Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves

Journal of Fluids and Structures xxx (xxxx) xxx

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Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves ∗

Qianlong Xu a,b , Ye Li a,b,c , , Yi-Hsiang Yu d , Boyin Ding e , Zhiyu Jiang f , Zhiliang Lin a,b , Benjamin Cazzolato e a

State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China Multi-function Towing Tank, Shanghai Jiao Tong University, Shanghai, China c Key Laboratory of Hydrodynamics(Ministry of Education) , Shanghai Jiao Tong University, Shanghai, China d National Renewable Energy Laboratory, Golden, CO, USA e School of Mechanical Engineering, University of Adelaide, Adelaide, SA 5005, Australia f Department of Engineering Sciences, University of Agder, 4879 Grimstad, Norway b

article

info

Article history: Received 16 May 2018 Received in revised form 8 February 2019 Accepted 12 March 2019 Available online xxxx Keywords: Wave energy Floating-point absorber Experimental test Reynolds-averaged Navier–Stokes equations Boundary element method Viscous damping Nonlinear effect

a b s t r a c t This paper presents experimental and numerical studies on the hydrodynamics of a twobody floating-point absorber (FPA) wave energy converter (WEC) under both extreme and operational wave conditions. In this study, the responses of the WEC in heave, surge, and pitch were evaluated for various regular wave conditions. For extreme condition analysis, we assume the FPA system has a survival mode that locks the powertake-off (PTO) mechanism in extreme waves, and the WEC moves as a single body in this scenario. A series of Reynolds-averaged Navier–Stokes (RANS) simulations was performed for the survival condition analysis, and the results were validated with the measurements from experimental wave tank tests. For the FPA system in operational conditions, both a boundary element method (BEM) and the RANS simulations were used to analyze the motion response and power absorption performance. Additional viscous damping, primarily induced by flow separation and vortex shedding, is included in the BEM simulation as a quadratic drag force proportional to the square of body velocity. The inclusion of viscous drag improves the accuracy of BEM’s prediction of the heave response and the power absorption performance of the FPA system, which agrees well with experimental data and the RANS simulation results over a broad range of incident wave periods, except near resonance in large wave height scenarios. Overall, the experimental and numerical results suggest that nonlinear effects, caused by viscous damping and interaction between waves and the FPA, significantly influence the system response and power absorption performance. Nonlinear effects were found to be particularly significant when the wave height was large and the period was near or shorter than the resonant period of the FPA system. The study is expected to be helpful for understanding the nonlinear interaction between waves and the FPA system so that the structure of the FPA can be adequately designed. © 2019 Elsevier Ltd. All rights reserved.

∗ Corresponding author. E-mail address: [email protected] (Y. Li). https://doi.org/10.1016/j.jfluidstructs.2019.03.006 0889-9746/© 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Q. Xu, Y. Li, Y.-H. Yu et al., Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.03.006.

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1. Introduction As one of the most promising alternative energy resources (Khaligh and Onar, 2010; Tester et al., 2005), wave energy has attracted a lot of interest. Since an oil crisis arose in late 1970 and Salter (1974) published a milestone paper in Nature in 1974, a wide variety of wave energy conversion technologies have been proposed Babarit et al. (2017) and Eatock Taylor et al. (2016). Among these technologies, the floating-point absorber (FPA) is one of the most common wave energy converter (WEC) designs that has been developed. FPAs are floating WEC systems, which consist of one body or multiple bodies that are designed to generate power by a power-take-off (PTO) system from the wave-induced relative motion between the body and a reference frame (e.g., based on the seabed). A number of FPA designs have been deployed and tested for open-ocean demonstrations. For example, Ocean Power Technology tested a 150-kW FPA system in Scotland in 2011 (OPT — ocean power technologies, 2018), and Northwest Energy Innovations deployed their Azura prototype design with grid connection at the U.S. Navy’s Wave Energy Test Site in Hawaii (NWEI — northwest energy innovations, 2016). During the past decade, as the oceanic technology was becoming more mature and investment was increasing, smallscale FPA models were built and tested in laboratory wave tanks (Yeung et al., 2010; Rhinefrank et al., 2010; Bjarte-Larsson and Falnes, 2006; Tyrberg et al., 2010). Specifically, Yeung et al. (2010) proposed and studied a single-body heaveonly point absorber. Rhinefrank et al. (2010) presented the performance results of a three-body system with multiple degrees of freedom (DOF). Bjarte-Larsson and Falnes (2006) tested the PTO control scheme in a wave flume. Tyrberg et al. (2010) presented an investigation on the power generation of an FPA device in a near-shore testing site. Zurkinden et al. (2014) tested and modeled a semisubmerged hemisphere point absorber and evaluated the nonlinear effects on the device system dynamics. Recently, Goteman (2017) proposed an analytical model for point-absorber WECs connected to floats of different geometries and topologies based on the potential-flow theory. Meanwhile, we recently used a Reynoldsaveraged Navier–Stokes (RANS)-based computational fluid dynamics (CFD) method to analyze a two-body FPA system’s power generation performance (Yu and Li, 2013), which was developed by the reference model (RM) project of the U.S. Department of Energy (Jenne et al., 2015). In general, these studies revealed that the device system dynamics followed linear potential theory well, except for steeper waves (i.e., the waves with wave height to wave length ratio larger than 0.04), where the influence of the nonlinear buoyancy and wave excitation forces became essential in predicting the device response. Despite the differences in their FPA designs, these studies mainly focused on the FPA dynamic behavior and maximizing power output under operational conditions or weakly nonlinear wave environments. FPA devices are generally designed to sustain large amplitude wave environments, wherein waves are steep and nonlinear wave–body interactions may have a great influence on the system’s response and power generation. Besides, the extreme loads from these critical wave environments are destructive to the device structures, mooring, and PTO systems. Therefore, these issues associated with survival conditions need to be evaluated carefully. In this study, we advanced our recent study in Yu and Li (2013) by systematically conducting experimental tests for both operational and survivable conditions and implementing a solver of boundary element method (BEM) developed based on the linear potential-flow theory. Consequently, experimental test results are extensively discussed and both RANS and BEM results are compared against the experimental results to demonstrate the viscous effects under various conditions. The major findings from the two sets of experimental tests are summarized in this article. Noting that RANS methods and part of experimental tests are presented in Yu and Li (2013), some information might be restated in this paper. Those readers who are interested in RANS methods used for operational conditions can refer to Yu and Li (2013), and this paper is a more systematic study of both operational and survivable conditions. This paper is organized as follows. In Section 2, we introduce the FPA model design, the wave tank setup, and the instrumentation system for both the operational and survival condition tank tests. In Section 3, a heave-constrained dynamic model of the FPA system in the frequency domain is developed, and the boundary conditions for the general wave-FPA interaction problems and the formulation of the BEM are described. Meanwhile, we describe the computational methodology of the RANS method for the simulations. Some preliminary experimental data and numerical results of regular wave tests with large wave heights for the locked FPA model are summarized in Section 4, and the comparative analyses of the experimental and numerical results on the power extraction performance with selected PTO damping under operational wave conditions are also presented here. A discussion of the present findings is provided in Section 5. Conclusions and expectations for future work are listed in Section 6. 2. Model design and experimental setup In this section, we study a two-body FPA system: RM3 WEC device (Yu et al., 2015). The FPA design has two major components: a float on the top, and a spar/plate at the bottom (Fig. 1). For this FPA system, the wave height of the designed operational conditions is much smaller than that of the survival conditions. For example, Ocean Power Technology’s PowerBuoy PB150 system was designed to operate in waves with heights less than 7 m (OPT — ocean power technologies, 2018). On the other hand, the typical 100-year significant wave height during storms on the U.S. West Coast is generally in the range between 8 m and 13 m (Mackay et al., 2010). We built two physical models: one for the survival conditions and the other for the operational conditions, and the model for the survivability study is smaller than that for the power generation capability study. Accordingly, two series of wave tank tests were conducted to investigate the behavior of the Please cite this article as: Q. Xu, Y. Li, Y.-H. Yu et al., Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.03.006.

