Parametric design methodology for maximizing energy capture of a bottom-hinged flap-type WEC with medium wave resources

Accepted Manuscript Parametric design methodology for maximizing energy capture of a bottom-hinged flap-type WEC with medium wave resources Yi-Chih Chow, Yu-Chi Chang, Da-Wei Chen, Chen-Chou Lin, Shaiw-Yih Tzang PII:

S0960-1481(18)30370-7

DOI:

10.1016/j.renene.2018.03.059

Reference:

RENE 9928

To appear in:

Renewable Energy

Received Date: 26 August 2017 Revised Date:

11 January 2018

Accepted Date: 22 March 2018

Please cite this article as: Chow Y-C, Chang Y-C, Chen D-W, Lin C-C, Tzang S-Y, Parametric design methodology for maximizing energy capture of a bottom-hinged flap-type WEC with medium wave resources, Renewable Energy (2018), doi: 10.1016/j.renene.2018.03.059. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Yi-Chih Chow1, Yu-Chi Chang1, Da-Wei Chen2, Chen-Chou Lin3, Shaiw-Yih Tzang2* 1

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Department of Systems Engineering and Naval Architecture, National Taiwan Ocean University, Keelung, Taiwan. 2 Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan. 3 Department of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, Keelung, Taiwan

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Parametric Design Methodology for Maximizing Energy Capture of a Bottom-Hinged Flap-Type WEC with Medium Wave Resources

*Corresponding Author: [email protected]

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Abstract

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This paper describes a parametric design methodology for maximizing the capture factor (CF) of a bottom-hinged flap-type wave energy converter (BHF-WEC). The general equation for CF is first derived using the damped-harmonic-oscillator model. Second, correspondences between the general and the 2-D ideal CF equations are established. Then, a scheme is proposed to account for any effects apart from the 2-D ideal modeling with three parameters, which constitute the basis for fitting any data series stemming from either numerical simulations or experiments. Once these three parameters are evaluated from data fitting, the maximum CF and its occurring conditions can be found. In the present study, WEC-Sim simulations are conducted for a series of finite rectangular BHF-WECs with effects of PTO and varying width (B) for two thicknesses (d) under two characteristic wave lengths (L) the medium wave resources that Taiwan possesses. It is found that for B/L smaller than about 0.30, the maximum CF in resonance mode, CFres, is greater than 1.0 and much higher than that not in resonance mode, CFopt, which is always below 1.0. The captured power index in resonance mode, CFres×(B/L), is almost invariant in B/L=0.11~0.30. Several BHF-WEC design guidelines can be deduced from these results.

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Keywords: capture factor, bottom-hinged flap-type WEC, flap width, parametric data-fitting scheme, resonance mode, wave resource

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1. Introduction

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In regard to installation, maintenance, power transmission, etc., types of wave energy converter (WEC) applied to coastal waters of depths between 10 to 20 meters are considered to be more technically feasible and economic than that to deeper waters. Furthermore, major risks due to the extreme typhoon waves should always be taken into account for every aspect of wave energy exploitation in typhoon-prone areas such as Taiwan. Therefore, the bottom-hinged, flap-type WEC (BHF-WEC) like the Oyster developed by the Aquamarine Power [1], as categorized into the type of OWSC (Oscillating Wave Surge Converter), is considered to be suitable for Taiwan regarding its applicability to coastal waters and survivability in typhoon waves by latching its prime mover (flap) to form a submerged body lying flat on the seabed. As a result, the performance of a BHF-WEC becomes the focus in the present study with the objective of developing a practical design methodology to maximize the energy capture of a BHF-WEC. In order to develop a WEC suitable for a specific region, the wave resources of that region must be surveyed beforehand. In Taiwan, potential yearly-averaged wave energy resources larger than 10 kW/m are mostly located offshore the northeast (NE) coasts of Taiwan’s main island [2]. Higher wave energy resources particularly occur in months starting from about October to next March due to dominant NE monsoons [3]. For design and test purposes of the WEC development, characteristic wave conditions in NE monsoon seasons should be first collected and analyzed from local wave stations. Since May, 2012, a WEC test site was set up offshore the seawalls of the National Taiwan Ocean University (NTOU) in Keelung, which is located at northeast Taiwan. At the boundary of the test site, a NTOU owned data buoy at a depth of 40 meters was installed and started to operate [4]. From Oct., 2013 to March, 2014, the first complete continuous wave data for six NE monsoon months were successfully collected. As will be shown below, the typical wave power at Taiwan is below 30 kW/m, i.e. significantly lower than that in Europe. Therefore, the WEC design for a relevant wave energy exploitation at Taiwan must address this issue and focus on capturing wave energy as much as possible. Regarding the flap-type WEC, Evans theoretically discussed the performances of a single top-hinged vertical plate [5], and two of that [6] oscillating in tandem. Whittaker and Folley [7] proposed the design of a sea-bed bottom-hinged wave surge converter with single flap operated in shallow-water depths of 10-15 m. Renzi and Dias [8, 9] focused on the resonance experienced by a BHF-WEC in a wave channel due to the lateral channel walls and further compared the results with that of an open sea. In [9], a complicated linear theory for a 3D BHF-WEC was developed using Green’s integral theorem to characterize its hydrodynamic parameters that the (3D) diffractive effects were shown so fundamental to the performance (capture factor) of the flap-type WEC. Sammarco et al. [10] and Michele et al. [11] analytically investigated an array of flap-gate OWSC and found its natural modes of synchronous resonance. Later, Michele et al. [12] developed a theory for such a flap-gate OWSC array at resonance by solving the diffraction and radiation potentials using elliptical coordinates and Mathieu functions. They obtained an analytical expression for the maximum capture factor that a single BHF-WEC could possibly attain. Sarkar et al.

