Comparison of Hydrodynamic Performances Between Single Pontoon and Double Pontoon Floating Breakwaters Through the SPH Method

A numerical study adopting the 2D δ-SPH model is performed to compare the hydrodynamic characteristics of a single pontoon floating breakwater and a double pontoon floating breakwater. Numerical simulations are performed using the δ-SPH model and experimental tests are conducted to validate the numerical model. The numerical results of both the free surface elevations and motions of the floating breakwater are in good agreement with the experimental results. Numerical results show that when the pontoon drafts are larger, the double pontoon floating breakwater performs better in wave attenuations compared with the single pontoon floating breakwater, and for all the drafts, the amplitudes of motions including sway, heave and roll of the double pontoon floating breakwater is always smaller. In addition, increasing the spacing between the two pontoons can further reduce the amplitudes of pontoon motions and improve the wave attenuation ability of the double pontoon floating breakwater.


Introduction
Breakwaters are used to protect harbors and beaches against wave action. Many types of breakwaters have been developed, such as bottom-mounted breakwater (Gao et al., 2020Wang et al., 2019;Li et al., 2020;Liu et al., 2021) and floating breakwater (FB) (Ren et al., 2017;Dai et al., 2018). Although the traditional bottom-mounted breakwater has a good wave reflecting/dissipating ability, it suffers from high cost and negative impact on the shore ecology. In response to these limitations, FB was introduced during last decades as an alternative to traditional bottom-mounted breakwater.
For the simple shape, the single pontoon floating breakwater (SPFB) has become a kind of common FB, and its hydrodynamic characteristics have been studied by many researchers. Rahman et al. (2006) developed a numerical model, based on the Volume of Fluid (VOF) method to estimate the nonlinear dynamics of a submerged SPFB and its mooring force. Peng et al. (2013) studied the interactions between waves and an SPFB moored by inclined tension legs, using the numerical wave tank proposed by Lee and Mizutani (2009). Christensen et al. (2018) experimentally studied the damping mechanisms of SPFBs with three different cross-sections: a regular pontoon, a regular pontoon with wing plates and a regular pontoon with wing plates and porous media. They found that the cross section with wing plates reduced the motions of the breakwater to the largest extend, while the cross section with wing plates and porous media reduced the transmission coefficient most effectively.
The wave dissipating performances of SPFBs are restricted by their simple geometries. To obtain better wave dissipating performances, some researchers proposed dual pontoon floating breakwaters (DPFBs) which have two floating pontoons fixed rigidly to each other based on SPFB. In this manner, the width of SPFB is increased and the total mass can be left unchanged. Ji et al. (2017) proposed a cylindrical DPFB, and studied its hydrodynamic behavior numerically and experimentally. They found that the proposed DPFB exhibited a better wave dissipating performance, especially for waves with larger period and amplitude. Ji et al. (2019) compared the hydrodynamic behaviors of a double-row DPFB and a single-row DPFB experimen-tally. They found that the double-row DPFB significantly reduced the transmission coefficient in comparison with the single-row DPFB, especially for waves with shorter period. However, the reflection coefficients were almost identical between two types. To improve the wave-attenuation ability of the SPFB, Cheng et al. (2021) proposed a new-type DPFB by adding an airbag on the leeward side of the SPFB. They studied the influences of the spacing between two pontoons on the hydrodynamic characteristics, and concluded that the proposed DPFB was superior to SPFB in reducing the wave transmission and the optimal pontoon spacing was 75% of the wavelength.
