Numerical and Experimental Investigation of Interactions Between Free-Surface Waves and A Floating Breakwater with Cylindrical-Dual/Rectangular-Single Pontoon

This paper investigates the hydrodynamic performance of a cylindrical-dual or rectangular-single pontoon floating breakwater using the numerical method and experimental study. The numerical simulation work is based on the multi-physics computational fluid dynamics (CFD) code and an innovative full-structured dynamic grid method applied to update the three-degree-of-freedom (3-DOF) rigid structure motions. As a time-marching scheme, the trapezoid analogue integral method is used to update the time integration combined with remeshing at each time step. The application of full-structured mesh elements can prevent grids distortion or deformation caused by large-scale movement and improve the stability of calculation. In movable regions, each moving zone is specified with particular motion modes (sway, heave and roll). A series of experimental studies are carried out to validate the performance of the floating body and verify the accuracy of the proposed numerical model. The results are systematically assessed in terms of wave coefficients, mooring line forces, velocity streamlines and the 3-DOF motions of the floating breakwater. When compared with the wave coefficient solutions, excellent agreements are achieved between the computed and experimental data, except in the vicinity of resonant frequency. The velocity streamlines and wave profile movement in the fluid field can also be reproduced using this numerical model.


Introduction
Nowadays, with the increasing utilization of marine resources and the development of ocean engineering technologies, coastal structures are widely applied in offshore regions. Among an array of coastal structures, floating breakwaters (FBs) have gained a favorable attention as the concept provides the protection against wind-driven waves' attack in coastal engineering. In other words, FBs have the mechanism of the wave attenuation by means of consuming or reflecting wave energy. Compared with traditional fixed breakwaters, such floating structures have the potential of cost-effective, material reducing and most importantly water can exchange freely satisfying environmental requirements. In this regard, numerous researchers pay much attention to understand the hydrodynamic performance of FBs.
Over the years, numerical studies about the hydrodynamic performance of the FB have been conducted (Sannasiraj et al., 1998;Lee and Cho, 2003;Loukogeorgaki and Angelides, 2005;Bai and Eatock Taylor, 2009;Ji C.Y. et al., 2017), but these works are based on the potential theory. Their discussions are under the assumption of a non-viscous effect and it is difficult to obtain the body motions caused by the damping effect in the vicinity of the floating body. As such, this paper devotes much more efforts to the development of viscous model and considers the influence on the hydrodynamic behavior of the FB due to the viscous effects of water.
With the viscous flow theory, a common approach is used to carry out the numerical simulations directly based on the N-S equations. Kawasaki (1999) proposed a numer-ical wave model for two-dimensional (2D) wave field in the vertical plane and the study demonstrated that the proposed model can reproduce the wave deformation at the onshore side of the submerged breakwater. The wave forces acting on a submerged permeable breakwater were discussed by Hur et al. (2004) who used the direct numerical simulation to solve the modified N-S equations. Young et al. (2007) developed a higher-order non-hydrostatic model to simulate wave cases, and this model used an implicit finite difference scheme on a staggered grid to solve N-S equations. Vanneste and Troch (2015) performed a numerical simulation of wave interaction with a large-scale Rubble-Mound Breakwater, built in a generic multi-physics CFD code. This numerical simulation employed the standard branch of VOF-based N-S model to predict the wave interaction with breakwater. In the same year, a numerical wave tank based on the N-S equations was established by Zhu et al. (2015) to study wave dissipation and transmission coefficient of a double curtain-wall breakwater. Ji Q.L. et al. (2017) presented a numerical multiphase flow model based on N-S equations to estimate the wave interaction with a submerged breakwater. The results show that the model could capture the free surface and simulate the wave deformation within acceptable accuracy. However, these studies focus on the wave interactions with fixed breakwaters without accounting for the floating coastal structures. In fact, many wavestructure interactions are related to the dynamics of the floating breakwater. In other words, the structures driven by water wave forces are usually accompanied by dynamic motion responses, which is more in accordance with the actual circumstances. With the fluid viscosity, the accurate prediction of wave-induced behaviors of the body is one of the main concerns for ocean engineering.