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[13] investigated the hydrodynamic behaviors of a modular OWSC system with 6 vertical round cylinders of total width of 24 m. Babarit et al. [14] developed wave-towire numerical models to simulate multiple types of WEC including a BHF-WEC based on the dimensions of Oyster 2 in five different field sea states at a depth of 13m. The capture factors obtained in their BHF-WEC simulations range from 0.52 to 0.72. Chang et al. [15] were also concerned with the energy capture of a BHF-WEC, and they analyzed the problem using the potential-flow-based 2-D linear wavemaker theory [16] and SPH (Smoothed Particle Hydrodynamics) simulations. They calculated the theoretical capture factor as a function of the non-dimensional damping coefficient of PTO (Power Take-Off) for three periods of the incident wave and three densities of the flap in a laboratory model scale. Their results show that the performance curves (capture factor vs. PTO’s damping coefficient) of the two cases with flap densities equal to or less than 1000 kg/m3, i.e. the density of water, are almost identical, and significantly higher than that of the case with flap density greater than 1000 kg/m3 (details follow). In this paper, a practical parametric design methodology and evaluations accounting for the effects of flap’s geometrical and inertial parameters and PTO’s damping in full scale with field wave conditions of Taiwan were developed and carried out for maximizing the capture factor of a BHF-WEC. Two-dimensional analytical and SPH modelings were first carried out to illustrate the dampingdominated performance of the WEC. Then, 3-D numerical simulations addressing the geometrical effects of both width and thickness of the flap were performed using the time-domain WEC-Sim (Wave Energy Converter SIMulator) program [17], which was recently developed by NREL (National Renewable Energy Laboratory) to facilitate fast and efficient computations for a complete WEC system as compared to time-consuming CFD schemes. The WEC-Sim directly acquires hydrodynamic parameters such as wave radiation and diffraction functions from the frequencydomain potential-flow Boundary Element Method (BEM) solver, e.g. WAMIT. Then combining both 2-D and 3-D results, practical ranges of the flap’s width and the wave resource in coastal waters in NE Taiwan will be adopted to evaluate the performance of present methodology in the end. The proposed parametric methodology is expected to be universally applicable to coastal waters with either low or high wave resources.

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2. Design Wave Heights and Periods

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Fig. 1 Joint probability of wave height H & period T in northeast monsoon in Keelung (Oct. 2013-Mar. 2014) (iso-wave power contour lines in unit of kW/m).

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For the practical application of a BHF-WEC, design field wave conditions at a targeted shallower-water depth of 10m could be obtained by transforming the wave data of the nearby wave stations. Based on the first set of the continuously collected data at the buoy station for six months in the winter monsoon (Oct.-Mar.), the joint probability of wave height H and wave period T was first calculated and plotted as shown in Fig. 1. Values of probability above 3% were highlighted and those above 7% were further highlighted in dark pink. In addition, the iso-wave power lines in unit of kW/m are also drawn in Fig. 1. It is clearly seen in the same figure that double-peak probabilities occur at periods of about 6.5 s and 8.5 s with wave heights ranging from about 0.75 m to 1.25 m and 1.5 m to 3.0 m, respectively. Both peak zones represent wave powers between 5 kW/m and 10 kW/m. Previously, data from a nearby AWCP station offshore Port of Keelung at a depth of 36 m were also analyzed for four seasons [18]. The analyses had shown characteristic waves in summer (June-Aug.) with heights less than 1.0 m and periods ranging from 5-8 s, while in winter (Dec.Feb.) with heights ranging from 1.0-2.5 m and periods from 7-9 s. The data from both stations gave quite similar results in the overlapping months. Clearly the wave resources in NE coastal waters of Taiwan are relatively smaller than that in

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For the present study, 2 sets of the highest-ranking-probability waves with different periods and associated powers larger than 5 kW/m from Fig. 1 are selected as the characteristic design waves as demonstrated in Table 1. By adopting Goda’s methods [19] with considerations of equivalent deep-water wave and refraction, the design wave heights at the depth of 10 m are further derived as displayed in Table 1. The results illustrate that both wave heights are only slightly smaller at the depth of 10 m for the two characteristic wave periods of 6.5 s and 8.5 s. Thus, associated wave heights ranging from about 1.2 m to 1.7 m shall be adopted in the following analyses.

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Table 1 Characteristic wave conditions in northeast monsoon of Taiwan NTOU Met/Ocean Data Buoy (on-site water depth = 40 m) Waves at On-site On-site Joint probability % Wave power depth = 10 m sea states sea states (highest ranking) (kW/m) H 10 (m) Hs (m) Tp (s) 5.80 (3) 7.50 (2)

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3. Analytical-Numerical Synergistic Modeling

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Fig. 2 Schematic of the analytical model for a bottom-hinged flap-type WEC, where θ ( t ) denotes the pitching angle, d the thickness, h the water depth, Ai the amplitude of the incident wave, and PTO a torque-based Power Take-Off device incorporated with the hinge. As shown in Fig. 2, a flap hinged on the bottom is incorporated with a PTO, which can be in general modeled as a torque-based device. A plane wave with frequency ω is incident to the flap, making it oscillate to form a sinusoidal pitching angle θ ( t ) with the same frequency, i.e.

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θ ( t ) = Re  Aθ e−iωt 

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where Aθ denotes the complex amplitude of θ ( t ) . Assuming that the amplitude of the

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incident wave ( Ai ) and the induced pitching angle ( Aθ ) are small, and the fluid is ideal, the flap motion can be modeled in the frequency domain as a damped harmonic oscillator [9, 20]

( −ω

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I − iω ν PTO + C ) Aθ = T p

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where I, C , ν PTO and T p respectively denote the flap’s moment of inertia and buoyancy torque, the damping coefficient of the torque exerted by the PTO, and the

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complex hydrodynamic pressure torque experienced on the flap which can be decomposed into

T p = T E + ω 2 µ A + iων R

(3)

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where T E is the complex exciting torque, µ A is the added torque due to acceleration,

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and ν R is the wave radiation damping. Substitute Eq. (3) into Eq. (2) and obtain

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Aθ =

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C − ω 2 ( I + µ A )  + ω 2 (ν R + ν PTO ) 2

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It can be shown that the average power P transmitted to the PTO over a period T is ([21])

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P=

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The so-called capture factor (CF) is defined as

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CF =

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(6)

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ω 2ν PTO 2

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ν PTO (θ&) dt =

where Pi is the incident wave power within the flap’s width, B; ρw is the water density; g is the acceleration of gravity; C g is the group velocity. Combining Eqs. (4), (5) and (6) and performing some algebraic manipulations, CF can be expressed as     2 *   TE  2ν PTO CF =    2 2  2 ρ w gν R Ai Cg B   C − ω 2 ( I + µ )  2    A *  + (1 +ν PTO )   ων R   

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∫

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(7)

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where

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* ν PTO =

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Here we arrive at a general equation for CF, i.e. Eq. (7) can be applied to cases of either 3-D or 2-D configurations. Next, we start with a 2-D analytical modeling for a BHF-WEC and validate it with a more realistic 2-D SPH modeling. Then, this 2-D

ν PTO νR

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analytical modeling is used as a baseline performance of the WEC to extrapolate into the 3-D configurations with 3-D numerical simulations combined with Eq. (7).