In the abovementioned research, the performances of FBs were investigated through numerical simulations and experimental test. Nowadays, numerical simulation is a very common research method. In general, there are two kinds of numerical schemes, one is mesh-based method and the other is meshless method. In mesh-based method, the simulation of waves and FB present a number of difficult issues. Some difficulties are related to the motion of FB inside the fluid domain. This is generally modelled by using sliding meshed (Hadžić et al., 2005), overtopping grids (Zhuo et al., 2014) or by representing the body through an immersed boundary technique (Yang and Stern, 2012). Compared with conventional mesh-based numerical schemes, Smoothed Particles Hydrodynamics (SPH) method (Gingold and Monaghan, 1977) uses a series of particles to simulate the fluid domain. These particles do not need grid connections and can move freely according to the governing equations. Therefore, it has the natural merits of easily treating violent free-surface flows, large hydrodynamic loads, and complex boundary conditions especially the fluid-floating body boundaries. SPH is generally categorized into incompressible SPH (hereinafter called ISPH) and weakly compressible SPH (hereinafter called WCSPH). In ISPH model, the fluid incompressibility is imposed strictly by solving a Poisson Pressure Equation (Khayyer and Gotoh, 2009) and it can yield a relatively precise pressure field. The WCSPH assumes a slightly variable density field and calculates the fluid pressure through an equation of state (Gingold and Monaghan, 1977), consequently leading to unphysical pressure fluctuation. However, compared with ISPH, WCSPH can be easily programmed and is particularly suitable for parallel computation. Recently, SPH has been extensively adopted in ocean engineering and coastal engineering such as slamming phenomenon (Gómez-Gesteira and Dalrymple, 2004;Lind et al., 2015;Altomare et al., 2015;Cheng et al., 2017;Ren et al., 2021), wave generation (Ni et al., 2018;Altomare et al., 2018;Wen and Ren, 2018;He et al., 2021) and waves interactions with breakwaters (Meringolo et al., 2015;Khayyer et al., 2018;He et al., 2018He et al., , 2019Tripepi et al., 2020;Quartier et al., 2021). Reviews of the applications of SPH and other particle methods in engineering fields have been presented by Gotoh and Khayyer (2018)  . Historically, FB have been addressed with the SPH method many times. Bouscasse et al. (2013) presented a complete algorithm which can compute fully coupled viscous Fluid-Solid interactions focused on the applications involving nonlinear interaction between waves and floating bodies. Ren et al. (2015) investigated nonlinear interactions between waves and floating bodies. In their work, an improved algorithm based on the dynamic boundary particles (DBPs) is proposed to treat the moving boundary of the floating body. Cui et al. (2021) developed a coupling model between SPH and MAP++, an open-source mooring model. The free-floating and the moored cases were employed as benchmarks to validate the accuracy and stability of the proposed numerical procedure. Then the coupling model was utilized to investigate the wave-attenuation performance of a newly-proposed DPFB under different wave conditions. Zhang et al. (2022) coupled the Moving Particle Semiimplicit (MPS) method and Finite Element Method (FEM) to solve the problem of hydroelastic response of floating structures. Khayyer et al. (2021) presented a 3D entirely Lagrangian meshfree solver for reproduction of fluid-structure interactions corresponding to incompressible fluid flows and elastic structures. In this work, MPS was coupled with a Hamiltonian MPS (HMPS) structure model and then, the accuracy, stability, convergence and conservation properties of this model were verified by many benchmark tests. Cui et al. (2022) designed a new type of FB with moon pool and studied its hydrodynamic behaviors through SPH method. They noted that the new designed FB possesses an outstanding wave energy dissipating capability compared with traditional SPFB.
To the best of our knowledge, although DPFB and SPFB have been studied extensively, the systematical examinations of the differences between them are still very limited. Comparing the difference between them is significant for engineers in choosing more appropriate breakwaters according to local wave conditions. Notably, Liu and Wang (2020) have compared the dual rectangular breakwater, single rectangular breakwater, dual circular breakwater and single circular breakwater numerically. But in their work, the FBs were submerged. In practical engineering, the surface piercing FBs are more common, and thus in the present work, the surface piercing FBs are discussed through an in-house code programmed based on the SPH method.
The remaining paper is organized as follows. Section 2 presents the SPH governing equations and the related numerical strategies. Section 3 describes the experimental tests of wave interaction with SPFB, and the numerical model is validated by the experimental data of the wave surface elevations and the motions of FB. Section 4 uses the numerical results to compare the hydrodynamic performances of SPFB and DPFB. Moreover, the influence of the spacing between two pontoons on the hydrodynamic performances of DPFB is investigated. Conclusions are drawn in Section 5.