Some efforts to understand the wave interactions with the movable structures have attracted plenty of researchers' attentions. Chen and Liu (2002) employed a Reynolds-Averaged Navier-Stokes numerical method to simulate largeamplitude ship roll motions in the time domain. The results could effectively predict the ship's roll resonance when the period of the incident wave coincided with its natural roll period. Rahman et al. (2006) developed a numerical model based on the volume of fluid method to estimate the dynamic responses of a pontoon type submerged floating breakwater. The computed results revealed that the numerical model could reproduce the dynamics of the floating breakwater. The interactions of water waves with submerged floating breakwaters were studied by Peng et al. (2013) based on the N-S solver. The study showed that the floating breakwater's motions could be reproduced and measured well in the numerical model. Jeong and Lee (2014) examined the floating breakwater's wave attenuation in large regular wave amplitude. In the numerical simulations, the modified markerdensity method was used to define the nonlinear waves nearby the moving breakwater. Although these works de-vote to the numerical simulations of floating breakwaters based on the N-S equations, they still suffered from two issues when the numerical model is related to a movable object. One is the realization of the motion response in every degree of freedom other than merely solving one of the single direction movements. The other is to determine the moving solid boundaries that are affected by multi-phase flow. These papers discussed above are about the structures that are submerged or below the surface of the water and solely affected by the liquid. Thus, this paper focuses on the fully free surface floating breakwaters that are influenced by both water and air simultaneously. Usually, this kind of model has large-scale movement for a moving body under severe wave forces. It is more complex to simulate this kind of free surface structures with large-scale motion response displacements.
The investigations on the severe dynamic interactions between waves and the FB using the viscous numerical model are relatively scarce. A Scale Adaptive Simulation (SAS) model was applied to the moving zones allowing the degrees of freedom in different directions (Egorov et al., 2010). In 2017, the SAS model was also employed by Zhan et al. (2017) to simulate the interaction between the inverse T-type floating breakwater and waves. The analysis indicated that the simulated sway motion responses of the floating breakwater were well measured. However, the SAS model usually generates excessive dense grids in the partial region resulting in too large gradient of the mesh unit volume or length scales due to the large-scaled movement, which have an adverse effect on the calculation accuracy. Therefore, in this study the whole computation domain grids use structured meshes, namely, quadrangular meshes and all the moving zones employ the dynamic layering technique to split or collapse the grids. The motion response of the FB in every direction can be realized based on the N-S equations. In addition, this kind of grids' updated method can avoid unexpected mesh deformation or distortion and provide the computation accuracy to the maximum extent.
This work mainly analyzes the dynamic interactions between the free surface water waves and a cylindrical-dual or rectangular-single pontoon floating breakwater from both scaled experiments and numerical simulations. For the analysis, a 2D numerical wave flume is established to investigate the hydrodynamic performance of a movable floating body based on the Navier-Stokes solver. An innovative fullstructured dynamic grid method is employed and the whole computation domain is divided into the quadrangular mesh elements which are updated by the dynamic layering technique. The VOF approach (Hirt and Nichols, 1981) used to capture the free surface is contained in the proposed viscous model. Moreover, the CFD code in this paper is also incorporated in the model which is based on a laminar equation carried out in an Eulerian reference frame. The main sections for the code programming include the use of trapezoid analogue founded on Euler's method to solve the motion equations, the mooring lines representation, the efficient approach for transient wave forces computation and the water waves' generation and absorption. A series of physical experiments are implemented in the water wave flume at the Ocean Engineering Laboratory to validate the numerical floating body model and verify the work's accuracy. The experiment is as an extension of our previous tests (Ji et al., 2015(Ji et al., , 2016a. The work is organized as follows: experimental facilities, floating breakwater model, wave conditions and mooring system are introduced in Section 2. Formulation of the FB system, mathematical model about the structure and the zonal setup method are provided in Section 3. In Section 4, the comparisons between the computed and experimental data are discussed in detail. Finally, Section 5 gives the conclusions.

