE-FEM 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 QALE-FEM 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 ΓO. The relative difference between the QALE-FEM 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.

figure parent remove

Figure 5. Comparison of the wave elevation recorded at WG5 (empty tank test, qaleFOAM: dsh = 0.05 m, dsv = 0.0175 m; QALE-FEM: ds = 0.075 m). (a) 1BT2, (b) 2BT2, (c) 3BT2

figure parent remove

Figure 6. Comparison of the wave elevation recorded at WG8 (empty tank test, qaleFOAM: dsh = 0.05 m, dsv = 0.0175 m; QALE-FEM: ds = 0.075 m). (a) 1BT2, (b) 2BT2, (c) 3BT2

3.2 Mesh convergent test

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 (dsh) and vertical grid sizes (dsv), the total number of grids, Nt and the number of grids on the structure surface, Ns, are summarised in Table 3. In order to capture the non-linear WSI and also small-scale 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.

figure parent remove

Figure 7. Illustration of the computational mesh near the WEC (model 1, dsh = 0.05 m, dsv = 0.0175 m, red: water; blue: air)

Table

Table 3. Summary of computational grids used in the convergent test

Table 3. Summary of computational grids used in the convergent test

Model Mesh dsh: m dsv: m Nt: M Ns
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 (Ep) 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.

figure parent remove

Figure 8. Comparison of the WEC motions and mooring force in the cases with different mesh sizes (case 3BT2, model 1). (a) Surge, (b) heave, (c) pitch, (d) mooring force

figure parent remove

Figure 9. Comparison of the WEC motions and mooring force in the cases with different mesh sizes (case 3BT2, model 2). (a) Surge, (b) heave, (c) pitch, (d) mooring force

Table

Table 4. Relative error of results of qaleFOAM with the medium mesh

Table 4. Relative error of results of qaleFOAM with the medium mesh

Error: % Model 1 1BT2 Model 1 2BT2 Model 1 3BT2 Model 2 1BT2 Model 2 2BT2 Model 3 3BT2
Ep (surge) 0.17 2.53 1.69 0.49 0.84 1.83
Ep (heave) 0.81 1.05 0.69 0.30 0.22 0.18
Ep (pitch) 0.27 0.39 0.71 0.44 1.12 2.48
Ep (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
3.3 Responses of WECs in extreme waves

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).

figure parent remove

Figure 10. Wave elevations near the WECs at t =  45 s (case 2BT2). (a) t = 45 s model 1, (b) t = 45 s model 2, (c) t = 45.5 s model 1, (d) t = 45.5 s model 2, (e) t = 46 s model 1 (e) t = 46 s model 2

The motions of the WECs and the mooring force acting on the WECs in the case shown in Figure 10 are illustrated in Figure 11. It is found that the profiles of the heave motions largely follow the wave motion (Figure 5(b)). This can be confirmed in Figure 12, which displays the amplitude spectra of the WEC motions and the corresponding wave spectrum at WG5 where the WECs are initially located. The spectra shown in Figure 12 are obtained using the time histories at the duration of 35.3–50.3 s with a sampling frequency of 128 Hz. As observed from Figures 12(c) and 12(d), the amplitude spectra of the wave and the heave motion are very close, suggesting a linear heave response to the incident wave. However, the surge motion and the pitch motion exhibit different features from the expected wave at the WEC sites. Specifically, the surge motions suffer from a long-period oscillation after the focused wave crest passes the WECs at
t45s
(Figure 11(a)), whereas the pitch motion exhibits a high-frequency response, which is gradually suppressed in the case of model 2. These are confirmed by the corresponding spectrum analysis shown in Figures 12(a) and 12(b) and 12(e) and 12(f), respectively.
figure parent remove

Figure 11. Comparison of the time histories of the WEC motions and the mooring loads (case 2BT2). (a) Model 1, (b) model 2

figure parent remove

Figure 12. Comparison of the amplitude spectra of the WEC motions (case 2BT2). (a) Surge spectrum (model 1), (b) surge spectrum (model 2), (c) heave spectrum (model 1), (d) heave spectrum (model 2), (e) pitch spectrum (model 1), (f) pitch spectrum (model 2)

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 free-decay 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.

figure parent remove

Figure 13. Comparison of the time histories of the WEC motions and the mooring loads (case 1BT2). (a) Model 1, (b) model 2

figure parent remove

Figure 14. Comparison of the time histories of the WEC motions and the mooring loads (case 3BT2). (a) Model 1, (b) model 2

Table

Table 5. Relative error of results of qaleFOAM with reference to the experimental data

Table 5. Relative error of results of qaleFOAM with reference to the experimental data

Error: % Model 1 1BT2 Model 1 2BT2 Model 1 3BT2 Model 2 1BT2 Model 2 2BT2 Model 3 3BT2
Ep (surge) 6.94 1.36 7.52 6.20 1.14 7.39
Ep (heave) 9.77 0.54 3.53 3.41 1.38 1.31
Ep (pitch) 44.0 16.4 19.9 6.38 8.51 6.75
Ep (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 CCP-WSI 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 non-linear 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 non-linear 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 8-processor MPI parallel computing in a workstation with Intel Xeon E5-2680, 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 UKIERI-DST project (DST-UKIERI-2016-17-0029).

Notation
d

water depth

k~

instantaneous wave number

nh

normal direction of the absorber surface

p

pressure

Uh

horizontal velocity component

Uz

vertical velocity component

u

fluid velocity

w

weighting function ranging from 0 to 1

η

free surface elevation

η~

recorded wave elevation at the wave absorber

ρw

density of the water

ϕ

velocity potential

ω~

instantaneous wave frequency

References

Cited By

Related content

Related search

By Keyword
By Author

No search history

Recently Viewed