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Fig. 1. Six degrees of freedom of a two-body FPA. Source: adopted from Li and Yu (2012). Table 1 LFPA properties (model scale). Model properties

Value

Center of gravity Moment of inertia (pitch) Mass

0.23 m below the mean free surface 72.18 kg cm2 313 g

Table 2 Mooring configuration for the survivability test (model scale). Mooring settings

Value

Top layer connection Bottom layer connection Spring stiffness

0.04 m below the mean free surface 0.15 m above the damping plate 0.7 N/m (each line)

FPA system in survival and operational conditions, using two different wave tanks: the Berkeley tank and the Scripps tank1 . 2.1. Survival condition test As the survivability of the FPA must be investigated in large wave scenarios, a relatively small model is necessary considering the wave-making capability of the wave tank. We built a 1/100-scale model (Fig. 2) and its properties are summarized in Table 1. Note that the total mass of the model includes the mass of the device and a target plate for tracking the motion. We assume that the two bodies are locked to increase the FPA’s survivability under the survival condition. Therefore, the locked floating-point absorber (LFPA) model can be regarded as a single body with no relative motion between the reaction body and the float. Table 1 lists the whole weight of the body, the center of gravity, and the moment of inertia for pitch. We also built a mooring system to keep the FPA in position with eight mooring lines (Fig. 3). These lines are divided into two layers. Each layer has four lines in the configuration of a cross, as shown in Fig. 3, with the properties of mooring lines listed in Table 2. There is no doubt that the mooring configuration and the connecting point of mooring lines can affect the hydrodynamic performance of the FPA. Through initial calculations, we found that the effect of the mooring 1 Although the scale of the Berkeley tank is similar to the scale of the Scripps tank, the wave-making capabilities of the tanks are different. Considering the significant differences between the operational and the survival conditions, we need a wide spectrum of wave-making capabilities that neither of the tanks can provide individually. Therefore, we used the two tanks for experimental tests. Please cite this article as: Q. Xu, Y. Li, Y.-H. Yu et al., Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.03.006.

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Fig. 2. LFPA model dimensions and geometry.

Fig. 3. Mooring line configuration (left: side view; right: top view).

system on the heave response was minimized to be negligible, and that the heave response was generally uncoupled with motions in other DOFs (Jenne et al., 2015; Yu et al., 2015). The survival condition test was conducted in the wave tank at the University of California, Berkeley. The Berkeley wave tank is 68 m long, 2.4 m wide, and the water depth is set to 1.5 m, with a view window installed at one side of the tank. A plunger-type hydraulic piston is located at one end of the tank, which can make both regular and irregular waves. This wave tank was designed and constructed a half century ago for testing ships and large offshore structures, and was never used for testing floating WECs. Therefore, there is no auxiliary equipment for the test. To implement the LFPA system and the mooring system appropriately, we installed four metal piles on both tank sidewalls for the mooring line connection at the view window and the LFPA model is located at the center of the window. Please cite this article as: Q. Xu, Y. Li, Y.-H. Yu et al., Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.03.006.

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Fig. 4. Two-body floating point absorber model dimensions and geometry. Table 3 TFPA properties (model scale). Component

Mass (kg)

Float

A. Float D. Top roller E. Bottom roller

3.220 0.115 0.117

Spar/plate

B. Central column C. Damping plate F. Cylinder and clamps H. Load cell I. Needle valve J. Linear pot (short) K. Linear pot (long)

0.745 1.108 0.205 0.012 0.015 0.077 0.114

Not on the testing model

G. Heave guide

0.116

To conduct a systematic study, we implemented the following instrumentation: a wave gauge, a two-dimensional (2D) motion tracking system, and a load cell. We used a generic wave gauge to measure the wave height, which is located at 5.2 m upstream from the LFPA. We also used a 2D motion tracking system to capture the surge, heave, and pitch motions. This system included a digital camera on one side of the tank and a target plate attached to the top of the buoy. Specifically, there are passive markers on the target plate with a sampling frequency of 45 Hz. 2.2. Operational condition test As the wave height of the operational condition is much lower than that of the survival condition, we can design a model that is larger than the LFPA model. We built a 1/33-scale model of a two-body floating-point absorber (TFPA, as shown in Fig. 4), but the design concept and model profile are the same as the LFPA. The model specifications are presented in Fig. 4 and Table 3. The operational condition test was performed in the wave tank at the University of California, San Diego. The wave tank is 44.5 m long, 2.44 m wide, and the water depth is 1.46 m. Fig. 5 shows the experimental setup for the wave tank test. In the TFPA model test, the purpose was to investigate the heave-only power generation capability of the TFPA. We installed a guidance structure connected to the carriage so that the TFPA model was restricted to move freely in heave only. A miniature hydraulic cylinder in a closed circuit with a needle valve provided damping to the relative motion between the float and the central column, representing the PTO mechanism. The PTO force was measured using a load cell, which was installed between the bottom of the float and the top of the hydraulic cylinder, as shown in Fig. 4 (right), with properties listed in Table 3. The load cell directly measures the PTO damping force that was provided by the hydraulic cylinder. The PTO damping was controlled by turning the needle valve, and we presented the damping value in kN/m for full scale. Please cite this article as: Q. Xu, Y. Li, Y.-H. Yu et al., Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.03.006.