Experimental facilities
According to our previous study (Ji et al., 2015), both the cylindrical and rectangular FBs are made of reinforced concrete and the main prototype parameters are listed in Table 1. The validation of the 2D numerical model is based on a series of experimental measurements conducted in the wave flume in the Ocean Engineering Laboratory of Jiangsu University of Science and Technology, China.
The detailed setup of the experiment is shown in Fig. 1. This wave flume used is 40 m long, 1.5 m deep and 0.8 m wide. The bottom of the wave channel is a smooth cement floor, so the bottom friction is negligible and is not included in the numerical simulation. At one side, the flume is equipped with a piston-type wave paddle to generate water waves. Every wave condition should not be smaller than 10 stable wave components. At the other end of the flume, a wave absorbing beach is presented to mitigate the effect of wave reflections. The slope of the wave-absorbing beach is 1:5. A scaled floating breakwater in the physical model is positioned at about halfway between the wave generator and the absorbing beach.
In order to record the free surface elevations in the wave flume, five wave gauges (WGs) are used in this experiment both at the windward and leeward sides of the model. The detailed measure point locations are presented in Fig. 1. WG1 is close to the wave generator to measure the incident wave amplitudes. WG2 and WG3 are used to distinguish the incident waves from the reflected waves. The distance between WG2 and WG3 is 0.4 m. The function of WG4 and WG5 is to obtain the transmitted wave elevation and the distance between those two WGs is also 0.4 m. Moreover, the motion response displacements of the free surface FB are recorded by using a digital camera.

Model scale
In Consideration of the size of the wave flume and the frequency range of wave that can be generated by computer controlled by a wave generator, the model is made based on geometrical similarity with the scale factor of 1:20.

Experimental floating breakwater
In this experiment, we employ the traditional cylindrical floating breakwater (CFB) and the rectangular floating breakwater (RFB), mentioned in Ji et al. (2015), to be the experimental models. The cross sections of the floating breakwater are the same as that used in Ji et al. (2015). Model 1 shows the traditional CFB which has two 0.2 m (diameter) 0.76 m (length) cylinders and nine 0.02 m (diameter) 0.1 m (length) cylindrical braces. Model 2 indicates the traditional RFB which is 0.76 m (length) 0.5 m (width) 0.2 m (height). The main parameters of the scaled model are listed in Table 1

Mooring system
The floating breakwaters are moored by four-slack-line stainless chains. Each mooring line has a length of 1.6 m. The mooring system is carefully adjusted to ensure the symmetry in the y and z direction, thus, the FB is in its equilibrium position. To measure the force exerted on the floating breakwater, the load cell is placed at the top end of the chain connecting the mooring line and the floating breakwater.

Experimental conditions
In the prototype, the studied water depth is 20 m; wave height H p is 2.0 m; the prototype wave periods T p are from 4.92 to 8.05 s. In this scaled model experiment, regular waves are studied at the water depth of h=1.0 m. The experimental wave periods T i vary from 1.1 s to 1.8 s and the experimental incident wave height is 0.1 m, as summarized in Table 2.

Formulation of the problem
This work discusses a free surface FB, as shown in Fig.  1. The FB moves with the water waves and the movement is restricted by the mooring lines. The problem of the fluid motion is formulated in a Cartesian coordinate system. The origin is at the still water level; the y-axis is in the horizontal direction along the wave propagation direction; the z-axis is in the vertical direction pointing upward. The motion response problem of the FB is formulated in this coordinate system as shown in Fig. 4. We assume that the center of gravity (COG) of the FB coincides with the center of rotation (COR). The deformation of the FB is neglected since it is extremely small. There are only three degrees of freedom, namely, sway, heave and roll.
3.1 Mathematical model

Description of the governing equation
The simulation is carried out in a 2D numerical flume in the vertical plane. In this model it is assumed to be 2D incompressible flow, which is governed by the following standard equation.
where t and denote the time and fluid density, respectively; is the dynamic viscosity of the fluid; , and are the generic variables; v represents the absolute fluid velocity vector; T is the viscous part of the stress tensor and . (2) All the variables mentioned above are with respect to the fixed reference frame. The correspondences between these symbols and the mass conservation equation or the momentum conservation equation are listed in Table 3.

