**1. Introduction**

Light buoys are equipped with a lighting function and navigation sign (Figure 1). The buoy guides vessels sailing nearby in the daytime with its shape and color, and at night with its light. It also plays a role in notifying vessels about the presence of obstacles such as reefs and shallows.

**Figure 1.** Light buoys.

Because conventional large buoys are mainly made of steel, they are heavy and vulnerable to corrosion and erosion by seawater. This makes the installation and maintenance of the buoys difficult. Moreover, vessel collision accidents with buoys and damage to vessels due to the light buoys' material (e.g., steel) are reported every year in Korea. Recently, light buoys adopting eco-friendly and lightweight materials have come into the spotlight to solve the previously mentioned problems. In Korea, a new lightweight light buoy with a 7-nautical-mile lantern, adopting an expanded polypropylene (EPP) and aluminum buoyant body and tower structure was developed by Jeong et al. [1]. Figure 2 shows the comparison of the conventional light buoy with the newly developed one. The total weight and manufacturing cost of the new buoys are approximately 40% and 27% lower than the conventional ones, respectively.


**Figure 2.** Comparison between a conventional light buoy and the lightweight one developed by Jeong et al. [1].

When the light buoy operates on the ocean, the visibility and angle of light from its lantern changes, which may cause it to function improperly. From this point of view, the pitch and roll motions of a light buoy are important. Moreover, large heave motions may cause structural damage to the mooring system. The motion of a floating body is greatly affected by external environmental loads, especially waves. To ensure motion stability and structural reliability, the natural frequency of the floating body needs to be very different from that of the dominant waves at the installation site. Because the mass distribution and center of gravity of the lightweight buoy are different from those of conventional one, the motion performance of the new type of buoy in waves should be assessed.

Therefore, after checking the static stability, Son et al. [2] carried out a motion analysis of a newly developed lightweight light buoy under various environmental conditions using potential-based commercial software ANSYS AQWA(Ansys, In., Canonsburg, PA, U.S.) that considers wind and current loads estimated by numerical simulations using the commercial computational fluid dynamics (CFD) software Siemens STAR-CCM+ (Siemens Industry Software Ltd., Plano, TX, U.S.) to increase the accuracy of the motion analysis. As a result, it was predicted that the pitch and roll motion were large and did not meet the design targets in specific conditions. As mentioned in several studies [3–7], one of the reasons might be that the viscous damping effect is ignored in the potential-based simulations commonly performed for the motion analysis.

A widely used way to consider the viscous effect in potential-based motion analyses is to evaluate and apply a viscous damping coefficient using the free decay test or force harmonic oscillation test [8,9]. These tests can also be conducted through CFD simulations. Wassermann et al. [10] estimated the roll damping of ships using CFD simulations of free decay and harmonic excited roll motion tests. They compared the advantages and disadvantages of both techniques. Wilson et al. [11] performed free roll decay tests on a surface combatant ship using CFD and compared the estimated damping coefficients with experimental results. Irkal et al. [12] carried out experiments and CFD simulations of free roll decay tests with different dimensions of a bilge keel. Song et al. [13] and Kianejad et al. [14] estimated

the roll damping coe fficient of a 2D section of a floating body and container ships through harmonic excited roll motion tests using CFD simulations.

Some studies based on CFD simulations were conducted to determine the resonance condition of a floating body as an energy harvesting device. Zhang et al. [15] investigated the influences of selected parameters such as incident wave condition, submerged depth, and power take o ff damping on the hydrodynamic performances of 2-D sections of rectangular heaving buoys. Luan et al. [16] estimated the hydrodynamic performance of a wave energy converter under various wave conditions and confirmed the relationship between optimal linear damping and incident wave conditions. Mohapatra et al. [17] formulated the mathematical modelling of wave di ffraction by a floating fixed truncated vertical cylinder based on Boussinesq-type equations in the application range of weakly dispersive Boussinesq model, and showed the fidelity of the model by comparing the results from the developed analytical model with those from experiments and their CFD simulations using OpenFOAM.