SPH fundamentals
In SPH method, the fluid domain is represented by a set of particles. The physical quantities of each particle are computed as an interpolation of the values of the neighbour particles. The value of the function is evaluated through the following convolution integral: where the value of the quantity in the position r and r′ are f (r) and f (r′), respectively. W represents the kernel function, and the area of influence of this function is defined using a characteristic length called smoothing length, h. For a discrete computational domain, Eq.
(1) can be rewritten as: where approximation at interpolation i-th particle is computed by summation over all j-th particle within the area of compact support of kernel function. V j denotes the volume associated with the j-th particle. In the present work, a Wendland C2 (Wendland, 1995) kernel is adopted: where γ is the cut-off radius. h is the smoothing length, and a ratio h/dx = 2.0 is adopted, where dx is the initial particle spacing. q = ||r ij ||/h.

Equation of fluid motion
The evolution of the flow field is described by Navier−Stokes equations. Following the δ-SPH framework (Antuono et al., 2010), in which a diffusive term is added into the continuity equation for smoothing high-frequency acoustic perturbations occurring in the numerical pressure field, the Navier−Stokes equations can be discretized as follows: In Eq. (4), ρ i , u i , p i , r i , m i and V i denote the density, velocity, pressure, position, mass and volume associated with the i-th particle, respectively. The gradient with respect to the coordinates of particle i is denoted as .
The quantity π ij in the viscous part is defined as: The quantity α is the parameter for the artificial viscosity that is usually chosen in the range [0.01 -0.05] for stability reasons. In the present work, it is set as 0.01. The term D i , diffusive contribution in the δ-SPH model, is given by: where parameter δ is the magnitude of the diffusive term, which is set as 0.1. is the renormalized density gradient. This term is calculated as: In weakly-compressible SPH method, an equation of state (Antuono et al., 2010) can be supplemented as closure of Eq. (4): where c 0 is the numerical speed of sound. In order to guarantee the weakly-compressible regime, the numerical sound speed should satisfy the following constraint (Colagrossi and Landrini, 2003): Eq. (4) is integrated in time by using a fourth order Runge−Kutta scheme with frozen diffusion as described by Meringolo et al. (2015). The time step, ∆t, is obtained as the minimum over the following bounds (Meringolo et al., 2019): where ν is the kinematic viscosity, and C u , C δ , and C c are given by 0.125, 0.44 and 2.0, respectively.

Motion of floating body
The FB is assumed to be rigid. According to the Newton's law of motion, the linear and angular equations of motion are expressed in the following form: where V and Ω are the linear velocity and the angular velocity, respectively; M and I are the mass and the moment of inertia; F f-s is the hydrodynamic force acting on the body and T f-s is the torque of the hydrodynamic force acting on the center of mass. F t is the mooring force; T t is the torque of the mooring force. Following the work of Bouscasse et al. 896 CHEN Yong-kun et al. China Ocean Eng., 2022, Vol. 36, No. 6, P. 894-910 (2013), F f-s and T f-s are calculated as follows, respectively: where r c is the position of the center of mass.

Computation of mooring forces
In the present work, the mooring forces are computed by the concentrated mass method (Cheng et al., 2020). In this method, the mooring line is divided evenly into N segments of spring connecting N+1 nodes. Each segment is regarded as a massless spring and its mass is equally distributed to the adjacent nodes. Through the static balance analysis of those nodes, the position of the top end point of the mooring line can be calculated as: where subscripts n and m are the node serial numbers; X N+1 and Z N+1 are the horizontal and vertical coordinates of the top end point of the mooring line, respectively; l 0 is the initial length of each spring; ω is the unit weight of the mooring line in water; k is the tensile stiffness of the mooring line. F x and F z is the horizontal tension and pull-up force on the mooring line, respectively. F n is the total mooring force, and it can be calculated as: (15) After Eq. (14) is calculated, Z N+1 should be compared with the height of anchor point on FB, Z: if Z N+1 > Z, increasing the number of nodes with Z n = 0 to increase the length of touch down of mooring line, on the contrary, if Z N+1 < Z, reducing the number of nodes with Z n = 0 to shorten the length of touching down. If all nodes (except the anchoring point on the seabed) are no touchdown, F z can be increased to reduce the bend of the mooring line. This scheme should be repeated until Z N+1 is approximately equal to Z. Then, X N+1 should also be compared with the horizontal position X of the anchor point on FB in the abovementioned manner. If X N+1 >X, F x decreases, and if X N+1 < X, F x increases. This also should be iterated repeatedly until X N+1 is approximately equal to X.