Governing equations of the FB motion
The dynamic motion responses of the rigid floating body supported by the mooring lines have been analyzed theoretically. The motion equations of the body are written as: ξθ where, m and M I denote the mass of the FB and the moment of inertia of the structure, respectively; and represent the instantaneous acceleration and roll angular acceleration of the body; F w and F ml show the hydrodynamic force and anchor tension acting on the rigid body; F g is the gravity; M w and M ml are the moments of the hydrodynamic force and mooring line tension with respect to the body center. The forces are shown in Fig. 4, where F wy and F wz are the hydrodynamic forces in the y-direction and z-direction, respectively. The hydrodynamic force F w is computed by integrating the hydraulic pressure and shear stress over the structure's surface S.
where I is the unit tensor; n is the unit normal vector of the body's surface element and directed outwards. The hydrodynamic force moment M w is obtained by integrating the moments of the pressure and shear stress acting on the FB surface. The ratio of the mooring line length to water depth is regarded as the slack mooring line scope, which has a slight influence on the hydrodynamic performance for floating breakwater. This is also discussed by Vijayakrishna Rapaka et al. (2004), He et al. (2012) and Ji et al. (2018). Thus, the effects of the slack mooring line scope are ignored in the numerical simulation. Given that the assumption (tension mooring lines) and the geometric relations are shown in Fig.  4, we can derive the tensions of the mooring chains.
where |F ml | y and |F ml | z stand for the component values of the mooring force in the y-direction and z-direction; is the position vector of the anchor point of the FB; k s is the restoring stiffness; and are the horizontal and vertical displacements of the structure from the initial position.
The mooring force moment M ml is obtained by calculating the moments of the mooring tensions acting on the FB surface.

Boundary conditions
(1) Inlet boundary The free surface floating breakwater is placed in a 2D wave flume. The ability of the FB's motion response in waves depends firstly on the ability of the water waves. The left end boundary of the flume is the wave generator which is specified in a unique movement form to control the water surface elevation. This is a plate-type wave making method and the equation is as follows: σ S o where, denotes the circular frequency; is the stroke of the piston wavemaker (Ursell et al., 1960) and approximately expressed as: where H and are the wave height and wavenumber, respectively; h represents the water depth ( ).
(2) Fluid-structure interface On the fluid-structure interface S (on S), the motions of the structures and water are fully coupled by the forces and velocities. The body is impenetrable and the water particles adhere to the fluid-structure interface. v s · n = v · n, on S ; ξθ where and are the velocity vector and roll angular velo- city vector of the FB, respectively.
(3) Numerical beach The wave propagates from the wave maker to the right end of the flume, and there will be wave reflections because of the right end wall boundary. In order to eliminate the numerical reflection, a numerical beach is applied by adding a momentum damping term in Eq. (1). The numerical beach zone is in the vicinity of the right end wall boundary and far away from the stern of the FB, as shown in Fig. 6. In this zone, coarse grids are employed and Eqs. (15)-(17) are also used to absorb the wave energy. [C] x min = 1, [C] x max = 0, (17) where [C](x) is a relaxation function related to the spatial coordinate to eliminate the right-going water waves. The variable quantities with the subscript c are the computational values from Eq. (1).

Fluid motion
For incompressible viscous flows, the mass conservation equation and the momentum conservation equation are discretized into rectangular cells utilizing the finite volume method based on the Eulerian structured meshes. The two equations in the integral form are described as (Hadžić et al., 2005): where CV is the control volume; S is the closed surface of the control volume; I is the unit tensor; n is the unit normal vector of the FB's surface element and directed outwards.
The volume of fluid (VOF) technique (Hirt and Nichols, 1981) is used to track the liquid-gas interface. The model supposes that the two phases in the control volume share the same pressure and velocities. Therefore, the governing Eqs. (18) and (19) are solved for an equivalent fluid.
An additional equation is taken into consideration to describe the movement of the fluid and is described by the conservation equation as: where, is the indicator phase function defining the percentage of water at each control volume. If a control volume is full of water, ; if a control volume is full of air, ; otherwise the cell will be categorized as the interface. Any physical property at each control volume can be computed as follows: where and are the liquid and gas properties, respectively.
In this work, the fluid motion is calculated by using the finite volume method package ANSYS FLUENT. The FB's motion responses, wave generator and numerical beach are defined by User Defined Function (UDF) in ANSYS FLU-ENT. The FB's motion responses are shown in the next section.