The adoption of a proper appendage, such as a (bilge) keel or a heave (damping) plate, is one of the options for improving the motion performance of a floating body in waves. Although it is di fficult to find motion reduction devices specifically designed for a light buoy, a heave damping plate for a vertical circular cylinder or a spar platform would be e ffective for a light buoy because of its geometrical similarity. Research on the e ffect of heave damping plates can be found in the following papers. Koh and Cho [18] carried out analytical and experimental studies to investigate the heave motion response of a circular cylinder according to the characteristics of dual damping plates as heave motion reduction appendages. Tao and Cai [19] investigated the vortex shedding pattern and hydrodynamics forces arising from the flow separation and vortex shedding around a damping plate of a circular cylinder. Through a series of experiments, some approximation equations were developed to calculate the added mass of the floating cylinder with a separate heave plate, and the motion response of a vertical circular cylinder with a heave plate to a series of regular waves was examined by Zhu and Lim [20]. Sudhakar and Nallayarasu [21,22] investigated the influence of single and double damping plates on the hydrodynamic response of a spar in regular and irregular waves by experimental studies. Koh et al. [23] performed free decay tests by experiments to obtain the viscous damping coe fficients of a circular cylinder with a heave damping plate changing the porosity of the damping plate. From their experiments for regular and irregular waves, the pronounced motion reduction was observed by applying a porous plate.

In this study, motion analyses of a newly developed lightweight light buoy in waves were performed to predict the motion performance and to check the e ffects of the conceptually developed appendages intended for improving the motion performance. First, free decay tests including benchmark cases using CFD were carried out to estimate the viscous damping coe fficients that cannot be obtained by potential-based simulations. Then, the results from potential-based simulations considering the viscous damping coe fficients estimated by CFD were compared with the results of motion simulations in regular waves using CFD simulations (Figure 3).

**Figure 3.** Scope and process of present study.

#### **2. Problem Formulation**

## *2.1. Governing Equation*

For incompressible turbulent flows, the governing equations are the continuity and Reynolds-averaged Navier-Stokes equations, as shown in Equations (1) and (2), respectively.

$$\frac{\partial \mu\_i}{\partial \mathbf{x}\_i} = 0 \tag{1}$$

$$\frac{\partial u\_i}{\partial t} + \frac{\partial \{u\_i u\_j\}}{\partial x\_j} = -\frac{1}{\rho} \frac{\partial p}{\partial \mathbf{x}\_i} + \frac{\partial}{\partial \mathbf{x}\_j} \left\{ (\mathbf{v} + \boldsymbol{\nu}\_t) \frac{\partial \mathbf{u}\_i}{\partial \mathbf{x}\_j} \right\} + f\_{i\boldsymbol{\nu}} \tag{2}$$

where *ui* and *xi* are the velocity component and coordinate in the *i*-direction; ρ is the density; *p* is the pressure; ν is the kinematic viscosity; ν*t* is the eddy viscosity; and *fi* is the external force per unit mass.

#### *2.2. Estimation Procedure of Viscous Damping Coe*ffi*cients from a Free Decay Test*

The 1-degree of freedom (DOF) motion equations of pitch and heave of a floating body are as follows.

$$(I + I\_a)\ddot{\theta} + b\mathfrak{s}\dot{\mathfrak{s}}\dot{\theta} + c\mathfrak{s}\mathfrak{s}\theta = M(t) \tag{3}$$

$$(m+m\_a)\ddot{z} + b\_{33}\dot{z} + c\_{33}z = F(t),\tag{4}$$

where *I* and *Ia* are the moment of inertia and added moment of inertia, respectively; θ is the angular displacement; *b* and *c* are the total damping and restoring coefficients, respectively; and *M*(*t*) is the pitch wave excitation moment. In Equation (4), *m* and *ma* are the mass and added mass, respectively; *z* is the vertical displacement; and *F*(*t*) is the heave wave excitation force.

The total damping coefficients of pitch and heave were calculated from the free decay test using Equations (5) and (6), respectively.

$$b\_{55} = 2\zeta\_{55}\sqrt{(I+I\_d)c\_{55}}\tag{5}$$

$$b\_{33} = 2\zeta\_{33}\sqrt{(m+m\_{\mu})c\_{33}},\tag{6}$$

where ζ is a non-dimensional total damping coefficient estimated by the method of Journée and Massie [9], for which four peaks of motion from the free decay test, as shown in Figure 4, were chosen to evaluate ζ through Equation (7).

$$\mathcal{K} = \frac{1}{2\pi} \cdot \ln\left(\frac{z\_{a1} - z\_{a2}}{z\_{a3} - z\_{a4}}\right) \tag{7}$$

**Figure 4.** Free decay curve.

The viscous damping coefficients were derived from Equations (8) and (9) proposed by Koh and Cho [18].

$$b\_{\text{55\\_vis}} = b\mathfrak{s}\mathfrak{s} - \nu\_{\text{55}}(\omega\_{\text{o}}, \mathfrak{s}\mathfrak{s}) \tag{8}$$

$$b\_{33,\text{vis}} = b\_{33} - \nu\_{33}(\omega\_{o,33}) \tag{9}$$

where <sup>ν</sup>55(<sup>ω</sup>*o*, 55) and <sup>ν</sup>33(<sup>ω</sup>*o*,<sup>33</sup>) are the radiation damping coefficients, which were evaluated by the ANSYS-AQWA software in this study, at the undamped natural frequencies of pitch and heave motions defined as <sup>ω</sup>*o*, 55 = *c*55/(*<sup>I</sup>* + *Ia*) and <sup>ω</sup>*o*, 33 = *c*33/(*m* + *ma*), respectively.