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Fig. 5. Experimental wave tank test settings at the University of California, San Diego (left: TFPA PTO design and connection; and right: whole tank setup, NREL 20364).

We also developed the test instrumentation system based on what we used in the Berkeley tank test as described in Section 2.1. Besides, we used a potentiometer to record the relative motion between the float and the central column and the motion tracking system with a four-camera device. Furthermore, the data acquisition system was updated by increasing the sampling frequency to 100 Hz and introducing the filter function that removes the signal noise induced by vibrations and disturbances. The power output is calculated by multiplying the PTO force and the relative velocity between the float and the reaction plate, and is defined as P = Fpto × urel

(1)

where P is the power output, Fpto is the measured PTO force, and urel represents the relative velocity between the float and the reaction plate. The averaged power can be evaluated by integrating the instantaneous power over a period after the transient response has damped out and only the steady-state response remains. 3. Numerical modeling 3.1. Potential flow method In operational conditions, wave heights are relatively small, and the linear potential flow theory is generally applicable to analysis of the hydrodynamics of the FPA system. In survival conditions, nonlinear hydrodynamic effects are significant and using the linear potential flow theory is not adequate (Li and Yu, 2012; Xu et al., 2018). In this section, we only focus on investigating the TFPA system using the linear potential flow method. 3.1.1. Boundary-value problems A linear, heave-constrained, dynamic model in the frequency domain is used to investigate the motion and power capture of a two-body floating-point absorber WEC in regular waves, which contains a linear spring–damper to represent the PTO mechanism. The spatial domain is assumed to have an infinite size. A simplified representation of a TFPA system is shown in Fig. 6. Based on Newton’s second law, for a rigid two-body system, the constrained heave motion can be written as (Falnes, 1999):

[K ]⃗z = f⃗e K1 (ω) + Kpto Kc (ω) − Kpto

[ [K ] =

(2) Kc (ω) − Kpto K2 (ω) + Kpto

] (3)

Kj = −ω2 [mj + Aj (ω)] − iωBj (ω) + kj

(4)

Kpto = kpto − iωcpto

(5)

where [K ] is the complex stiffness matrix given by Eq. (3). The wave excitation force in heave mode on body j (j = 1 for the float and j = 2 for the reaction body) is generally written as Fej = Re[fej e−iωt ], where fej and ω are the complex Please cite this article as: Q. Xu, Y. Li, Y.-H. Yu et al., Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.03.006.

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Fig. 6. A schematic representation of a TFPA model.

excitation force amplitude and the incident wave frequency. mj and kj are the inertia mass and the hydrostatic stiffness of body, j. Aj (ω) and Bj (ω) represent the added mass and radiation damping of the body, j, in heave. The hydrodynamic radiation coupling term, Kc (ω), between the float and the reaction body, as a small quantity, can be neglected (Falnes, 1999; Beatty et al., 2015). For each body, the wave excitation force and hydrodynamic coefficients are obtained from either a BEM or diffraction and radiation tests in the wave tank. The complex amplitudes of body displacements are written as ⃗ = −iω⃗z . z⃗ = [z1 , z2 ]T , and the complex velocity amplitudes can be expressed as u In Eqs. (2)–(5), a spring–damper force, Fpto = Re[fpto e−iωt ], between the float and the reaction body is quantified as: fpto = −cpto ur − kpto zr

(6)

where kpto is the spring stiffness and cpto is the power absorption damping. ur = u1 − u2 and zr = z1 − z2 are the relative velocity and displacement between the float and the reaction body, respectively. For this PTO mechanism, the time-averaged power extracted by the TFPA is equal to the time-averaged power dissipated across the resistance of the PTO, which results in the following equation: P =

1 2

cpto ∥ur ∥2 =

ω2 2

cpto ∥zr ∥2

(7)

As defined in Fig. 6, we consider the wave–body interaction problem of the TFPA model. The water depth for the analysis is h. The spatial domain is assumed to have infinite size. It is supposed that O − xyz is a right-handed Cartesian coordinate system, with O − xy plane coinciding with the undisturbed free surface. The z-axis is along the central axis of the TFPA and positive upward against gravity. In the context of potential flow, we can introduce a velocity potential, Φ (x, y, z , t), to express the fluid field, which is defined as

[ ] Φ = Re (φI + φd + u1 φr1 + u2 φr2 )e−iωt

(8)

It is based on the linear superposition principle. φI is the incident wave potential, and φd is the diffraction potential when these bodies are held fixed, φrj is the unit radiation potential related to the body velocity, uj , in heave. With the assumption of the linear potential flow theory, the diffraction and radiation potentials satisfy the following boundary conditions:

∇ 2 φ(d,r) = 0,

(x, y, z) ∈ Ω

∂φ(d,r) ω2 − φ(d,r) = 0, ∂z g ∂φ(d,r) = 0, ∂z

(9)

z=0

(10)

z = −h

( ) 1 φ(d,r) = O √ eik0 R , R

∂φd ∂φI =− , ∂n ∂n

(11)

R=

on Sb1 & Sb2

√

x2 + y 2 → ∞

(12)

(13)

Please cite this article as: Q. Xu, Y. Li, Y.-H. Yu et al., Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.03.006.

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Fig. 7. Panel meshes used for BEM analysis of the TFPA.