FB's motion
The discrete form of Eqs. (3) and (4) at the n-th time step is shown as: ∆t where is the time step of the simulation; superscripts n and n-1 represent the n-th and (n-1)-th time step.
The velocities , displacements , rotational velocities and rotational angles of the structure are computed by using the trapezoid analogue integral method as follows: T n = T n−1 + ∆t; where, , and , are the calculated function vectors at the n-th and (n-1)-th time step.
The hydrodynamic force ; mooring line forces and ; hydrodynamic force moment M w and mooring force moment M ml in Eqs. (5)-(9) are obtained as: Those unknown variables , , , , , , , , and are related to Eqs. (22)-(32) and are calculated through the fixed-pointed iteration method. The algorithm described above is presented in Fig. (5). ξ n When the structure's displacements at the n-th time step have been determined at the end of the n-th time step, each mesh in the movable section (as shown in Fig. 7) will be updated as long as one of the following criteria is satisfied.
(1) The mesh length scale is smaller than the specified minimum length size; (2) The mesh length scale is larger than the specified maximum length size.

Zonal setup method
Generally, the 2D numerical flume in the vertical plane is divided into two stationary parts and one movable section as shown in Fig. 6. As the waves pass through the floating breakwater, the rigid body will be in the moving state. At the same time, the movable section will follow the breakwater's movement by re-meshing its structured grids. The stationary parts remain stable while the movable section is updated. In another way, from left to right, the computation domains are the wave-generating zone (WGZ), working area (WA) and sponge zone (SZ), respectively.
To realize the FB's response displacements (sway, heave and roll) during the interaction with waves, the movable part needs to be divided into several zones as shown in Fig. 7. Every single zone corresponds to specific kinds of the motion responses. The longitudinal movement (LM) zones are only allowed to have heave motion; the transversal movement (TM) zones are only allowed to have sway motion; the longitudinal and transversal movement (LTM) zone can achieve the response displacements of heave and sway. The zone within the circle area including the free surface floating breakwater is called LTRM zone which enables three degrees of freedom, namely, heave, sway and roll motion responses. Because of the specific shape of the FB, the blue grid in the LTRM zone is designed in order to improve the mesh quality. In this work, the whole computation domain grids are quadrangular structured meshes. The wave-structure interaction process is simulated by using dynamic mesh technique based on the evolutionary grids updating method. At the end of each time step, the quadrangular meshes in the moving zones will be split or collapsed synchronously by using the dynamic layering technique. Compared with the Scale Adaptive Simulation (SAS) model used by many researchers in the past, this innovative approach can avoid unexpected mesh deformation or distortion and guarantee computation accuracy to the maximum extent. The N-S equations are solved for all the mesh components simultaneously. ∆x ∆z ∆x = ∆z ∆t Both the time-step and grid convergence tests have been performed to achieve a sufficient numerical accuracy. In the two stationary parts, each grid element length is =L/100 (L is the incident wave length) and the grid element height nearest to the free water surface is =H/25 (H is the incident wave height). In the movable section, the mesh size of the zones close to the floating body is about =H/50. In order to enhance the computation stability in the whole numerical process, the time step is set as =T/1000 (T is the incident wave period).

Results and discussions
The proposed 2D numerical model is validated by using the experimental data. In this section, the simulated time series results are measured and discussed, including the dynamic motion responses, transmitted coefficient (K t ), reflected coefficients (K r ), the energy dissipation coefficient (K d ) and mooring forces acting on the FB, etc.