Experimental setup
The laboratory investigation is performed in a wave flume at the Ocean University of China. The wave flume is 60.0 m long, 3.0 m wide and 1.5 m high. A piston-type wave maker is installed at one end of the flume and at the other end, a wave absorber is installed to dissipate incident waves. The wave flume is divided into two channels by a thin glass wall, with one channel being 2.2 m wide and the other 0.8 m wide. Both the FB model and the wave gauges are placed in the 0.8 m wide channel. To obtain the target waves, a 1:22.5 slope with the length of 4.5 m is set at a distance of 23 m from the wave maker. The water depth is fixed at d = 0.514 m. Three wave heights (H = 0.02, 0.05, and 0.10 m) and five wave periods (T = 1.0, 1.2, 1.4, 1.7 and 2.0 s) are adopted.
The experimental layout is depicted in Fig. 1. At a distance of about 35.15 m from the wave paddle, the FB model is restrained in its equilibrium position by a four-slack-line mooring system, and the draft w = 0.15 m. The mooring line is made of stainless steel and has the total length of 0.62 m with a stiffness of 2.6 N/mm. The horizontal distance between two anchor points of one mooring line (one is on the FB and the other one is on the flume bottom) is 0.358 m. Five wave gauges (denoted as G1, G2, G3, G4 and G5) are used to record the surface elevations. G1 is placed about 3.378 m from the seaside surface of the FB model along the opposite wave incident direction. The spacing between G1 and G3 along the wave incident direction is 1.258 m, and the distance between G1 and G2 is one quarter of the wavelength. We should note here that for wave conditions of T = 1.7 s and T = 2.0 s, the wave gauge G2 is not placed because of too small distance between G2 and G3. G4 and G5 are placed behind the FB. G4 is 2.55 m from the leeside surface of the FB, and G5 is 1.55 m behind G4. The motions of FB are recorded using the NDI Optotrak Certus, a kind of threedimensional (3D) motion measurement system. In this work, the surface elevations at G1, G2 and G3 are used to estimate the reflected wave height H r through two-point method (Goda and Suzuki, 1976). G4 and G5 are used to obtain transmitted wave height H t . Reflection and transmission coefficients, K r and K t , are defined as K r = H r /H and K t = H t /H, respectively. The Response Amplitude Operators of the sway motion R x , heave motion R y and roll motion R r of the FB are defined as: where A x , A y and A r are the amplitudes of the sway, heave and roll motion of the FB, respectively. The geometric details of the FB model used in the experiment are shown in Fig. 2. It is 0.745 m long, 0.5 m wide and 0.3 m high. The total mass and the moment of inertia at its center of gravity are 55.875 kg and 2.20795 kg•m 2 , respectively. The center of gravity is (0.25 m, 0.3725 m, 0.0729 m). The way of establishing the coordinate system is shown in Fig. 2a. Notably, the mooring system consists of four stainless chains which are carefully arranged to ensure a good symmetry along x and y directions. Meanwhile, the width of FB model (0.745 m) is nearly the same as the width of wave flume (0.8 m). Then, the 3D experimental tests can be approximately regarded as 2D cases.

Numerical setup
In the present numerical work, regular waves are generated using the momentum source method (Wen and Ren, 2018), and sponge layer technology (Ren et al., 2015) is applied to absorb unwanted reflected waves. The detailed setup of the numerical flume is shown in Fig. 3, where x is the horizontal coordinate positive in the direction of wave propagation with x = 0 m at the left side of the flume, and y is the vertical coordinate positive upward. The flume is 22 m long with a source zone whose horizontal position is x = 1.3L and its width is L (L is the wavelength). Two sponge layers of 1.3L length are set at both sides of the flume. The locations of numerical wave gauges, pertaining to FB, is the same as that in the experimental tests.