{ ∂φr1 n3 , on Sb1 = 0, on Sb2 ∂n

(14)

{ ∂φr2 0, on Sb1 = n3 , on Sb2 ∂n

(15)

where the simplified notation, φ(d,r) , is the diffraction potential or the radiation potential, Ω is the whole fluid domain, g is the gravity acceleration, k0 is the incident wave number, Sb1 and Sb2 are the mean wetted surfaces of the float and the reaction body, respectively, n is the normal unit vector on body surfaces directed into the float and the reaction body, where n3 is the Cartesian component along the z-axis. 3.1.2. Boundary element method To solve the boundary value problems, the boundary integral equation derived from the Green’s third identity is applied for the evaluation of φd and φr :

α (p)φ (p) =

∫∫

[G(p, q)φn (q) − φ (q)Gn (p, q)]ds

(16)

S

where G(p, q) is the free surface Green function with respect to a field point, p(x, y, z) and a source point, q(ξ , η, ζ ). α (p) is an interior solid angle at the corresponding point, p(x, y, z), on the body surface S(Sb1 + Sb2 ). Subscript ‘n’ denotes the normal derivative with respect to q(ξ , η, ζ ) (φn = ∂φ/∂ nq , Gn = ∂ G/∂ nq ). φ could be either diffraction potential or radiation potential depending on the problem to solve. A BEM-based code solver, SJTU-WEB,2 was used for calculation (Xu et al., 2018). For the simplified geometry, the panel meshes of the TFPA model are generated as displayed in Fig. 7, which include 1940 panel elements with 1719 nodes. An analysis of each body in the whole system was performed to determine the hydrodynamic effects of the float and reaction body in the presence of the other body. For each body, the added mass, radiation damping, and excitation forces were generated as a function of frequency using the BEM solver. 3.1.3. Additional viscous damping Fluid viscosity has a significant impact on the dynamics of the TFPA in heave motion, particularly at resonance. To account for the viscous effects, a drag force is introduced within the present dynamic model. In general, the total drag force includes the form drag as a result of flow separation and vortex shedding, and the friction drag as a result of fluid shear stress on body surfaces. In this paper, we assumed a nonlinear drag term to model the drag forces on the float and 2 SJTU-WEB is a constant boundary-element-method-based numerical model. It solves the radiation and diffraction problems and is developed for modeling the linear hydrodynamic interaction between waves and various types of floating and submerged bodies. Please cite this article as: Q. Xu, Y. Li, Y.-H. Yu et al., Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.03.006.

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the reaction plate, which is proportional to the square of the heave velocity and expressed as follows (Tao and Cai, 2004)

⏐ ⏐ 1 Fdrag = − ρ As Cd Z˙ ⏐Z˙ ⏐ 2

(17)

where As is the cross-sectional area of the float or the reaction plate perpendicular to the heave direction, Cd is the drag coefficient, and ρ is the density of water. Generally, the value of the drag coefficient depends on the body geometry, the Reynolds number (Re), and the Keulegan–Carpenter (KC) number, KC = 2π A/D, where A is the motion amplitude and D is the characteristic diameter of the body. For the present TFPA system, KC is generally less than 10. Within this range, Graham’s study (Graham, 2006) showed that the variation of drag coefficient was primarily induced by the strength of shed vortices. At low KC numbers, Graham assumed that vortex flow was dominated by the local flow near sharp edges, and the drag coefficient for plates became KC dependent Cd ∝ KC −1/3

(18) −1/3

Based on Graham’s assumption, we suppose Cd,j = δj KC (j = 1 for the float and j = 2 for the reaction plate), where the proportional coefficient, δ , depends on the plate geometry and dimension. Because the reaction plate is much thinner than the float, it is clear that vortices are shed more frequently and intensively near the edges of the reaction plate. Therefore, the proportional coefficient, δ1 , should be larger than δ2 . In the frequency domain analysis, the nonlinear drag term can be substituted by a linear term using the Lorentz linearization

⏐ ⏐

Z˙ ⏐Z˙ ⏐ ≈

8 3π

( ) ∥− iωz ∥ Re −iωze−iωt

(19)

thus fdrag ,j =

4iω2 3π

ρ As,j Cd,j ∥zj ∥ zj

(20)

The complex drag force, fdrag ,j (j = 1 for the float and j = 2 for the reaction plate) is included in the constrained heave Equation (2). To solve the equation and overcome the nonlinearity, we proceed through an iterative scheme so that a linear problem is solved at each iteration. 3.2. Reynolds-averaged Navier–Stokes method The unsteady incompressible flow field is described by the continuity equation and the Navier–Stokes equations:

∇ ·U=0

(21)

ρ (Ut + U · ∇ U) = −∇ p + Fb + ∇ · T

(22)

where ρ is the water density, U is the flow velocity vector, and Ut is its time derivative, Fb is the body force vector (e.g., gravity), and T is the stress tensor. For the survival condition analysis, the translation and rotation responses of the LFPA can be obtained by solving the equations of motion after the excitation force is obtained from the RANS simulations. The equations of motion at the center of gravity are mb a t = F

(23)

Ig aΩ + Ω × Ig Ω = M

(24)

where mb is the mass of the body, at is the acceleration vector for the translation, Ω and aΩ are the angular velocity and acceleration vectors, Ig is the moment of inertia tensor at the center of gravity, and F and M are the resulting force and moment acting on the body. In the numerical simulation, the sway, roll, and yaw motions are constrained, and the FPA is only allowed to move freely in surge, heave, and pitch. For the operational condition studies, the TFPA includes two parts: a float and a reaction body, which are connected through a linear spring–damper system representing the PTO mechanism. Because the effect of the mooring system on the power extraction performance is negligible (Muliawan et al., 2013), this effect is not considered in the study. As the selected FPA system predominantly operates in heave, the FPA system is limited to the heave motion. Eqs. (23) and (24) are then reduced to a 1-DOF equation for each body. As a result, the equations of motion in heave for the float and the reaction body become Float: mF aF = (Fz )F + Fpto

(25)

Please cite this article as: Q. Xu, Y. Li, Y.-H. Yu et al., Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.03.006.