Motion responses
The computed motion responses of the free surface FB are recorded during the interaction with the water waves. The motion response is an important measurement target for this study to validate the developed numerical models. Therefore, the measured sway, heave and roll motion responses are compared with the experimental data in the time series. Figs. 8 and 9 show the compared rectangle-type breakwater's motion responses in the cases of H=0.1 m, T=1.1 s and H=0.1 m, T=1.6 s both numerically and experimentally.
The positive sway amplitudes stand for the horizontal displacements in the positive y direction. The positive heave magnitudes represent the vertical displacements along the direction of positive z-axis. Similarly, the anticlockwise rotation radians of the free surface FB indicate the positive roll values in this numerical simulation.
As these figures depicted, the simulated sway and heave body motion responses show good agreements with the experimental results. The results from Figs. 8c and 9c indicate that the predicted body roll motion response values are slightly smaller than the recorded experimental data. This is more likely caused by the simplified spring mooring lines in the simulation work. The study focuses on the static analysis of the mooring lines and has not taken the dynamic nonlinear analysis into consideration, which may lead to the difference.
Furthermore, the estimated dimensionless motion response displacements are compared with the experiment results. The sway, heave and roll motion responses are normalized by the wave amplitude (A) and the influence of the FB's relative breadth (B/L) is presented in Figs. 10 and 11; L denotes the wave length. The motion amplitude of the FB is defined as our previous study (Ji et al., 2016b).
The results in Figs. 10a and 11a show that the sway motion of the box-type FB declines as B/L increases. This is due to the fact that the influence of the slow drift motion is removed in low wave frequency. The result also indicates that there is larger sway response displacement when the wave steepness is smaller or the longer period waves containing more wave energy can induce larger sway motions. By contrast, the roll motion of the RFB (see, Fig. 10c) increases as B/L increases when the wave height is 0.1 m. This is because the wave steepness grows as B/L increases   and there is higher roll motion response when the wave steepness rises. In Fig. 11c, due to the influence of the CFB's roll resonance, the peak value appears at B/L =0.224.
Figs. 10b and 11b reveal the changes of the heave motion, which shows that the motion responses increase first to the peak value and then decrease with the increase of the relative breadth. Those peak values are more likely to be caused by the resonance of the heave. Observing these data carefully, we can find that the peak value of the heave mode for the RFB shifts towards smaller relative breadth region than that for the CFB. This may be due to that the RFB's mass is bigger than the CFB's mass, which results in a lower natural frequency.
Figs. 10c and 11c show that the numerical model results smoothly underestimate the dimensionless roll motion response values. Those results confirm the analysis mentioned above that the simplified spring mooring system and the static analysis lead to underestimating the roll motion response displacements. However, the comparisons between the numerical and experimental results show a good agreement generally. Therefore, the developed numerical model can generally predict the motion response displacements of the floating rigid body during the interaction with water waves.

Wave coefficients
From the recorded data that the wave gauges measured during the numerical simulation and experiment, the transmitted wave height H t and the reflection wave height H r can be obtained by using a two-point method proposed by Goda and Suzuki (1976). In this section, the wave coefficients include the transmission coefficient K t , the reflection coefficient K r and the energy dissipation coefficient K d .
where H i is the tested incident wave height. Figs. 12 and 13 show the variations of K t , K r and K d with respect to B/L when the wave height is H=0.1 m. When the FB moves back to the windward side or the leading waves approach the FB, the water waves are partially reflected to the weather side. Thus, the front and bottom sides of the rigid body have the function of reflecting the water waves (see, Figs. 12b and 13b). Observing those data, we can find that the difference of the motion responses has little impact on the accuracy of the reflection coefficients. In Fig. 12b, the reflection coefficients K r are seen to increase as the B/L grows for the RFB. It is revealed that when the wave frequency increases, the RFB's reflection effect becomes stronger. However, for the CFB, the maximum reflection coefficient can be found at B/L=0.224, as shown in Fig. 13b. The reason probably is that the joint influence of the heave and roll resonances (see, Figs. 11b and 11c) results in the peak value of K r .
In Figs. 12c and 13c, the higher values of the relative breadth of the FB have better performance of dissipating water wave energy, which is mainly due to the vortexes shedding around the corner area of the FB. During the wave interactions with the floating body, the vortexes are generated, split up, move away from the structure, and then gradually vanish. This phenomenon is caused by the vis-  cous damping effect of the water flow for the corner structure, which also is reported by Jung et al. (2006). The increasing wave frequency enhances the generation of eddy and the energy dissipation also increases, which explains the energy dissipation coefficient K d rises for shorter water waves.
Figs. 12a and 13a obviously illustrate that the transmission coefficient K t shows a down trend as the relative breadth B/L increases. The higher values of the B/L have more efficient effect of the wave attenuation because with the increase of the wave frequency, more water waves are reflected and wave energy is dissipated.
In Figs. 12a and 12c, there is difference when B/L =0.112 which is close to the RFB's natural period (see, Fig. 10b) and it may be caused by the resonance effect. Similar phenomenon can be found in Figs. 13a and 13c when B/L =0.224 which is correlated with the CFB's natural frequency, as shown in Figs. 11b. Therefore, the resonance of the floating body has a direct impact on K t and K d but has a minor influence on the reflection coefficient K r . Generally, the variations of the wave coefficients can be reproduced well by the numerical model, except the situation where the wave frequency is in the vicinity of the rigid structure's natural frequency.

