Geometrical Evaluation on the Viscous Effect of Point-Absorber Wave-Energy Converters

The fluid viscosity is known to have a significant effect on the hydrodynamic characteristics which are linked to the power conversion ability of the wave energy converter (WEC). To overcome the disadvantages of case-by-case study through the experiments and numerical computations employed by the former researches, the viscous effect is studied comprehensively for multiple geometries in the present paper. The viscous effect is expressed as the viscous added mass and damping solved by the free-decay method. The computational fluid dynamics (CFD) method is employed for the calculation of the motion and flow field around the floater. The diameter to draft ratio and bottom shape are considered for the geometrical evaluation on the viscous effect. The results show that a slenderer floater presents a stronger viscous effect. Through the comparisons of the floaters with four different bottom shapes, the conical bottom is recommended in terms of low viscous effect and simple geometry for manufacture. A viscous correction formula for a series of cylindrical floaters is put forward, for the first time, to help the engineering design of outer-floaters of point-absorber WECs.


Introduction
Wave energy convertors (WECs) can be divided into three categories according to the relative position between the predominant wave direction and WECs, namely attenuator, terminator, and point absorber (PA-WEC) (Drew et al., 2009). A PA-WEC possesses small dimension relative to the incident wavelength, so that it is easy for array arrangement. McCabe and Aggidis (2009) argued that the PA-WEC is the most efficient in terms of the wave-power conversion per unit volume. This feature makes it highly suitable for the seas with relatively low wave energy density, e.g., Chinese adjacent seas (Wu et al., 2015). In these areas, the wave energy may not be able to produce enough electricity steadily for main-land grids, while it could be an effective supplement for the net-off microgrids of islands, oil platforms, or other offshore marine structures (Babarit et al., 2006).
For PA-WECs that work in the heave mode, axial-symmetrical floaters are normally adopted to reduce the sensibility of wave directions, such as the CETO (Australia) (Penesis et al., 2016), PowerBuoy (USA) (Edwards and Mekhiche, 2014), and Wavebob (Ireland) (Weber et al., 2009). The hydrodynamic characteristics of PA-WECs need to be studied in detail to maximize the wave power absorption. Generally, there are mainly three types of methods for solving hydrodynamic properties of PA-WECs: analytical method, Boundary Element Method (BEM), and computational fluid dynamics (CFD). A comprehensive review can be found in Li and Yu (2012). The linear potential flow theory can highly overestimate the motion and power response of a PA-WEC (Jin and Patton, 2017) because of neglect of the viscous effect. Especially when it is around the resonance frequency, the response simulated by non-viscous linear potential flow theory could be 10 times or more larger than that of the experiment, see examples in Tom (2013). The viscous effect can inevitably reduce the ability of the wave power conversion of a PA-WEC as shown in Son et al. (2016), Li and Yu (2012), and Tom (2013).
The viscous effect of PA-WECs has been studied experimentally or numerically by many researchers. Through the experimental study, Vantorre et al. (2004) argued that, a floater with the rounded-edge bottom has less energy dissipation due to viscosity. Yeung and Jiang (2011) explored the viscous damping and added mass of four two-dimensional heaving floaters by a viscous method called Free-Surface Random-Vortex Method (FSRVM). Jin and Patton (2017) studied three cylindrical floaters by the viscous CFD software LS-DYNA and the results demonstrated that the rounded-and conical-bottom floaters had less viscous damping than that with the flat-bottom. Palm et al. (2016) investigated a PA-WEC with a slack-moored cylinder through the OpenFOAM in consideration of the viscosity and green-water effect. Bhinder et al. (2011) and Caska and Finnigan (2008) studied PA-WECs with generic cylindrical floaters working in the heave and pitch modes, respectively, by introducing a Morison-like non-linear quadratic damping term. There are also many studies on other types of WECs that considered the fluid viscosity, such as the OWC (oscillating water column) (Ning et al., 2015(Ning et al., , 2016, a flap-type terminator (Chen et al., 2015), and a Rolling WEC (Jiang and Yeung, 2015).