#### **3. Numerical Simulations**

#### *3.1. Simulation Method*

STAR-CCM+ 11.04 was used to simulate the free decay test and for the motion simulations in regular waves. To capture the free surface, the volume of fluid (VOF) method was used. The realizable k-ε model was applied as a turbulent model. In addition, the overset grid and dynamic fluid body interaction techniques were used to handle the motion of the floating body. 3-DOF (surge, heave, and pitch) and 6-DOF simulations were performed for the free decay tests and motion simulations in regular waves, respectively.

With the application of the viscous damping coefficients, which were estimated by CFD simulations, the motion analyses were performed using ANSYS-AQWA, which is based on panel methods.

#### *3.2. Modeling of Lightweight Light Buoys*

To improve the motion performance of the recently developed lightweight light buoy, named "Base," two kinds of appendages were conceptually designed and assumed to be installed on the light buoy, as illustrated in Figure 5. The first addition is similar to a heave damping plate for an offshore structure; it is named "Plate" hereafter. The other addition is a conical shape similar to a ship's bilge keel, which is marked as "Cone" in the present paper. From the research of Koh et al. [23], the damping coefficient of a vertical circular cylinder with a porous damping plate is larger than that with

**Cone**

a non-porous damping plate. Therefore, the effect of the porosity of the appendage was also evaluated in this study. The models considering the porosity are named "Porous Plate" and "Porous Cone."

**Figure 5.** Cross-section views of target lightweight light buoys.

The simplified geometries of the lightweight light buoys with and without the developed appendages for the numerical analysis are shown in Figure 6, where only the major parts affecting the motion of the buoy were modeled considering the total mass and mass moment of inertia of the tower structure. The particulars and hydrostatic properties of the buoys are listed in Table 1. The mass of the light buoy with the appendages is approximately 7% higher than that of the Base model.

**Figure 6.** Geometries of lightweight light buoy with and without developed appendages.



#### *3.3. Estimation of Viscous Damping Coe*ffi*cient Using CFD Simulation*

## 3.3.1. Validation

To confirm the accuracy of the present numerical schemes and methods, CFD simulations of the free pitch and heave decay tests of a circular cylinder were performed under the same conditions as Palm et al. [24], who carried out experimental and numerical tests. A vertically truncated cylinder was tested in a wave tank with a water depth of 0.9m. The mass and diameter of the cylinder were 35.85kg and 0.515m, respectively. The moment of inertia around the center of gravity was 0.9kgm<sup>2</sup> and the center of gravity was placed 0.0758m above the bottom of the buoy along the symmetry z-axis.

Figure 7 shows the comparisons of the pitch and heave time histories obtained from the present CFD simulations with the results of the reference. The results of the present study are in good agreemen<sup>t</sup> with those of the reference. The reason for the discrepancy in pitch motion between the numerical and experimental results is thought to be the limitation of a small-scale experiment, in which generally allowable errors in the controlling and measured variables, such as the draft and center of gravity, may result in considerable di fferences in the motion response, as pointed out by Palm et al. in their work [24].

**Figure 7.** A comparison of the time histories of (**a**) free pitch and (**b**) heave decay curves of a circular cylinder between the present results and those of the reference.

#### 3.3.2. Computational Domain, Boundary Conditions, and Grid System

Figure 8 shows the computational domain and boundary conditions of the free decay test for the light buoys using CFD. The size of the computational domain was set to 25 m in the depth direction below the free surface. The length and width are 30.0 *D* based on the diameter *D* (2.40 m) of the light buoy. To suppress the radiated waves from the light buoys, a numerical wave damping scheme is applied at the ends of the side boundaries to approximately one third of the computational domain.

**Figure 8.** Computational domain and boundary conditions of the free decay test of lightweight light buoys using CFD.

Figure 9 shows the grid system of the free decay test using CFD. The grid system was generated using surface remesher, trimmer mesh, and prism layer mesh in STAR-CCM+. Near the light buoy and free surface, grids are refined to accurately capture the complicated flow around the buoy including the appendage and free surface.

**Figure 9.** Grid system for the free decay test of lightweight light buoys using CFD.