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Q. Xu, Y. Li, Y.-H. Yu et al. / Journal of Fluids and Structures xxx (xxxx) xxx Table 4 RANS simulation parameters. Setting

Method

Turbulence model Turbulence wall model Time integration Pressure–velocity coupling Free surface Mesh motion Wave absorption

k-ω SST All y+ wall treatment model Second-order implicit scheme SIMPLE algorithm Volume of fluid method Morphing Damping zone (2 wavelengths)

Table 5 Natural frequencies in decay tests (full scale). Decay test

Natural period

Estimation method

Heave Surge Pitch

9 s 21 s 21 s

Heave-decay test Mass, added mass, and mooring stiffness Pitch-decay test

Reaction body:

mR aR = (Fz )R − Fpto

(26)

where Fpto is the PTO force and Fz is the force component in heave. Subscripts F and R indicate the float and the reaction body, respectively. Fpto is represented by a spring–damper force, given by Fpto = −cpto (uF − uR ) − kpto (zF − zR )

(27)

where kpto is the spring stiffness, cpto is the power absorption damping coefficient, uF and uR are the heaving velocities of the float and the reaction body, respectively, and zF and zR are the heaving displacements of the float and the reaction body, respectively. The CFD simulations were performed using a finite-volume method-based unsteady RANS model (STARCCM+). The mesh resolution, computational domain size, and time-step size were determined based on the CFD analysis performed by Yu and Li (2013), and the simulation parameters are listed in Table 4. 4. Results The measured data obtained from the survival condition test and operational condition test are given in this section. Comparative analyses of the present numerical models with the experimental data for the LFPA and TFPA are also presented. 4.1. Survivability condition test In the survival condition test, we mainly focused on the LFPA’s motion response in regular waves, with wave heights of 10 m and 15 m in full scale. For the wave tank test, the root-mean-square peak-to-trough response amplitude operators (RAOs), defined as the ratio of response magnitude to wave height, were calculated from five oscillations. For each run, the five-oscillation time series was selected after the transient response had damped out and only the periodic response remained. Besides, we also conducted free-decay tests of the LFPA model in pitch and heave. In the heave decay test, the LFPA device was lifted with an initial displacement of 0.02 m (model scale). In the pitch-decay test, the device was initially rotated with an angle of 5.7 degrees. To investigate the system dynamics of the LFPA device, we evaluated the natural period of the system in heave, surge, and pitch, based on the results from the free-decay tests, the mooring stiffness, and estimated added mass in surge. It is assumed that the added mass in surge is approximately equal to the mass for the device. The values are listed in Table 5. Here we compare the RAO results of heave, surge, and pitch motions of the LFPA predicted from the RANS simulations to those obtained from the experimental measurements in 10-m wave height scenarios. The quantity is plotted against the wave period in Fig. 8. As shown in the figure, the RANS prediction of heave motion shows a good agreement with the experimental data in the 10-m wave height scenarios, but smaller surge amplitude and larger pitch amplitude from the RANS simulation than the experimental test. These variances could be attributed to the differences of the model geometries and gravity centers. In particular, the LFPA model used in the experimental test had a larger float height than that used in the RANS simulation. As a result, in the experimental wave tank test, wave overtopping did not occur as often as we observed in the RANS simulations. It is anticipated that wave overtopping could result in an additional damping force that restrains the LFPA motion. Fig. 8 also shows the motion response data from the experimental test in the 3-m and 15-m wave height scenarios. As shown in the figure, it is obvious that heave response of the LFPA decreases as the wave height increases. We Please cite this article as: Q. Xu, Y. Li, Y.-H. Yu et al., Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.03.006.

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Fig. 8. Heave, surge, and pitch RAOs for the LFPA wave tank test (in full scale).

believe that a larger damping force restrains the heave motion in larger wave height scenarios, which is induced by the nonlinear interactions between the waves and the LFPA device, including fluid viscosity and other important effects, such as wave overtopping and slamming. It is anticipated that these nonlinear effects are more significant in larger wave height scenarios. For surge, the response generally follows the same surge-to-wave-height ratio trend, except for high wave-frequency scenarios. For pitch, the RAO decreases much more when the wave height increases. Note that the WEC model has a peak response in both surge and pitch when the wave frequency is close to the model natural frequency 0.3 rad/s (wave period 20 s). A minimum RAO is also observed in both surge and pitch when the wave frequency is close to 0.5 rad/s (wave period 12 s), particularly for the designed wave heights of the 10-m scenarios, which is most likely attributed to the two-layer mooring configuration. Please cite this article as: Q. Xu, Y. Li, Y.-H. Yu et al., Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.03.006.

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Fig. 9. Motion response histories of the LFPA from RANS and experimental measurements for the 10-m and 15-m waves with wave periods T = 12.5 s and T = 17.5 s (in full scale).

To investigate the dynamic response of the LFPA system, we plot the motion response histories of the 10-m and 15-m wave height scenarios from the experimental measurements and RANS simulations in Fig. 9. Specifically, the results of T = 12.5 s correspond to the case where the minimum pitch response occurs, and the results of T = 17.5 s correspond to the case where maximum pitch response occurs. Two observations of Fig. 9 are made as follows. First, the RANS simulations predict the motion responses reasonably well, with two exceptions: for the case of T = 12.5 s and H = 10 Please cite this article as: Q. Xu, Y. Li, Y.-H. Yu et al., Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.03.006.

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Fig. 10. Vertical force on the LFPA under extreme waves.

Fig. 11. Hydrodynamic pressure (scaled by ρ gH) contour around LFPA (t /T = 5.38; T = 7.5 sec; H = 8 m).

m, the experimental pitch response is underpredicted, and, for the case of T = 12.5 s and H = 15 m, a phase shift of surge response between the experimental data and the RANS simulation result is observed. Second, for the case of T = 12.5 s and H = 15 m, wave overtopping is often observed both in the experimental wave tank test and the RANS simulations. Note that the wave steepness is quite large in this case, therefore, the nonlinear effects including high-order wave excitation force, the viscous drag caused by flow separation and vortex shedding, wave overtopping force, and the impact load of piercing the free surface have a more significant influence on the motion responses, which result in the inharmonic response. Overall, the agreement demonstrates that the RANS method is able to model the complex nonlinear interaction between waves and the floating bodies. We also evaluated the wave-induced force and the hydrodynamic pressure distribution on the LFPA system. The vertical force component (Fy ) is plotted against the wave periods for various wave heights in Fig. 10. The vertical force is proportional to the wave height when the wave period is sufficiently long. As the wave period decreases, the ratio of force to wave height for shorter period waves becomes smaller under extreme waves. The results are consistent with the heave response analysis. In addition, it is essential to analyze the local pressure distribution on the LFPA device. Fig. 11 shows the hydrodynamic pressure distribution around the LFPA model. Because the motion of fluid particles decreases rapidly with increasing depth below the free surface, the hydrodynamic wave impact on the float is more significant than that on the damping plate. Fig. 12 plots the pressure distribution on the LFPA at two time instances. The out-of-water float is re-entering the water surface at t /T = 3.12, where the slap load can locally create a high-pressure impact on the structure. On the other hand, the float is completely submerged when t /T = 3.47, and the top of the float is subject to additional hydrodynamic pressure given by the weight of the overtopping waves. Please cite this article as: Q. Xu, Y. Li, Y.-H. Yu et al., Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.03.006.