3.1 Two-Dimensional Analytical Modeling

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Evans [5] and Mei [22] started the groundbreaking, theoretical works for extraction of wave energy by a 2-D floating body. They concluded the maximum capture factor of wave energy to be 50% if the body motion being axis-symmetrical. A series of continuing efforts made by, for example, Falnes and Budal [23] and Evans [24], contributed to the theoretical background of buoy-type WECs. Chang et al. [15, 25] had developed a 2-D analytical model for the BHF-WEC as briefly described below. As shown in Fig. 2 with d → 0 , an infinitely wide (2-D) and infinitesimally thin flat flap-type WEC is hinged on the bottom with a 2-D PTO, whose damping coefficient (denoted as fPTO) indicates the amount of torque delivered by the PTO per unit angular velocity per unit flap width (i.e. unit = [N·s]) and is assumed to be constant in accordance with the damped harmonic oscillator model adopted in this paper. Assuming the amplitude of the incident wave ( Ai ) and the induced pitching

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angle ( θ ( t ) ) are small, and the fluid is irrotational and incompressible, potential flow theory can be applied to this problem and all the formulations can be linearized. Adopting the wavemaker theory [16], this boundary-value problem is solved to yield [15, 25]

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CF =

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The terms in Eq. (9) can be interpreted as non-dimensional mechanical impedances: the impedance associated with progressive waves, W1, representing the damping effect of wave radiation; the impedance associated with standing waves, W2, representing the effect of the inertia and stiffness of the wave field; the impedances, P1 and P2, respectively representing the effects of the inertia and the stiffness of the PTO; the impedance, P3, representing the damping effect of the PTO that takes energy off the flap. Finally, the impedances, F1 and F2, respectively represent the effects of the inertia (moment of inertia) and the hydro-mechanical stiffness (restoring moment) of the flap body. Among these impedances, both W1 and P3 usually dominate the others. To help elucidate their physical meanings, they are expressed as follows

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( W2 + F1 + F2 + P1 + P2 ) + ( W1 + P3 )

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 0 (1 + z * )  ∫ cosh ki* (1 + z * ) dz *  2 * * *  −1 2  k sinh k − cosh k + 1 ( ) i i i  = W1 =  0 4 * 2 * * * ki ∫ cosh ki (1 + z ) dz ( ki* ) ( 2ki* + sinh 2ki* ) -1

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P3 =

f PTO 8ρ wω h 4

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where ki* = kih and z* = z/h are the non-dimensional wave number of the incident wave and z coordinate, respectively.

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In Eq. (9), P3 can be normalized with W1 to form a 2-D version of the CF equation in Eq. (7) as follows

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where P3* =

P3  1   f PTO  =   W1  8ρ w h 4   ω W1  (13)

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C − ω2 ( I + µA )

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    2P3*   C F = (1)  2    W2 + F1 + F2 + P1 + P2  + (1 + P* )2   3   W1    (12)

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W2 + F1 + F2 + P1 + P2 W1

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Assuming the 2-D (damping-less) resonance of the BHF-WEC is attained, i.e. (W2 + F1 + F2 + P1 + P2) vanishes, then

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CF =

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As shown in Fig. 3(a), C F ( P3* ) is a single-peak function starting from zero at P3* = 0 and asymptotically approaching zero as P3* approaches infinity. Equation (15) represents a baseline performance of the BHF-WEC that will be further used as a

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functional for fitting data obtained from the 3-D modeling later on (details follow). This function has an interesting property that any P3* and its reciprocal corresponding to the same CF value. Therefore, the axis of supposedly infinite range of P3* in Fig.

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3(a) can be condensed into a compact range that 0 ≤ P3* or ( P3* ) ≤ 1 as shown in Fig.

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3(b). It can be further found that the maximum CF is equal to 0.5 at P3* = 1, i.e. the

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state of impedance matching where W1 = P3 is reached [15, 25], and P3* = 1 can be

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termed as the optimum P3* . Also, it is clearly evident that CF varies with P3* much faster in 0 ≤ P3* < 1 than in P3* > 1 . It can be readily shown that the absolute value of

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−1

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advised from this analytical result that the P3* variation should be designed or

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controlled to avoid the P3* range whose values are smaller than the optimum value, since CF drops very quickly in this P3* range. Considering the shallow-water (SW) condition where

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g ( ki * )

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the average slope of the CF curve in 0 ≤ P3* < 1 (i.e.  C F (1) − C F ( 0 )  (1 − 0 ) = 1 2 ) is nine times that in 1 < P3* ≤ 2 (i.e.  C F (1) − C F ( 2 )  ( 2 − 1) = 1 18 ). Therefore, it can be

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, ki* sinh ki* ≈ ( ki* ) , cosh ki* ≈ 1 + 2

1 * 2 ( ki ) 2

ω2 ≈

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and combining Eqs. (16) and (10) to calculate ωW1 , we obtain

h (16)

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g ( ki h

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which indicates that (ω W1 )SW is independent of ω . With a reasonable assumption

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(ω W1 )SW ≈

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1 * 2   * 2 ( ki ) − 1 − 2 ( ki ) + 1 1 ⋅ = 4 16 ( ki * ) ( 2 ki * + 2 k i * )

that fPTO is not a function of ω , it can be drawn from Eq. (13) that P3* is independent of ω under the shallow-water condition. This result is quite remarkable in that the period (or wavelength) variability of the incident wave under the shallow-water condition cannot affect P3* , and hence, C F ( P3* ) . In other words, the damping control

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(and/or the product selection) for a BHF-WEC’s PTO doesn’t need to respond to the incident wave variation since the BHF-WEC is a shallow-water or coastal application in nature. In this paper, the wave-period variation is from 6.5 s to 8.5 s and the water 0.0609 ≤ ωW1 ≤ 0.0616 depth 10 m, resulting in and

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(ω W1 )SW = (1 16 )

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g 10 = 0.0619 (Eq. (17)), i.e. ω W1 is almost a constant and very

close to (ω W1 )SW . Combining this result and Eqs. (13) and (15),

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(f

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f wave = 8ρ w h 4 (ω W1 )SW (18)

(

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And CF f PTO peaks at 0.5 with f PTO = f wave = 4.95 × 106 ( N ⋅ s ) .