In the published literatures, most studies on the effect of fluid viscosity were specific for a few given floaters. Even when different bottom shapes ( Jin and Patton, 2017;Yeung and Jiang, 2011) were considered, no detailed slenderness parameter studies have been provided. In the present study, we consider not only different bottom shapes, but also the slenderness. The viscous effect is expressed by the linearized viscous damping and added mass corrections. The viscous hydrodynamic quantities are acquired by free-decay curves calculated by the Reynolds-Averaged Navier-Stokes (RANS) and Volume of Fluid (VOF) method based on a CFD commercial software StarCCM+. And then the viscous corrections can be obtained by comparing the viscous and potential radiation forces. Most importantly, through the curve fitting technology, a correction formula has been derived for both viscous damping and added mass for the first time. This formula can be directly applied to the performance evaluation and the geometrical design of the absorber with fast speed. An example application of the viscous correction formula to the floater geometry design is demonstrated.

Methodology
The viscous effect of the floater of a PA-WEC considering only the heave mode is simplified and expressed in this section. The numerical and experimental studies conducted by Tom (2013) and Son et al. (2016) demonstrated that the excitation forces can be well predicted by the linear potential flow theory, while the radiation forces (especially the damping term) are significantly affected by the viscous effect. Therefore, the viscous effect should be studied mainly on the radiation force.
The radiation force is the hydrodynamic force acting on the floating body by the radiation wave field generated by the body motion, which can be expressed as: where 33 and 33 are the potential added mass and radiation damping calculated by AQWA in the frequency domain based on the Boundary Element Method (BEM). The velocity and acceleration of the body are denoted by and , respectively. Similarly, the radiation force considering the viscosity can be expressed as: μ λ where vis and vis denote the linearized added mass and damping in the viscous fluid, respectively.
As shown in Fig. 1, the radius of the floater is a, the draft is d, and the water depth is h. The motion equation of a floater in the heave free-decay motion can be written as: where M is the mass of the floater, x 3 is the heave motion of the floater, and C 3 is the hydrostatic restoring force coefficient. For cylindrical floaters, with the water density and gravity acceleration g. By setting the initial velocity as zero and the initial excursion as x 30 , the displacement can be obtained by is the damped resonance frequency, is the undamped resonance frequency, is the decay factor, and is the phase angle. For a larger x 30 , the equivalent viscous damping is larger (Tom, 2013). The experimental study of Tom (2013) proved that, when setting , the viscous hydrodynamic coefficients from free-decay tests matched very well with that from the regular-wave experiments. For engineering applications, the prediction of the performance of a PA-WEC in waves should be conservative and therefore a relatively large value of x 30 is chosen, i.e., x 30 =0.4d. ν The decay factor can be derived by the logarithmic decrements of the peaks of a free-decay curve (Tom, 2013) where is the damped resonance period, x 3, k is the amplitude of the k-th peak of a free-decay curve, and N is the number of peaks. Accordingly, the added mass and damping in the viscous fluid can be calculated by The non-dimensional linearized viscous corrections are defined as: f μ,visfλ,vis The physical meaning of and shows the ratio of the viscous added mass or damping to the potential added mass or damping. Similar notation method can be found in Son et al. (2016), Tom (2013), andWang et al. (2016).
Consequently, the radiation force in viscous flow fluid Eq. (2) can be expressed in the form of The CFD software Star CCM+ is used to simulate the free-decay motion of the cylindrical floater in the heave mode. The free surface is tracked by the VOF method and the Dynamic Fluid/Body Interaction (DFBI) module with the overset mesh adopted to simulate the motion of the body.
The numerical wave tank (NWT) is shown in Fig. 2. Most of the regions are hexahedron structural meshes, only around the corner or the bottom with complex geometry are tetrahedron non-structural meshes. To avoid the non-structural mesh at rounded boundaries, a rectangular NWT is adopted instead of a cylindrical one. The length and width of the numerical domain are equal because there is no need of incident wave generation. To avoid wave reflection from the NWT boundary, the length and width are set more than 20 times the radius of the cylinder and 1/3 the NWT from both wall boundaries are damping zone for absorbing radiation waves.
Four layers of prismatic meshes are used near the surface of the floater to increase the simulation quality of the boundary layer (Fig. 2a). For the balance of the simulation efficiency and accuracy, the mesh size grows larger as the distance to the floater increases as demonstrated in Fig. 2. The finer meshes are used around the floater and the free surface. Through the convergence tests of space and time as shown in Fig. 3, the minimum mesh length is taken as d/30, and the time step is T res /200, where T res is the non-damped resonance period and can be calculated by the linear potential theory. For the example shown in Fig. 3 (d/30 and T res /200), the number of the total cells is 84767 and the total CPU time is 1.8 h with a quad-core Intel Core i7-6700 CPU (3.40 GHz, 64-bit).