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Fig. 12. The pressure distribution (scaled by ρ gH) on the LFPA surface at t /T = 3.12 (top), and t /T = 3.47 (bottom) with T = 7.5 s; H = 8 m.

4.2. Operational condition test 4.2.1. Heave response in waves The heave RAOs of the float and the spar/plate (reaction section), and the relative motion between them in the 2.5-m wave height scenarios from the experimental wave tank test and BEM simulations are shown in Fig. 13. In the experiment, the PTO damping was controlled by adjusting the needle valve, and the averaged damping value for the results presented in Fig. 13 were about 1800 kN/m. In the BEM simulation, drag coefficients for the float and the spar/plate are determined by the proportional coefficients δ1 = 1.0 and δ1 = 3.5. In general, the heave RAOs of the float and the spar/plate, and the relative motion RAO derived from the present BEM analyses are in agreement with the experimental results. The relative motion response between the two bodies is reduced when wave energy is extracted by the PTO mechanism, and it also decreases as the PTO absorption damping increases. To verify the BEM prediction on the phase shift between the float and the reaction section predicted, we compare the results from BEM simulations to those from the RANS simulations, presented by Yu and Li (2013), for a different PTO damping value (CPTO =1200kN/m). The RAOs and the phase shift are plotted against wave period in Fig. 14. The BEM-derived heave RAOs of the float and the reaction section and the relative motion RAO agree fairly well with the RANS simulation results, indicating the nonlinear drag term is capable of representing viscous effects on the TFPA system. The heave response of the float has a peak period of about T = 10 s, whereas the data of the reaction section gradually increase with the incident wave period, and no peak occurs in the heave response, which implies that the reaction section is overdamped. Compared to Fig. 13, it is anticipated that the heave motion of the reaction section tends to follow the motion of the float section as the PTO damping increases. The BEM-predicted phase shifts also agree well with those from the RANS simulations when the wave period is larger than 10 s. The phase shift is reduced as the incident wave period increases. We anticipate that the angle will vanish when the wave period is sufficiently longer, whereas the float and the spar/plate will follow the motion of the incident wave. On the other hand, the BEM and RANS results of the phase shift between the float and the reaction section differ by approximately 20% at wave periods less than 8 s. The difference could be attributed to other nonlinear hydrodynamic and damping effects, which are more significant when the wave period is smaller. In this case, the phase shift results in a relative motion peak period about T = 8 s, which differs from the peak period of the float or the reaction spar/plate. As discussed in Section 3.1.3, the drag coefficients, Cd , used in the BEM simulations are obtained when the iterative calculation of the motion equation converges, based on Graham’s assumption (Section 3.1.3). The influence of varying Please cite this article as: Q. Xu, Y. Li, Y.-H. Yu et al., Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.03.006.

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Fig. 13. Heave RAO and phase difference between the float and the reaction spar/plate from TFPA experimental test and BEM with viscous damping (H = 2.5 m and CPTO = 1800 kN/m).

Fig. 14. Heave RAO and phase difference between the float and the reaction spar/plate from the TFPA experimental test, RANS simulation, and BEM with viscous damping (H = 2.5 m and CPTO = 1200 kN/m).

Fig. 15. Drag coefficients for the float and the reaction plate with different PTO damping (H = 2 m).

wave environment and PTO damping on the estimation of the drag coefficient is investigated. The viscous drag coefficient for the float and the reaction spar/plate are plotted against the wave period with a range of PTO damping coefficients (i.e., 600 kNs/m, 1200 kNs/m, 1800 kNs/m) in Fig. 15 and a range of wave heights (i.e., 2 m, 4 m, 6 m) in Fig. 16. PTO damping and wave height have little effect on Cd for the float, whereas for the reaction plate, it slightly decreases with PTO damping Please cite this article as: Q. Xu, Y. Li, Y.-H. Yu et al., Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.03.006.

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Fig. 16. Drag coefficients for the float and the reaction plate with PTO damping of 1200 kNs/m in different wave height scenarios.

and wave height. At short wave periods, an increase in wave period slightly decreases Cd for the float. A similar trend is evident for the reaction plate: an increase in wave period greatly decreases Cd . In fact, based on Graham’s assumption, Cd decreases as the response amplitude increases, which results in the visible trends in Figs. 15 and 16. Therefore, for the TFPA system, we can conclude that Cd for the float ranges from about 1 to 1.5 and Cd for the reaction plate ranges from about 3 to 6 in operational wave conditions. 4.2.2. Wave power absorption Fig. 17 plots the wave power extraction performance of the TFPA system against the wave period in the 2.5-m and 4-m wave height scenarios across a range of PTO damping between 200 kNs/m and 2000 kNs/m, where the BEM prediction is compared to the RANS simulation data. The TFPA system has an optimal power extraction performance when the PTO damping ranges between 700 kNs/m and 1200 kNs/m. Based on the RANS simulations, a maximum time-averaged power of 330 kW in the 4-m wave height scenarios is generated by the TFPA system when the PTO damping is equal to 1000 kN/m. The results from the BEM simulations agree well with those from the RANS simulations in the 2.5-m wave height scenarios. Whereas, in the 4-m wave height scenarios, the BEM simulations overpredicted the power performance, particularly for the wave period near resonance. The difference could be attributed to other nonlinear hydrodynamic effects, including wave overtopping and the changes of the cut waterplane area for the float. In the RANS simulations, particularly near resonance, these nonlinear effects occurred much more frequently in the 4-m wave height scenarios than in the 2.5-m wave height scenarios, which further restrain the TFPA motion and reduce wave power output. However, the BEM model does not account for these effects. 5. Discussion When the FPA system is deployed, the system is subjected to various forces, including hydrodynamic wave load, restoring force, viscous damping force, and the mooring line tension force. The dynamic motion highly depends on the wave environment and the natural frequency of the FPA system. As shown in Fig. 9, the interaction between waves and the oscillating FPA can be nonlinear, particularly in extreme waves. A systematic experimental wave tank test is commonly used for evaluating the WEC design during the design process, and the results of both our survivability and operational condition tests are presented in Section 4. The results are useful for understanding the hydrodynamics and the power generation capability of the FPA system. Nevertheless, we also learned that a few features are worth further discussions. The survivability analysis presented in this paper shows that the pitch RAO is less than 1 degree/meter under the 8-line mooring configuration for most extreme conditions, except for very large wave period scenarios. The mooring configuration used in the survivability wave tank test was designed to mitigate the motions in surge and pitch (see Fig. 8). However, the two-layer design makes the dynamic of the system more complicated because the loads on the two layers are significantly different and unsteady. We anticipate that the result of the minimal pitch response at certain wave frequencies (0.5–0.6 rad/s) was driven by this particular two-layer mooring configuration. As shown from the time domain analysis (Fig. 9), higher harmonics of the system were also observed. The mooring design is essential because it can significantly influence the hydrodynamic response of WECs. Moreover, the mooring system for a floating WEC is often one of the primary cost drivers, and the modeling of the mooring system in the wave tank can also be challenging, such as anchoring and characterizing the scaled mooring line properties in the wave tank, particularly for the small-scale model, which is often the case for the survivability test because of the wave-making limitation at the tank. Please cite this article as: Q. Xu, Y. Li, Y.-H. Yu et al., Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.03.006.