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2f wave ⋅ f PTO

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C F ( f PTO ) =

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range of P3* , and (b) CF ( P3* ) in a compact range of 0 ≤ P3* or ( P3* ) ≤ 1 −1

When considering both inertial and damping effects, Eqs. (9) or (12) should be used. As described in [15] and shown in Fig. 4, three performance curves corresponding to three densities (555 kg/m3, 1000 kg/m3, and 7000 kg/m3) of a rectangular flap were calculated using Eq. (9). It is clearly evident that the curves for flap densities 555 kg/m3 and 1000 kg/m3 at each wave period are almost the same with the maximum capture factors of about 0.5, i.e. damping effects dominate that of flap inertia. But for much larger density of 7000 kg/m3, the BHF-WEC has the maximum capture factors much lower than that of the other two densities. As a result, the BHF-WEC with flap density of 555 kg/m3 is adopted for the following analyses.

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(b) Fig. 3 Two-dimensional analytical capture factor plotted as (a) CF P3* in a finite

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Fig. 4 Theoretical capture factor vs. dimensionless damping coefficient of PTO, for three wave periods 1.11 s, 1.33 s and 1.57 s, and three flap densities of 555 kg/m3, 1000 kg/m3, and 7000 kg/m3. [15]

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WFz sin θ

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uuuuv uuuuv In Fig. 5, H 0 denotes the hinge; WFx and WFz are the resultant forces exerted on the flap by waves in the horizontal and vertical directions, respectively; TPTO is the toque exerted by the PTO; mg denotes the weight of the flap. The detailed numerical schemes for the operations of a BHF-WEC and that with a PTO are referred to [15] and [27], respectively. For considering the prototypal OWSC (in full scale) in the field, the 2-D numerical wave flume with a length of 300 m and a water depth of 10 m is integrated with a piston-type wavemaker at the upstream boundary and a beach of 1:5 slope at the downstream boundary for reducing reflected waves, as shown in Fig. 6. The OWSC system includes a flap of density of 555 kg/m3 hinged at x = 100 m and above the bottom at z = 0.5 m with a fixed thickness of 2 m and height of 10 m. With particle spacing resolution of 0.4 m, the initial setting of the model includes 98 assigned boundary particles for the wavemaker and 64 for the flap, and about 13,000 fluid particles. The empirical coefficient ( α ) of 0.05 for the artificial viscosity term is adopted [28].

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Fig. 5 Definition schematic for a bottom-hinged flap-type WEC [15, 27].

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WFx cos θ + mg sin θ

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Fig. 6 Schematic for the SPH simulation domain (not to scale).

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derived from the designated P3* and wave condition using Eq. (5). The CF (=EPTO/Ei)

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1 ω with the total incident wave energy ( Ei =  ρ w gH i2  8  ki

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captured energy ( E PTO = ∫ TPTO

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the angular velocity.

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Table 2 Comparisons of 2-D SPH and analytical results at different wave conditions. Analytical SPH fPTO ID H T E EPTO i i ( N ⋅ s) CF (%) CF (%) (m) (s) (J/m) (J/m) 20 Case 2.11× 9.24× 1.10 8.46 22.8 6.28×105 * ( P3 = 0.127 ) A-1 104 104 40 Case 1.10× 4.15× 1.19 8.52 37.8 1.89×106 * 5 ( P3 = 0.383 ) A-2 10 104 50 Case 1.13× 5.59× 1.20 8.53 49.4 4.92×106 * ( P3 = 1.000 ) A-3 105 104 40 Case 1.20× 4.00× 1.25 8.48 33.3 1.29×107 * ( P3 = 2.618 ) A-4 105 104 Case 1.07× 2.18× 20 1.40 6.89 20.4 6.24×105 5 B-1 10 104 Case 8.70× 3.70× 40 1.27 6.86 42.6 1.87×106 4 B-2 10 104 Case 7.29× 3.60× 50 1.17 6.80 49.3 4.92×106 B-3 104 104 Case 6.99× 2.45× 40 1.16 6.75 35.0 1.28×107 B-4 104 104 Case 7.53× 1.91× 20 1.36 5.95 25.3 6.24×105 4 C-1 10 104 Case 6.97× 2.88× 40 1.35 5.80 41.3 1.84×106 4 C-2 10 104 Case 5.57× 2.99× 50 1.20 5.84 53.9 4.84×106 C-3 104 104

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0

1  2k i h    1 +   ) and the  2  sinh 2ki h   ( t )θ&( t )dt ) in a wave period by [15], where θ&denotes

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For comparing values of the simulated capture factor of the WEC with the analytical results, three analytical CF values and their corresponding P3* : 0.2 ( P3* = 0.127 ), 0.4 ( P3* = 0.383 ) , 0.5 ( P3* = 1.000 ) and 0.4 ( P3* = 2.618 ) (as shown in Fig. 3 and Table 2) are adopted. Incident wave periods for the present SPH simulations are set to be around 8.5 s (Case A), 7.0 s (Case B), and 6.0 s (Case C) with wave heights ranging from 1.1 m to 1.4 m, according to the aforementioned typical ranges of data (Table 1) acquired from the AWCP and Data Buoy of the wave stations offshore Keelung. The PTO’s damping coefficient f PTO of each case is

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5.70

5.21× 104

1.78× 104

With Table 2, it is clearly evident that the simulated CF values generally agree with their analytical counterparts for the same wave conditions. For the analytical CF value of 0.4, the simulated CF values with smaller P3* agree better than that with larger P3* . Also all these simulated CF values with P3* = 2.618 > 1.000 (Cases A-4, B-4, and C-4) are less than 0.4, i.e. their analytical counterpart. This deficit may be attributed to the insufficient resolution of SPH for resolving the much slower and smaller movement of the flap caused by the much higher P3* . Considering the fact that the analytical modeling neglects the fluid viscosity and the flap’s thickness whereas these two parameters are included in the SPH modeling, the agreement between the analytical and SPH results as shown in Table 2 clearly suggests that the effects of real fluid (i.e. viscous effect) and the flap’s thickness don’t play a significant role in the 2D BHF-WEC problems. However, as will be shown in the following analyses, the flap’s thickness as well as its width has significant effects on the performance of the WEC of 3-D configuration.