To verify the accuracy of the present numerical method, we compare the numerical results with experimental data of Tom (2013) for two cylindrical floaters with different bottom shapes. One is with the flat bottom (2a=0.273 m and d=0.613 m) and the other is with the rounded bottom (2a=0.273 m and d=0.706 m). The draft of the rounded bot-tom in the literature (Tom, 2013) refers to the distance from the mean water plane to the lowest point of the rounded bottom. As illustrated in Fig. 4, the free-decay curves match very well between the experimental data and the present numerical results. The largest differences are around the peaks, while other areas are perfectly matched. The mean differ-  ence of the amplitudes is smaller than 4.0%. Therefore, the numerical method is confirmed to be capable of simulating the free-decay motion with high accuracy.

Geometrical evaluation on viscous effect
3.1 Diameter to draft ratiō visfλ,visf μ,visfλ,vis The floaters considered in this paper are axial-symmetric, so that the characteristic of the geometry can be denoted by only one variable, i.e., the diameter to draft ratio 2a/d. The floater becomes fatter as 2a/d increases. Fig. 5 shows that the viscous effect (both added mass and damping) are greatly influenced by 2a/d. As 2a/d increases, both and decrease. This reveals that a fatter floater has less viscous effect. and are both asymptotic to 1.0 when 2a/d increases. This means that the viscous effect of a very fat floater is inappreciable. λ The damping of a floater in the viscous fluid vis comes from two parts: One is the potential radiation damping and the other is the viscous dissipation, which mainly consists of the viscous friction and vortex shedding (Bhinder et al., 2011). The expression of the viscous effect in the presentf paper is the ratio of the total damping in viscous fluid to the damping in potential fluid, . By the nondimensionalization of damping as , the viscous damping correction coefficient can also be in the form of . The comparison between and is shown in Fig. 6. It reveals that and both increase with the increase of 2a/d . The difference between and denotes the contribution of the viscous dissipation. Fig. 6 illustrates that the increment of is relatively small compared with that of which is the denominator of . This means that for a fat floater. That is, the viscous effect for a fat cylindrical floater can be neglected.

Bottom shape
This section studies the influence of the bottom shape on the viscous effect. As illustrated in Fig. 7a to Fig. 7d Madhi et al. (2014) and meant to diminish the viscous damping in the heave mode. In this paper, we pivot the two-dimensional BW curve to form a three-dimensional cylindrical floater with a BWB. For a non-FB floater, the submerged part under the mean water line consists of two parts: one is vertical cylindrical parts with the height of d cylin and the other is the non-flat bottom part with the volume of V bottom . To study the effect of the bottom shape, the displacements (or masses) are taken the same for floaters with different bottom shapes. Therefore, d cylin of different non-FB floaters can be calculated as: where V and a are the displacement and the radius of the floater, respectively. For example, for floaters with 2a/d=1.0 (Fig. 7), d cylin are 0.50d, 0.67d, and 0.57d for the CB, RB, BWB floaters, respectively. Moreover, for the convenience in the discussion of the parameter 2a/d for non-FB floaters, the concept of equivalent draft d is defined, which is the same as the FB floater, see examples in Fig. 7.

μ λ
The free-decay curves for these floaters are illustrated in Fig. 8 and the corresponding viscous corrections are shown in Table 1 with 2a/d=0.33 as an example. The studies of Tom (2013) and Son et al. (2016) for vertical axisymmetric floaters with different bottom shapes have proven that the added mass and damping characteristics are similar for the heave mode (the only difference is the magnitude). This means that the added mass and damping of a floater with a non-FB bottom shape can be estimated by those of a FB floater (which has the same diameter and displacement) with a linear factor correction. For the convenience of comparison, 33 and 33 of the FB cylindrical floater are used tof μ,visfλ,visf μ,visfλ,vis nondimensionalize and for all floaters with different bottom shapes. Therefore, and for non-FB cylindrical floaters not only contain the viscous effect information, but also have the information of the geometry difference (different bottom shapes).