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Fig. 17. Power absorption performance (scaled by the square of wave height) of the TFPA system in the 2.5-m wave height (top) and 4.0-m wave height (bottom).

For a practical FPA design, the natural period of the system in surge and pitch is often designed to avoid the range of real sea periods to provide a more stable system. However, the natural period of the WEC in pitch is within the range, and a practical system is expected to avoid that. In addition, the FPA system is designed to have a natural frequency in heave close to the typical peak period of real seas to maximize the power generation performance. The large heave response in extreme waves can result in the essential nonlinear interaction between waves and the oscillating FPA device. Therefore, the reference FPA system contains a damping plate, which provides an additional damping force to stabilize the system. The overall damping load of the system varies with the incident wave condition. To study the nonlinear restraint effects on the FPA system dynamics, particular in heave, we analyze the system through the use of linear dynamic theory. Given that the heave motion is generally uncoupled from the other motions, we can model the heave response by using a simple spring–damper system and assume that the effects of pitch and surge are negligible. If the FPA is regarded as a single-body WEC, the equation of heave motion can be given as (m + A)Z¨ + c Z˙ + kZ = Fd

(28)

where A is the added mass, Fd is the excitation force, c is the total damping coefficient and, k is the total spring stiffness. The magnification factor β is known as the magnitude of the frequency response and can be given as

β= √ ζ =

1

[1 − (ω/ωn ] + (2ζ ω/ωn )2 )2 2

c 2ωn (m + A)

(29)

(30)

where ζ is the damping ratio, and ω and ωn are the wave frequency and the system natural frequency, respectively. In regular linear waves, the restoring spring stiffness is equal to k = ρ gS, where S is the cross-section area of the float. Given that Fd is proportional to wave height, H, the magnification factor is equal to the ratio of heave to wave height. Please cite this article as: Q. Xu, Y. Li, Y.-H. Yu et al., Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.03.006.

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Fig. 18. Comparison of damping ratio in heave.

Through the use of linear dynamic theory, the estimated natural period from the heave-decay test is equal to 9 s, and ζ is 0.114 (Yu and Li, 2013). The damping ratio is expected to increase when the FPA system is subjected to incident waves. To evaluate the effect of nonlinear damping, we calculate ζ from Eq. (30). The damping ratio is plotted against the wave period in Fig. 18. The damping ratio varies with the incident wave conditions, and it tends to increase as the incident wave height increases, particularly for shorter wave period scenarios. It is known that the peak period will disappear when ζ is greater than 0.707, and the system will be overdamped when ζ is greater than 1. As shown in the study, the damping ratio is greater than 0.707 for most of the H = 10-m wave scenarios. The dynamic motion for the FPA system is also overdamped in extreme waves with periods shorter than the resonant period of the FPA system. As a result, the maximum heave response for the 10-m waves is close to the wave height, and the maxima occurs when the incident wave approaches T = 17.5 s, which is significantly longer than the natural period of the FPA system. We anticipate that the additional damping is induced by the nonlinearity of waves and the effect of nonlinear interaction between waves and the FPA system, where viscous damping force and the wave-overtopping-induced force become more essential. 6. Conclusions This paper shows our recent efforts in studying the hydrodynamics of the FPA system in survivability and operational conditions. Two scaled models were built and tested in the wave tanks, and both the BEM-based potential flow theory model and the RANS simulations were applied for analyzing the WEC behavior in regular waves. For the survivability condition analysis, we considered the extreme sea states at possible deployment locations, and two extreme wave heights were given in the study. The RANS simulations for LFPA were performed, where the heave, surge, and pitch motions of the system were evaluated, and the simulation results were validated against the measurements from the experiment. Because the FPA system is generally designed to have a natural period in heave close to the wave peak period at the deployment site location, the nonlinear interaction between waves and the FPA system, including wave overtopping and the float re-entering water surface, is expected to be significant when the incident wave period is close to the resonance. To quantify the influence of the nonlinear effects, we evaluated the overall damping ratio of the system in heave. As shown in the study, except for the wave period that is sufficiently longer than the natural period of the FPA system, the damping ratio increases with the incident wave height, particularly for short period waves, where the waves are steep. For the operational condition analysis, after the viscous drag term is included following the Graham’s approach, the results of heave responses and power extraction performance of the TFPA derived from the BEM agree well with the experimental measurements and the RANS simulation results in smaller wave height scenarios, where the viscous drag force induced by flow separation and vortex shedding is the dominant damping source. When the wave height is larger, our predicted results compare well with the RANS simulation results over a broad range of incident wave periods, but exhibit larger values near resonance. This is where wave overtopping and the changes of the cut waterplane area for the WEC occur frequently and their influence on the hydrodynamics of the system become more significant. In this study, the effects of a different mooring configuration and the behavior of structural material are beyond the scope of work, which need to be considered in a practical FPA design. Although we found our model design to be acceptable, the mooring line system and the PTO system have considerable room for improvement. A more simple mooring system can be developed with less cost and a comprehensive PTO system replicating a more practical design so that an appropriate control algorithm can be applied. For the power performance analysis, we estimated the averaged PTO damping coefficient based on the averaged power and the averaged relative motion amplitude, assuming the system is linear. However, the measured PTO force from the wave tank test showed a higher force on the compression stroke Please cite this article as: Q. Xu, Y. Li, Y.-H. Yu et al., Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.03.006.