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3.3 Three-Dimensional Numerical Modeling and Data Fitting

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&& =F + F + F + F + F + F mX ext rad PTO v B m

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The WEC-Sim generally solves the dynamic equation of a floating body

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(19)

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&& are respectively the mass (moment of inertia) and the where m and X acceleration (angular acceleration) of the floating body, Fext and Frad respectively the exciting and the radiation forces (torques) from the wave, FPTO the force (torque) exerted by the PTO, Fv the viscous force (torque) of the fluid, F B the effective buoyancy force (torque), and Fm the mooring force (torque). Figure 7 illustrates the operating framework of WEC-Sim. It is divided into three blocks in series from left to right: User Inputs, WEC-Sim Executable, and PostProcessing Modules. Input data include device specs and geometry, wave spectra, and time series of ocean current if any. The WEC-Sim Executable includes two module blocks: Pre-Processing Modules and Time-Domain Simulation. The PreProcessing Modules employ a BEM (Boundary Element Method) solver to perform added-mass and radiation damping matrix computations in the frequency domain. In this paper, we use WAMIT (Wave Analysis MIT) to compute these matrices as the inputs to the Time-Domain Simulation for solving Eq. (7) in the time domain. Finally, the statistical results and visualizations about the motion and power output of the WEC investigated are generated by the PostProcessing Modules. WEC-Sim is constructed on Simulink/MATLAB.

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As described in Section 2, the periods (T) of incident wave chosen for WEC-Sim simulations are 6.5 s and 8.5 s, which correspond to wavelengths (L) of 54.12 m and 76.24 m with water depth of 10.00 m, respectively, with a fixed wave height of 1.40 m. We then use the wavelength to scale the width (B) and the thickness (d) of the flap (as shown in Fig. 8) and to respectively form two non-dimensional parameters: B/L and d/L. We compute CF ( P3* ) with various (B/L)’s where B varies from 5 m to 40 m with increment of 5 m, at either of (d/L)1 = 0.026 and (d/L)2 = 0.032 (as used in [7]), corresponding to d = 1.42 m (L = 54.1 m) or 2.00 m (L = 76.2 m), and d = 1.73 m (L = 54.1 m) or 2.44 m (L = 76.2 m), respectively. If a thickness is chosen between 1.73 m and 2.00 m, i.e. the middle of the thickness setting of our computations, the variability of the results due to the d/L variation can be reasonably accounted for with (d/L)1 and (d/L)2. Therefore, for the following analyses, 1.73 m < d < 2.00 m. Therefore,

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Fig. 7 Operating framework of WEC-Sim [17]

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Fig. 8 Schematic of the 3-D BHF-WEC’s flap geometry

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Based on Eq. (14), i.e. the correspondences between the general hydrodynamic model of the BHF-WEC (finite width and thickness, arbitrary shape, viscous fluid effect, etc.) and the 2-D, ideal one (infinite width and infinitesimal thickness, rectangular shape, ideal fluid effect), a scheme to fit data stemming from either numerical simulations or experiments with parameterizations using the 2-D, ideal hydrodynamic model as described in Section 3.1 can be proposed. Therefore, in order to account for the 3-D effects and find the mechanical behaviors of the BHF-WEC, the data generated from the WEC-Sim simulations conducted for a series of finite rectangular BHF-WECs with the inviscid-fluid assumption will be fitted using the proposed scheme as described below: 1. Assume that the variables in the 3-D configurations (e.g. ν *pto ) can be decomposed into parts of 3-D and/or non-linear effects (e.g. finite width and thickness) and the 2-D variables (e.g. P3* ) in separable forms, i.e.

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518

B d

ν *pto = f  ,  ⋅ P3*  L L

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C − ω2 ( I + µA )  B d   = q ,  ων R  L L  

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(22)

where q denotes the function expressed in terms of non-dimensional shape parameters.

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(21)

2. Adjust the inertial parameters (e.g. density) of the flap close to the state where the 2-D resonance of the flap is achieved. Therefore, 2

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(20)

where f and g denote the functions expressed in terms of non-dimensional shape parameters.

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TE B d = g  ,  ⋅1 2 2 ρ w gν R Ai Cg B  L L

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3. Substitute Eqs. (20)~(22) into Eq. (7) and perform some algebraic manipulations, yielding

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CF =

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where

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2α1 P3*

α 3 + (α 2 + P

)

* 2 3

(23)

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α1 =

g 1 q ,α 2 = ,α3 = 2 f f f

(24)

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In particular, α3 represents the inertial effect of the flap on CF. As a result, the 3-

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D resonance of the flap can be achieved when α3 vanishes.

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(i.e. B L → ∞ ) should bring the CF back to the 2-D analytical solution as shown in Eq. (15). In light of the limiting behaviors just described, one can expect that at a fixed d/L,

α1 , α 2 → 1, α 3 → 0 as B L → ∞ α1 , α 2 , α 3 → 0 as B L → 0

CFopt =

α1 α 2 + α 22 + α 3

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when P3* is optimized as

(P )

* 3 opt

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(26)

= α 22 + α 3

(27)

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(25)

It can be shown for Eq. (23) that the maximum CF without attaining to the 3-D resonance (i.e. α3 ≠ 0 ), C Fopt , is

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4. Perform WEC-Sim simulations to obtain the CF values for series of B/L and P3* at a specific d/L, and use Eq. (23) to find the best fits for the values of α1 ~ α3 at a specific B/L and that d/L. One can envisage on one end of the spectrum that an infinitesimally narrow flap-type WEC neither captures the wave energy nor interferes the wave field even when it is in motion through the water. In other words, when B L → 0 , there should be no radiation damping and added-mass effects or mechanical impedances, and CF → 0 . On the other end, an infinitely wide flap-type WEC

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If certain ways to make α3 vanish for attaining to the 3-D resonance are found,

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then the maximum CF at the 3-D resonance (i.e. α3 = 0 ), CFres , is

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CFres =

α1 2α 2

(28)

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when P3* is tuned as

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Equation (28) shows the ideal maximum CF without considering any loss due to PTO, viscosity of water around the WEC, etc.