Zhang et al. (2016) studied the performance of heaving PA-WECs with different bottom-shape floaters by the potential semi-analytical method without any viscous effect being considered. They concluded that the FB cylindrical floater had the largest motion and power response. However, clearly, the sequence is FB > RB > CB > BWB-H > BW in terms of the viscous damping. Again, this reveals that the neglect of the viscous effect can lead to big errors or even wrong results when studying the performance of heave PA-WECs. The declinations of the viscous damping are 42.7% (RB), 64.4% (CB), and 71.2% (BWB), respectively compared with those of the FB. The reasons are discussed as follows. The velocity fields are shown in Fig. 9 at t≈0.25T res when the vertical velocity of a free-decaying floater reaches the maximum. The variations of the velocity fields of around the bottoms of other three cases are relatively    Fig. 9a. Relatively, the BWB with the needlelike tip has the smoothest velocity field with the smallest velocity because of the smoothest streamlined curvature. Fig. 10 demonstrates the distributions of the wall shear stress on the surfaces of these floaters. The maximum wall shear stress appears at t≈0.25T res when the vertical velocity of a free-decaying floater reaches the maximum. Due to the inverse velocity area derived from the eddy (Fig. 9a), there is a region on the surface of the FB floater that has small or even zero shear stress (Fig. 10a). Overall, the shear stress of the FB is relatively small compared with the other three. Therefore, the large viscous damping of the FB demonstrated in Fig. 8 and Table 1 is mainly contributed by the vortex shedding. For the RB, CB, and BWB, there are no obvious large eddies around the bottom, which means that the disturbances to the flow field are weaker. As shown in Table 1, the viscous damping and added mass are smaller than those of FB. For the RB and CB, the areas where connect the convex bottoms have the largest wall stress due to the geometry change. Besides, for the BWB, the wall stress all along the submerged surface of the floater is very smooth because of the four-order streamline shape. Consequently, the BWB has the smallest viscous damping as illustrated in Table 1.
As the studies shown above, the BWB has the smallest viscous damping. However, the elimination of the needlelike tip brings 21.5% increase of the viscous damping, while changing very little to the added mass, as demonstrated in Table 1. This reveals that the performance of the small viscous effect of the BWB is highly depended on the sharp needlelike tip which is not practical. Moreover, the difference between the CB and BWB-H in terms of viscous damping is only 1.8%. Thus, in consideration of easy manufacturing, the floaters with CB are recommended for PA-WECs.f μ,visfλ,visf μ,visfλ,vis To further study the performance of cylindrical floaters with the CB, the influence of its taper angle coefficient TAC on the viscous effect is investigated. The profiles of the CB floaters with different TAC are shown in Fig. 11 and the corresponding and are illustrated in Fig. 12. A larger TAC represents a sharper CB and the viscous effect is smaller. With the increase of TAC, and decrease quickly at the beginning, and then slow down and finally the trend becomes constant. Therefore, TAC=3.0 is favorable.
From Eq. (9), with TAC=3.0 (i.e. ) and , the expression of d cylin for CB floaters can be derived as: For a fat floater (i.e. with relatively large 2a/d), d cylin is small or even not exists when a>d. The cone part of the floater may be out of water during the heave motion in waves. Then the nonlinearity of the hydrostatic restoring force is relatively large, which is out of the scope of this paper. Many experiments (Tom, 2013;Son et al., 2016;Madhi et al., 2014) have proven that the results of the experiment and the linear theory are well agreement, when the motion of the floater in waves is under (0.4-0.5)d. Therefore, TAC is set to fulfill the relationship Eq. (11) to ensure that d cylin ≥0.4d for floaters with different 2a/d. To emphasize, during the real operation of a PA-WEC, the floater may have heave motion that is larger than 0.4d in strong nonlinear waves. That phenomenon requires a non-linear wave-body interaction theory, which is not discussed in this paper. For the discussion of the viscous effect of CB floaters in Section 4, TAC is chosen by Eq. (11) for the floaters with dif- This section establishes a viscous correction formula for the CB and FB floaters with 2a/d as the independent variable. and curves of FB and CB floaters are shown in Fig. 5 and Fig. 13, respectively. The viscous added mass is found to be smaller than 1.0 for the CB floaters, which is due to the shape effect of the CB. As illustrated in Fig. 9, the ability of the flow-field-disturbance of the CB floater is smaller than that of the FB. The increase of of CB floaters as shown in Fig. 13a is due to the decreasing TAC when 2a/d>1.2.
of CB floaters are significantly smaller than those of the FB. Taking 2a/d=0.8, 1.0, and 1.25 as examples, the declinations of of CB floaters compared with those of FB floaters are 70.6%, 66.9%, and 59.8%, respectively.
After many tries, for the viscous correction of both added mass and damping, the function for the curve fitting is chosen as a combination of the exponential and rational functions as illustrated in Eq. (12). The example of the curve fitting results of the CB floaters is shown in Fig. 13.
α β σ δ where , , , and are the coefficients. For the CB and FB floaters, these coefficients can be found in Table 2. x denotes the independent variable 2a/d. For this formula, the scope of the application is 0.2<x<5.0, which covers the most possible fatness of floaters for PA-WECs.