Mooring forces
The mooring forces acting on the FB are some of the parameters in this floating body system. The chain loads can be obtained by solving the equations of the FB's motions which consider all the dominant external forces during the wave interactions.
Figs. 14 and 15 reveal the comparison of the mooring forces acting on the RFB and CFB between numerical and experimental results. For the RFB in Fig. 14, the mooring line forces decline firstly and then have an upward trend slightly. In Fig. 15, the mooring forces of the CFB increase first and then keep stable. When the wave period is larger than 1.4 s, the mooring forces are overestimated numerically than the measured data in the experiment, whereas the numerical results are under-predicted than the experimental wave loads as the wave period is smaller than 1.4 s.
Obviously, there is difference through the comparisons of the mooring forces; however the discrepancy has little influence on the accuracy of the motion response displacements and the wave coefficients of the rigid structure, which is similar to the results reported by Vijayakrishna Rapaka et al. (2004).

Velocity streamlines around the FB
For more details about the wave-structure interaction, the calculated water surface movement and flow streamlines around the RFB and CFB in one wave cycle are displayed in Figs. 16 and 17. In these pictures, the water phase domain is colored by blue. The corresponding wave period and wave amplitude are 1.2 s and 0.05 m, respectively.
The results show that the streamline and water flow near the FB are less turbulent. Despite the combined effect of the incident and reflected waves, the wave profile remains relat-   ively regular. When the progressive waves interact with the FB, the shift and rotation movements of the FB disturb the velocity streamlines. In Figs. 16 and 17, as the FB moves back to the downstream side or the leading waves approach the FB, small vortices are generated and developed in the left and right areas of the structure. The eddy generation is correlated with the wave dissipation during the interaction between the FB and water waves. Moreover, part of the waves flow to the bottom of the breakwater leading to wave reflection. The wave energy at the leeside of the FB is reduced to contribute the wave attenuation. It can be found that this numerical model can reproduce the velocity streamlines and water wave's movement during the wave-structure interaction.

Conclusions
This paper discusses the free-surface wave interactions with a cylindrical-dual and rectangular-single pontoon floating breakwater both numerically and experimentally. The experiment has been carried out in a two-dimensional (2D) 40 m-length glass-wall wave flume. The wave-structure interaction model in the time domain is based on the N-S solver coupled with a multi-physics CFD code. An innovative full-structured dynamic layering grid method is realized through dividing the movable region into different zones. The good achievements between the numerical results and measured data confirm that the proposed fully coupled Fluid-Solid model is reasonable. Based on the comparisons, some conclusions can be summarized as follows.
(1) Since the influence of the resonance frequency, the peak values appear for the heave and roll motions. The energy dissipation is enhanced by the vortexes shedding around the fluid field of the FB.
(2) Although the resonance phenomenon of the floating breakwater has an adverse impact on the simulation work, leading to a difference between the numerical and experimental results, it has little impact on the accuracy of the reflection coefficients.
(3) The discrepancy between the mooring forces obtained from the numerical model and experimental data can be attributed to the fact that the dynamic characteristics of the mooring lines are not taken into account, but this difference has a weak influence on the calculated accuracy of the motion responses and wave coefficients.
(4) The velocity streamlines and water wave's movement during the wave interaction with the FB can be accurately reproduced using this proposed model. The bottom of the breakwater makes a contribution to the wave reflection.
The present model is a useful tool to predict the dynamic fluid-structure problems, thus, the developed model can be extended to analyze the interaction effects of a breakwater with nets and even the multi-FBs system. A further studies and results will be reported in the near future.