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(P )

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4. Results

( )

Figures 9 and 10 show the C F P3* results of 3-D WEC-Sim simulations and

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their least-square fittings (Eqs. (20) ~ (24)) at sixteen (B/L)’s and at two (d/L)’s: (d/L)1 = 0.026 and (d/L)2 = 0.032, respectively. The results for both (d/L)1 and (d/L)2 are quite similar to each other, as comparing Figs. 9 and 10. It is evident that the C F increases quickly with P3* until the maximum C F ( C Fopt ) is reached, and afterwards decreases slowly toward zero, as expected from the 2-D analytical results as shown in Fig. 3. It is also evident on one hand that with B/L exceeding certain value (in-between 0.131 and 0.185 for (d/L)1 as shown in Fig. 9(a), and 0.092 and 0.131 for (d/L)2 as shown in Fig. 10(a)), C Fopt exceeds 0.5 (i.e. the 2-D analytical CF max ) due to the wave diffraction effect which is of 3-D nature. On the other hand, with B/L smaller than that certain value, the very wave diffraction effect would otherwise inhibit the capture of wave energy. For B/L = 0.739 (Figs. 9(d) and 10(d)), CF P3* is very close to its 2-D analytical

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result already (i.e. B L → ∞ ). It can also be found that the series of the

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maximum CF , CFopt ( B L ) , occur as P3*

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( )

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opt

approaches 1.0, i.e. the 2-D analytical

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(P )

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roughly stay within 0.7~0.8, while P3*

* 3 opt

. From B/L = 0.328 to 0.462, it can be observed that the values of C Fopt

( )

opt

is not far from 1.0.

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Fig. 9 CF P

(d) results of 3-D WEC-Sim simulations (symbols) and their least-square

fittings (lines) with Eqs. (20)~(24) at d/L = 0.026 in ranges of (a) 0.063 ≤ B L ≤ 0.185 , (b) 0.197 ≤ B L ≤ 0.328 , (c) 0.369 ≤ B L ≤ 0.462 , and (d) 0.525 ≤ B L ≤ 0.739 . The

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(( P ) ) (Eq. (26)) with P

633

diamond symbols represent CFopt = C F

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which are obtained using Eq. (27), at B/L = 0.131, 0.277, 0.459 and 0.647.

* 3 opt

( )

equal to P3*

opt

,

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(c)

( ) * 3

Fig. 10 CF P

(d) results of 3-D WEC-Sim simulations (symbols) and their least-square

fittings (lines) with Eqs. (20)~(24) at d/L = 0.032 in ranges of (a) 0.063 ≤ B L ≤ 0.185 , (b) 0.197 ≤ B L ≤ 0.328 , (c) 0.369 ≤ B L ≤ 0.462 , and (d) 0.525 ≤ B L ≤ 0.739 . The

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(( P ) ) (Eq. (26)) with P

( )

651

diamond symbols represent CFopt = C F

652

which are obtained using Eq. (27), at B/L = 0.131, 0.277, 0.459 and 0.647. Figure 11 shows the least-square fitting results for α1 ( B L ) ~ α 3 ( B L ) at sixteen (B/L)’s formed by B (flap width varying from 5m to 40m with increment of 5m) divided by L (wavelength of 54.12m or 76.24m), and at two (d/L)’s: (d/L)1 = 0.026 and (d/L)2 = 0.032, respectively. For both (d/L)1 and (d/L)2, α1 ( B L ) ~ α 3 ( B L ) similarly exhibit single-peak curve forms. It is clearly evident that their asymptotic behaviors with B/L follow Eq. (25). From B/L = 0.00 to about 0.31, α1 ( B L ) ~ α 3 ( B L ) increase with B/L. Peak values of α1 , α 2

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and α3 are about 1.40, 0.85 and 0.55 at B/L = 0.42, 0.44 and 0.31, respectively. It is shown in Fig 12 that the calculated results of the maximum capture factor CF and the captured power index C F × ( B L ) (representing the total power captured under CF) are quite similar for both values of d/L. The differences are generally less than 2~3 %. Thus for discussion purposes, typical values in Fig. 12(b) for the case of d/L = 0.032 shall be adopted. It is noted that the maximum value of C Fopt is about 0.80, which is higher than the 2-D analytical CFmax (i.e. 0.5), and occurs at B/L = 0.34. This result is very similar to what was obtained from the analysis in [9] for the Oyster. Furthermore, CFopt × ( B L ) monotonically increases with increasing B/L, indicating that the wider the flap, the greater the captured power. However, in cases of resonance the maximum capture factor CFres and the

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opt

captured power index CFres × ( B L ) exhibit very different trends. The value of CFres becomes amplified to reach as high as about 3.71 at B/L = 0.074 and rapidly decreases with further increasing B/L. But for B/L smaller than about 0.30, CFres is still greater than 1.0 and much higher than C Fopt , which is always below 1.0. For B/L

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equal to P3*

in between about 0.11 and 0.30, CFres × ( B L ) varies only slightly around the value of 0.33; the minimum value within this B/L range is 0.316 and occurs at about B/L = 0.18. Thus for B/L greater than 0.18, CFres × ( B L ) increases with increasing B/L as

EP

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* 3

C Fopt × ( B L ) does, but not as rapidly. Meanwhile, CFres in the B/L range from 0.11

to 0.30 monotonically decreases from 3.02 to 1.12, which is still more than twice the 2-D analytical CFmax. From design perspective, the fact that in real sea states a BHFWEC easily encounters waves with lengths longer than the design wavelength results in decrease of its B/L (or d/L). Thus, a design guideline to keep the variation of B/L in between 0.11 to 0.30 can be advised since in this B/L range CFres × ( B L ) , i.e. the captured power index during the mode of resonance, varies slightly. The 3-D theory for the flap-gate OWSC developed by Michele et al. [12] yields

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* 3 opt

−1

an analytical expression for the maximum capture factor: CFmax = π ( B L )  (expressed in accordance with the present paper). As shown in Fig. 12(b), the C Fmax and CFres curves almost collapse onto each other in the B/L range