The geometry parametric study of a general single-body PA-WEC is taken as an example for the application of the viscous correction formula. The schematic of a general single-body PA-WEC is shown in Fig. 14 and the power take-off (PTO) can be taken on the sea bed (e.g. Ulvgård, 2017) or on a fixed structure above water (e.g. Tom, 2013). The optimal damping of a general single-body PA-WEC in Fig. 11. Profiles of the cones with different TAC. CHEN Zhong-fei et al. China Ocean Eng., 2018, Vol. 32, No. 4, P. 443-452 449 regular and irregular waves had been well studied in many literatures, such as Tom (2013), Son et al. (2016), Ulvgård (2017), andWang et al. (2016). In irregular waves, the optimal damping can be achieved by simple one-variable searching algorithms (Brent, 2013). We adopt the MAT-LAB one-variable searching function "fminibnd" which is a combination of the golden section search and parabolic interpolation algorithms.C

w,year
The annual capture width ratio of a PA-WEC can be defined as: where P m, year is the annual averaged power, and P w, year is the annual averaged wave-power transportation rate per unit wave crest width for a given sea area. P m, year and P w, year can be calculated by adopting the methodology in the paper of Babarit et al. (2012). for CB and FB floaters with different geometries are calculated for the seas around Zhejiang, China (Wu et al., 2015) as an example. The results are shown in Fig. 16 and the corresponding long-term sea states (a joint distribution of the significant wave height H s and wave energy period T e ) are shown in Fig. 15. The viscous correction formula is obtained based on the assumption of small wave amplitude and small motion amplitude. Therefore, only the normal operational sea states are in the research scope of this paper. Any sea state with H s larger than 5.0 m, which may cause large non-linearity (e.g., green water, wave breaking, etc.), is not considered. Ignoring the viscous effect can lead to an overestimation of the wave energy absorption ability. For FB PA-WECs with and without viscous effect being considered (denoted by "FB-no-vis" and "FB-vis", respectively), the results of are shown in Figs. 16a and 16b. The maximum of the FB-no-vis is 0.33 and the corresponding diameter and draft of a floater are 2a=18.5 m and d=6.0 m. The maximum of the FB-vis is 0.28 which has a 15.2% declination relative to that of the FB-no-vis. Moreover, to achieve the maximum , the diameter and draft of a FB-vis floater are found to be 2a=27.5 m and d=5.5 m, in which the draft is similar to that of the FB-novis (d=6.0 m) but the diameter is 48.6% larger than that of the FB-no-vis.C For FB and CB PA-WECs with viscous effect being considered (corresponding to "FB-vis" and "CB-vis", respectively), the results of are shown in Figs. 16b and 16c. Due to the low viscous effect geometry, the maximum of the CB-vis is 35.7% larger than that of the FB-vis with a smaller CB floater, 2a=16.5 m and d=6.0 m, which means more cost-effective. Even compared with the FB-novis, the CB-vis still has 15.2% increase of with a smaller floater. Therefore, the CB cylindrical floater has better wave energy absorption ability and is more cost-effective.

Conclusions
The viscous effect of three-dimensional PA-WECs with cylindrical floaters working in the heave mode is studied through the free-decay curves of the body motion by use of CFD software Star CCM+. Through a comprehensive research, the following conclusions are drawn.
(1) The diameter to draft ratio 2a/d has a significant influence on the viscous effect of the floater. A fatter floater (with large 2a/d) has less viscous effect and for very fat floaters, the viscous effect can be neglected.
(2) In consideration of low viscous effect and easy manufacture, floaters with conical bottom (CB) are recommended for PA-WECs. The favorable taper angle coefficient TAC is 3.0. The viscous damping of floaters with conical bottom (CB) is smaller than that with flat bottom (FB). The usage of CB can greatly improve the hydrodynamic performance of PA-WECs.
(3) A viscous correction formula for floaters with both FB and CB is put forward with the diameter to draft ratio 2a/d as the independent variable. This formula can help researchers to design the floaters and study the performance of PA-WECs with a fast speed in consideration of fluid viscosity.C w,year (4) An example application, the geometry parameter study for a general PA-WEC, is presented at last. Because of the low viscous effect of CB floaters, the maximum annual capture width ratio of a PA-WEC with the CB in a given long-term sea state is 35.7% larger than that of FB, and the corresponding size of the floater is smaller, which means more cost-effective.