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 wave energy converter (WEC). To overcome the disadvantages of case-by-case study through the experiments and numerical computation employed by the former researches, the viscous effect is studied comprehensively for multiple geometries in the present paper. The viscous effect is expressed as 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 et al. (2009) argued that the PA-WEC is the most efficient in terms of 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 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), Wavebob (Ireland) (Weber et al., 2009), etc. The hydrodynamic characteristic of PA-WECs is needed 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 et al. (2012). The linear potential flow theory could highly overestimate the motion and power response of a PA-WEC (Jin & Patton., 2017) because of ignoring the viscous effect. Especially when it is around the resonance frequency, the response simulated by nonviscous linear potential flow theory could be more than 10 times larger than that of the experiment, see examples in Tom (2013). The viscous effect can inevitably reduce the ability of wave power conversion of a PA-WEC, as shown in Son et al. (2016), Li &Yu (2012), andTom (2013).
The viscous effect of PA-WECs was studied experimentally or numerically by many researchers.
Through the experimental study, Vantorre et al. (2004) argued that, a floater with rounded-edge bottom has less energy dissipation due to viscosity. Yeung & Jiang (2011) explored the viscous damping and added mass of four two-dimensional heaving floaters by a viscous method called the Free-Surface Random-Vortex Method (FSRVM). Jin & 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 with the consideration of viscosity and green-water effect. Bhinder et al. (2011) andCaska et al. (2008) studied PA-WECs with generic cylindrical floaters working in 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 and, a flap-type terminator , and a Rolling WEC (Jiang, 2015), etc.
In the published literatures, most studies on the effect of fluid viscosity were specific for a few given floaters. Even when different bottom shapes (such as Jin &Patton, 2017 andJiang, 2012) were considered, no detailed slenderness parameter studies have been provided. In our 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 freedecay curves calculated by 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 is 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 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 could 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 μvis and λvis denote the linearized added mass and damping in the viscous fluid, respectively.
where M is the mass of the floater, 3 x is the heave motion of the floater, C3 is the hydrostatic restoring force coefficient. For cylindrical floaters, C3=ρgπa 2 with water density ρ and gravity acceleration g. By setting the initial velocity as zero and the initial excursion as x30, the displacement can be obtained by For a larger x30, the equivalent viscous damping is larger (Tom, 2013 The decay factor ν can be derived by the logarithmic decrements of the peaks of a free-decay curve (e.g.
where T3,vis=2π/ω3,vis is the damped resonance period, x3,k is the amplitude of the k-th peak of a free-decay curve, N is the number of peaks. Accordingly, the added mass and damping in the viscous fluid can be calculated by The CFD software Star CCM+ is used to simulate the free-decay motion of the cylindrical floater in 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 0. 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 of the radius of the cylinder and 1/3 of the NWT from both wall boundaries are damping zone for absorbing radiation waves.  To verify the accuracy of the present numerical method, we compare the numerical results with data from the experiment of Tom (2013) for two cylindrical floaters with different bottom shapes. One is with the flat bottom (2a=0.273m and d=0.613m) and the other is with the rounded bottom (2a=0.273m and d=0.706m). The draft of the rounded bottom in the literature (Tom, 2013) means the distance from the mean water plane to the lowest point of the rounded bottom. As illustrated in Fig.4, the free-decay curves matched very well between the experimental data and the present numerical results. The largest differences are around the peaks, while other areas are matched perfectly. The mean difference of the amplitudes is less than 4.0%. Therefore, the numerical method is confirmed to be capable of simulating the free-decay motion with high accuracy.