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approximately from 0.11 to 0.30, resulting in values of CFres × ( B L ) almost identical

690

to CFmax × ( B L ) = π −1 ≈ 0.32 in that B/L range. However, the disparity between

691

C Fmax and CFres increases excessively as B/L decreases since CFmax → ∞ and

692

CFres → 0 as B L → 0 . This unphysical trend of C Fmax is due to the breakdown of

695

asymptotic value should be 0.5. In summary, CFres is in an excellent agreement

696

with C Fmax within B/L = 0.11~0.32, indicating the high fidelity of the analyses presented in this paper. The almost invariant CFres × ( B L ) in this particular range of B/L = 0.11~0.30 also suggests that for a fixed wave period (length), a BHF-WEC with narrower width could still capture similar amount of power to that of those with wider width. Once the mode of resonance being achieved, the advantage of a narrower BHF-WEC just mentioned could help motivate more installations of BHF-WEC. In Taiwan’s coastal waters, this means installations of BFH-WECs with widths of 5.95 m to 22.9 m for waves with periods in between 6.5 s and 8.5 s at a water depth of 10 m. For example, for the design wave period of 6.5 s (i.e. the design wavelength of 54.1 m), the width B could be designed at 9.74 m as the minimum CFres × ( B L ) occurs at B/L = 0.18. Once the BHF-WEC encounters longer waves up to the wave period of 8.5 s, its B/L would reduce to a value still greater than 0.11. As a result, the BHF-WEC could even capture slightly higher wave power.

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one assumption (i.e. d = B ) employed by the 3-D theory. Toward the other extreme as B L → ∞ (i.e. BHF-WEC becomes 2-D), CFmax → 0 whereas the

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(a)

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(b) Fig. 11 Least-square fitting results for α1 ( B L ) ~ α 3 ( B L ) at (a) d/L = 0.026, and (b) d/L = 0.032.

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Fig. 12 Curves of C Fres (CF

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(b) maximized by attaining to 3-D resonance),

CFres × ( B L ) (representing the total power captured under CFres ), C Fopt (CF

optimized by only adjusting PTO’s damping), CFopt × ( B L ) (representing the

724

total power captured under C Fopt ), C Fmax (the maximum CF obtained from the 3-D

725

theory developed by Michele et al. [12]) and CFmax × ( B L ) (representing the total

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power captured under C Fmax ) at (a) d/L = 0.026, and (b) d/L = 0.032.

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5. Conclusions

EP

This paper proposed a parametric design methodology that aimed to maximize the capture factor (CF) for designing a bottom-hinged flap-type WEC (BHF-WEC). The damped-harmonic-oscillator model was used to derive the general equation for CF. Three correspondences involving important parameters such as the PTO’s damping were established between the general and the 2-D ideal CF equations, leading to forming three parameters ( α1 ~ α 3 ) to account for effects apart from the 2-D ideal modeling. In this paper, these three parameters were then evaluated by fitting the data of WEC-Sim simulations with characteristic wave climates in coastal waters of northeast Taiwan. The analyses of field data had shown characteristic waves with periods of 6.5s and 8.5s, corresponding to wavelengths of 54.1m and 76.2m, respectively at an installation water depth of 10m. From previously derived 2-D theoretical formulations, it first illustrated that the CF was mainly related to a single parameter P3* consisting of the damping effects of wave and PTO. The resulting peak value of 0.5 at P3* = 1 indicated equal mechanical impedances by waves and the PTO. The analytical results were also confirmed with

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ACKNOWLEDGEMENTS

This study was sponsored by the National Science Council (now the Ministry of Science and Technology) of Taiwan with two grants of the National Science and Technology Project of Energy (NSTPE): NSC 101-3113-E-019-002, NSC 101-3113P-019-002.

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numerical simulations by an in-house 2-D SPH model illustrating P3* being independent of the wave conditions in coastal waters. Through the knowledge learned from the 2-D ideal modeling, the maximization of the CF for a 3-D BHF-WEC was derived through a parametric design methodology and further compared with the WEC-Sim modelling. The methodology involves three selected non-dimensional parameters consisting of effects of the flap’s shape and the PTO damping. The three parameters were evaluated for optimized and resonance conditions with different non-dimensional width B/L and thickness d/L by incident characteristic wavelength L. The simulated data by WEC-Sim clearly demonstrated the maximum values of CF in 3-D domains being able to be several times greater than 0.5. This is obviously due to the diffraction effects, which were certainly missing in a 2-D domain. For fixed thickness values of d/L = 0.026 & 0.032, it is found that for B/L smaller than about 0.30, the maximum CF in resonance mode, CFres, is greater than 1.0 and much higher than that in optimized mode, CFopt, which is always below 1.0. The captured power index in resonance mode, CFres×(B/L), is almost invariant in B/L = 0.11~0.30. These results have given quite practical guidelines for designing a BHF-WEC in coastal waters of Taiwan. That is, at a typical depth of 10m for waves with periods ranging from 6.5 to 8.5 s, a 10 m wide BHF with thickness of 1.73 m to 2.0 m can operate in resonance with an equivalent total captured power and with values of CF higher than 1.73 as those for wider BHF to an extent of 20 m. For practical considerations to approach resonance, it is recommended that the design criteria can be realized in two ways. The first way is by adjusting the flap’s center of mass to a very low position, even below the hinge location. The second one is by adjusting the restoring force by connecting a spring to the flap with a properly designed spring stiffness constant. In addition, the viscosity was not considered in the present paper, but it can be carried out through the presently proposed parametric methodology to assess its effects. Further results by taking into accounts of these effects shall be given in the subsequent paper.