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than the extension stroke. We believe this nonlinear PTO force was more likely attributed to the nonlinear needle valve PTO design used in the tank test. Therefore, a controllable-motor-generator-connected PTO model that can provide a constant torque-to-speed ratio and linear damping force to the WEC system is recommended in the future wave tank tests, particularly for the purpose of experimental data set development for numerical model validation. Acknowledgments We appreciate the funding support from the U.S. Department of Energy for the experimental tests. We also thank the staff members from MarineiTech for their help in the Berkeley tank test and the Scripps tank staff members for their help in the Scripps tank test. In addition, we would like to acknowledge the National Natural Science Foundation of China for supporting the numerical simulation work. This work was authored, in part, by the National Renewable Energy Laboratory, operated by Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC3608GO28308. Funding provided by the U.S. Department of Energy Office of Energy Efficiency and Renewable Energy Wind Energy Technologies Office. The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for U.S. Government purposes. References Babarit, A., Singh, J., Mélis, C., Wattez, A., Jean, P., 2017. A linear numerical model for analysing the hydroelastic response of a flexible electroactive wave energy converter. J. Fluids Struct. 74, 356–384. Beatty, S.J., Hall, M., Buckham, B.J., Wild, P., Bocking, B., 2015. Experimental and numerical comparisons of self-reacting point absorber wave energy converters in regular waves. Ocean Eng. 104, 370–386. Bjarte-Larsson, T., Falnes, J., 2006. Laboratory experiment on heaving body with hydraulic power take-off and latching control. Ocean Eng. 33 (7), 847–877. Eatock Taylor, R., Taylor, P.H., Stansby, P.K., 2016. A coupled hydrodynamic–structural model of the m4 wave energy converter. J. Fluids Struct. 63, 77–96. Falnes, J., 1999. Wave-energy conversion through relative motion between two single-mode oscillating bodies. J. Offshore Mech. Arct. Eng. 121 (1), 32–38. Goteman, M., 2017. Wave energy parks with point-absorbers of different dimensions. J. Fluids Struct. 74, 142–157. Graham, J.M.R., 2006. The forces on sharp-edged cylinders in oscillatory flow at low keulegan-carpenter numbers. J. Fluid Mech. 97 (2), 331–346. Jenne, D.S., Yu, Y.-H., Neary, V., 2015. Levelized cost of energy analysis of marine and hydrokinetic reference models. In: 3rd Marine Energy Technology Symposium, METS, Washington, DC, United States. Khaligh, A., Onar, O.C., 2010. Energy Harvesting: Solar, Wind, and Ocean Energy Conversion Systems. CRC Press Inc. Li, Y., Yu, Y.-H., 2012. A synthesis of numerical methods for modeling wave energy converter-point absorbers. Renew. Sustain. Energy Rev. 16 (6), 4352–4364. Mackay, E., Challenor, P., Bahaj, A., 2010. On the use of discrete seasonal and directional models for the estimation of extreme wave conditions. Ocean Eng. 37 (5–6), 425–442. Muliawan, M.J., Gao, Z., Moan, T., 2013. Analysis of a two-body floating wave energy converter with particular focus on the effects of power take-off and mooring systems on energy capture. J. Offshore Mech. Arct. Eng. 135 (3), 317–328. NWEI — northwest energy innovations, 2016. URL http://azurawave.com/. OPT — ocean power technologies, 2018. URL http://www.oceanpowertechnologies.com/. Rhinefrank, K., Schacher, A., Prudell, J., Hammagren, E., Stillinger, C., Naviaux, D., Brekken, T., von Jouanne, A., 2010. Scaled wave energy device performance evaluation through high resolution wave tank testing. In: IEE (Ed.), Oceans 2010. IEEE, pp. 1–6. http://dx.doi.org/10.1109/OCEANS. 2010.5664056. Salter, S.H., 1974. Wave power. Nature 720–724. Tao, L., Cai, S., 2004. Heave motion suppression of a spar with a heave plate. Ocean Eng. 31 (5), 669–692. Tester, J.W., Drake, E.M., Driscoll, M.J., Golay, M.W., Peters, W.A., 2005. Sustainable Energy: Choosing Among Options. The MIT Pr. Tyrberg, S., Waters, R., Leijon, M., 2010. Wave power absorption as a function of water level and wave height: Theory and experiment. IEEE J. Ocean. Eng. 35 (3), 558–564. Xu, Q., Li, Y., Lin, Z.L., 2018. An improved boundary element method for modelling a self-reacting point absorber wave energy converter. Acta Mech. Sinica 34 (6), 1015–1034. Yeung, R.W., Peiffer, A., Tom, N., Matlak, T., 2010. Design, analysis, and evaluation of the uc-berkeley wave-energy extractor. In: 29th International Conference on Ocean, Offshore and Arctic Engineering. OMAE. Yu, Y., Lawson, M., Li, Y., Previsic, M., Epler, J., Lou, J., 2015. Experimental Wave Tank Test for Reference Model 3 Floating- Point Absorber Wave Energy Converter Project. Tech. Rep., National Renewable Energy Laboratory (NREL), Golden, CO. Yu, Y.-H., Li, Y., 2013. Reynolds-averaged Navier–Stokes simulation of the heave performance of a two-body floating-point absorber wave energy system. Comput. & Fluids 73, 104–114. Zurkinden, A., Ferri, F., Beatty, S., Kofoed, J., Kramer, M., 2014. Non-linear numerical modeling and experimental testing of a point absorber wave energy converter. Ocean Eng. 78, 11–21. http://dx.doi.org/10.1016/j.oceaneng.2013.12.009.

Please cite this article as: Q. Xu, Y. Li, Y.-H. Yu et al., Experimental and numerical investigations of a two-body floating-point absorber wave energy converter in regular waves. Journal of Fluids and Structures (2019), https://doi.org/10.1016/j.jfluidstructs.2019.03.006.