Diameter to draft ratio
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

Bottom shape
This section studies the influence of bottom shape on the viscous effect. As illustrated in Fig.7 (a) 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), dcylin are 0.50d, 0.67d, and 0.57d for the CB, RB, BWB floaters, respectively. The free-decay curves for these floaters are illustrated in Fig.8     The velocity fields are shown in Fig.9 at t≈0.25Tres when the vertical velocity 3 x of a free-decaying floater reaches the maximum. The variations of the velocity fields of other three around the bottoms are relatively smooth compared with that of the FB. Due to the elimination of the bluff bottom edges, the sudden change of the velocity at the corner disappears, as shown in Fig.9 (a). Relatively, the BWB with the needlelike tip has the smoothest velocity field with the smallest velocity values because of the smoothest streamlined curvature. reaches the maximum. Due to the inverse velocity area derived from the eddy ( Fig.9(a)), there is a region on the surface of the FB floater that has small or even zero shear stress ( Fig.10 (a)). 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 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 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 For a fat floater (i.e. with relatively large 2a/d), dcylin is small or even not exist 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 matched very well, when the motion of the floater in waves is under 0.4~0.5d. Therefore, the TAC is set to fulfill the relationship Eq.(11) to ensure that dcylin≥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 strongly nonlinear waves. That phenomenon requires a non-linear wave-body interaction theory, which is not discussed in the present paper. For the discussion of the viscous effect of CB floaters in Section 4, the TAC is chosen by the Eq. (11) Fig.5 and Fig.13, respectively. The viscous added mass is found to be less 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 flow-field-disturbance of the CB floater is smaller than that of the FB. The increase of ,vis f  of CB floaters as shown in Fig.13 (a) where α, β, σ, δ 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 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 regular and irregular waves had been well studied by many literatures, such as Tom (2013), Son et al. (2016), Ulvgård (2017), and Wang et al. (2016), etc. In irregular waves, the optimal damping can be achieved by simple one-variable searching algorithms (Brent, 2013). We adopt the MATLAB one-variable searching function "fminibnd" which is a combination of the golden section search and parabolic interpolation algorithms.
The annual capture width ratio where Pm,year is the annual averaged power. Pw,year is the annual averaged wave-power transportation rate per unit wave crest width for a given sea area. Pm,year and Pw,year can be calculated by adopting the methodology in the paper of Babarit et al. (2012).
, w year C 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 significant wave height Hs and the wave energy period Te) are shown in Fig.15. The viscous correction formula is obtained based on the (a) added mass (b) damping assumption of small wave amplitude and small motion amplitude. Therefore, only the normal operational sea states are in the research scope of the present paper. Any sea states with Hs larger than 5.0m, which may cause large non-linearity (e.g., green water, wave breaking, etc.), are not considered.
Ignoring the viscous effect can lead to an overestimation of 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 , w year C are shown in Fig.16 (a) and (b). The maximum , w year C of the FB-novis is 0.33 and the corresponding diameter and draft of a floater are 2a=18.5m and d=6.0m. The maximum , w year C 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 , w year C , the diameter and draft of a FB-vis floater are found to be 2a=27.5m and d=5.5m, in which the draft is similar to that of the FB-no-vis (d=6.0m) but the diameter is 48.6% larger than that of the FB-no-vis.
For FB and CB PA-WECs with viscous effect being considered (corresponding to "FB-vis" and "CBvis", respectively), the results of , w year C are shown in Fig.16 (b) and (c). Due to the low viscous effect geometry, the maximum , w year C of the CB-vis is 35.7% larger than that of the FB-vis with a smaller CB floater, 2a=16.5m and d=6.0m, which means more cost-effective. Even compared with the FB-no-vis, the CB-vis still has 15.2% increase of , w year C with a smaller floater. Therefore, the CB cylindrical floater has better wave energy absorption ability and is more cost-effective.

Conclusion
The viscous effect of three-dimensional PA-WECs with cylindrical floaters working in heave mode is studied through the free-decay curves of body motion by use of CFD software Star CCM+. Through a comprehensive research, the conclusions are obtained as follows: (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) Considering low viscous effect and easy manufacturing, 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 the consideration of fluid viscosity and a fast speed.
(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 , w year C 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.