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References

[1] M. Folley, T. Whittaker, J. van't Hoff, The design of small seabed-mounted bottom-hinged wave energy converters, Proceedings of the 7th European Wave and Tidal Energy Conference, Porto (Portugal), 2007. [2] P.H. Hsu, C.W. Yen, Prospects for marine energy development in Taiwan, Physical bimonthly 29 (2007) 718-726. (in Chinese)

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[3] S.Y. Tzang, C.C. Wang, D.W. Chen, J.Z. Yang, C.M. Hsieh, J.H. Chen, Wave Energy Resources on Coastal Waters of Northeast Taiwan, 4th International Conference on Ocean Energy, Dublin (Ireland), 2012. [4] S.Y. Tzang, Y.L. Chen, T.W. Hsu, D.W. Chen, C.C. Wang, C.C. Lin, Numerical Assessment on Wave Energy Resources in Coastal Waters of Northeastern Taiwan, 33rd International Conference on Ocean, Offshore and Arctic Engineering, San Francisco (U.S.), 2014. [5] D.V. Evans, A theory for wave-power absorption by oscillating bodies, J. Fluid Mech. 77 (1976) 1-25. [6] D.V. Evans, A theory for wave-power absorption by two independently oscillating bodies, J. Fluid Mech. 90 (1979) 337-362. [7] T. Whittaker, M. Folley, wave surge converters nearshore oscillating and the development of Oyster, Philos. T Roy. Soc. A 370 (2012) 345-364. [8] E. Renzi, F. Dias, Resonant behaviour of an oscillating wave energy converter in a channel, J. Fluid Mech. 701 (2012) 482-510. [9] E. Renzi, F. Dias, Hydrodynamics of the oscillating wave surge converter in the open ocean, Eur. J. Mech. B/Fluids 41 (2013) 1-10. [10] P. Sammarco, S. Michele, M. d’Errico. Flap gate farm: from Venice lagoon defense to resonating wave energy production. Part 1: natural modes. Appl. Ocean Res. 43 (2013) 206–213. [11] S. Michele, P. Sammarco, M. d’Errico, E. Renzi, A. Abdolali, G. Bellotti, F. Dias. Flap gate farm: from Venice lagoon defense to resonating wave energy production. Part 2: synchronous response to incident waves in open sea. Appl. Ocean Res. 52 (2015) 43–61. [12] S. Michele, P. Sammarco, M. d’Errico. Theory of the synchronous motion of an array of floating flap gates oscillating wave surge converter. Proc. R. Soc. A 472 (2016): 20160174. [13] D. Sarkar, K. Doherty, F. Dias, The modular concept of the Oscillating Wave Surge Converter, Renew. Energy 85 (2016) 484-497. [14] A. Babarit, J. Hals, M.J. Muliawan, A. Kurniawan, T. Moan, J. Krokstad, Numerical benchmarking study of a selection of wave energy converters, Renew. Energy 41 (2012) 44-63. [15] Y.C. Chang, D.W. Chen, Y.C. Chow, S.Y. Tzang, C.C. Lin, J.H. Chen, Theoretical analysis and SPH simulation for the wave energy captured by a bottom-hinged OWSC, J. Mar. Sci. Technol.-Taiwan 23 (2015) 901-908. [16] R.G. Dean, R.A. Dalrymple, Water Wave Mechanics for Engineers and Scientists, World Scientific, 1991. [17] Y.H. Yu, M. Lawson, K. Ruehi, C. Michelen, Development and demonstration of the WEC-Sim wave energy converter simulation tool, Proceedings of the 2nd Marine Energy Technology Symposium, Seattle (U.S.), 2014. [18] S.Y. Tzang, J.H. Chen, J.Z. Yang, C.C. Wang, The analysis on wave power potentials on the northeast coastal waters of Taiwan, Proceedings of the 11th International Conference on Fluid Control, Measurements and Visualization, Keelung (Taiwan), 2011. [19] Y. Goda, Irregular wave deformation in the surf zone, Coast. Eng. in Jpn. 18 (1975) 13-26.

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[20] C.C. Mei, M. Stiassnie, D.K.-P. Yue, Theory and Application of Ocean Surface Waves, World Scientific, USA, 2005. [21] J. Falnes. Ocean Waves and Oscillating Systems. Cambridge University Press (2002). [22] C.C. Mei, Power extraction from water waves, J. Sh. Res. 20 (1976) 63-66. [23] J. Falnes, K. Budal, Wave power conversion by point absorbers, Nor. Marit. Res. 6 (1978) 2-11. [24] D.V. Evans, Arrays of three-dimensional wave-energy absorbers, J. Fluid Mech. 108 (1981) 67-88. [25] Y.C. Chang, Y.H. Lee, Y.C. Chow, C.C. Lin, S.Y. Tzang, J.H. Chen, Mechanical impedance effects on the capture factor of OWSC, J. Taiwan Soc. Nav. Archit. Mar. Eng. 33 (2014) 183-192. (in Chinese) [26] D.W. Chen, S.Y. Tzang, C.M. Hsieh, J.H. Chen, N.Y. Zeng, R.R. Hwang, Numerical simulation of hydrodynamic behavior on wave-flap structure interactions with a SPH model, Proceedings of the 6th International Conference on Asian and Pacific Coasts, Hong Kong (China), 2011. [27] D.W. Chen, S.Y. Tzang, C.M. Hsieh, Y.C. Chow, J.H. Chen, C.C. Lin, R.R. Hwang, Numerical modeling of wave-induced rotations of a bottom-hinged flapper with a SPH model, J. Mar. Sci. Technol.-Taiwan, 22 (2014) 372-380. [28] J.J. Monaghan, Simulating free surface flows with SPH, J. Computational Phys. 110 (1994) 399-406.

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ACCEPTED MANUSCRIPT March 23, 2018 Shiaw-Yih Tzang Professor Dept. Harbor & River Engineering National Taiwan Ocean University Keelung 202, Taiwan (R. O. C.)

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Editor-in-Chief Prof. S.A. Kalogirou Editorial Office for Renewable Energy Cyprus University of Technology Lemesos, Cyprus

Dear Prof. Kalogirou:

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For the submitted paper entitled “Parametric Design Methodology for Maximizing Energy Capture of a Bottom-Hinged Flap-Type WEC with Medium Wave Resources” I have prepared highlights as follows:

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1. A parametric design methodology is proposed to maximize WEC’s energy capturing 2. Analytical and numerical modelings are performed on a bottom-hinged flap-type WEC

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3. Wave climates of Taiwan are used in the evaluations as a medium wave resource 4. The CF in resonance mode is well above 1 with flap width (B) < 0.30 wavelength (L) 5. The captured power in resonance mode is almost invariant in 0.11 < B/L < 0.30

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Sincerely yours,

Shiaw-Yih Tzang