Convergence study
To determine the particle resolution dx, a convergence study is conducted. In the present work, the numerical convergence of three particle resolutions dx = 0.017 m, 0.012 m and 0.007 m are examined. Under the various dx values, 46000-250000 particles are used, resulting in runtimes of 3-7 h on an Intel Core i9-9900X CPU @ 3.10 GHz. A wavealone test is conducted. The target wave condition with water depth d = 0.514 m, wave height H = 0.05 m and wave period T = 1.4 s is examined. The numerical results of the wave height history (in the absence of the FB model) at the beginning and ending of the FB model, x = 12.8 m and 13.3 m, respectively are compared with those of the theoretical solution, as shown in Figs. 4a and 4b. η is the free surface displacement, and it is calculated through Linear wave theory. We can note that the numerical results converge toward the theoretical solution as dx decreasing and dx = 0.007 m is accurate enough. Comparing Figs. 4a and 4b also shows that when dx = 0.007 m the waveform is nearly invariant during wave propagation. This finding shows that the dx = 0.007 m can generate stable target waves and that the wave generation by the momentum source method (Wen and Ren, 2018) adopted in this study is valid. The particle resolution dx = 0.007 m is chosen in the present work.

Comparison between numerical and experimental results
In this section, we validate the present numerical model by comparing the numerical results of the free surface elevations and the FB motions with the experimental results.

Wave surface elevation
The time profiles of wave surface elevations during waves-structure interaction in the simulation and experiment are compared. The comparisons, under the abovementioned three cases, are shown in Figs. 5-7, where it can be observed that the resultant elevations of the incident and reflected waves measured at G1, G2 and G3 and the transmitted waves measured at G4 and G5 in the simulation are all in good agreement with the corresponding experimental data. However, the degree of agreement between the numerical and experimental results in Fig. 5 is not as high as those shown in Fig. 6 and Fig. 7. This is mainly because higherfrequency second-order waves are generated during the interaction between waves (with T = 1.2 s) and FB. For all cases, there is a slight phase difference between the SPH results and the experimental results. This may be attributed to the adopted source width in momentum source wave gen-   erator. To the best of the authors' knowledge, there is not an exact value for the source width. It is suggested to be similar to the target wavelength (Wen and Ren, 2018). In the present work, we just set it equal to the wavelength. To validate the numerical results of other wave condi-tions, the hydrodynamic coefficients, including the reflection coefficient K r and the transmission coefficient K t , of the numerical and experimental results for different wave height are compared in Fig. 8. Again, a good agreement is observed.  900 CHEN Yong-kun et al. China Ocean Eng., 2022, Vol. 36, No. 6, P. 894-910

FB motion
The time profiles of FB motions in the simulation and experiment are compared in Figs. 9-11, where δ x , δ y and θ represent sway, heave and roll, respectively. δ x is positive in the wave propagation direction. δ y is positive in the vertical upward direction and the roll θ is positive in the counterclockwise direction. It can be seen that the numerical heave and roll motions correlate well with the experimental results, while there is a slight discrepancy between the predicted and measured sway motion. The abovementioned source width may still contribute to this discrepancy.
Furthermore, Fig. 12 shows the comparisons between the numerical and experimental results of FB motion within a wave period under wave condition T = 1.4 s, H = 0.05 m. In those figures, t 0 is equal to 23.65 s. We can find that not only the numerical results of the FB motion are in good agreement with the experimental results, but also the shapes of mooring lines are consistent with each other. Notably, in those figures, some particles at the free-surface are detached from the main flow. This phenomenon should be attributed to the absence of exact satisfaction of dynamic free-surface boundary condition in SPH method (You et al., 2021;Luo Fig. 8. Comparisons between the experimental and numerical results of the hydrodynamic coefficients.   CHEN Yong-kun et al. China Ocean Eng., 2022, Vol. 36, No. 6, P. 894-910 et al., 2021 and the kernel function truncation at free-surface boundary (Colagrossi et al., 2011). By the way, Fig. 13 presents the instantaneous particle snapshots together with the reproduced pressure field of t = t 0 +0.25T and t 0 +0.75T of Fig. 12. We can note that the present model yields a stable pressure field.
In the above, we just compared the shapes of mooring lines with numerical and experimental results during the process of wave interaction with FB. This validation for mooring line system may not be enough. In the present experiment, the mooring line force is not measured. So, the experimental data presented in Liang et al. (2022) are used for validation. In that work, there are three different lengths of mooring lines, 1.567 m, 1.125 m, and 0.809 m, and the mooring line with a length of 0.809 m is used in the present work. Here we do not go into details of the experiments, which can be found in Liang et al. (2022) . Fig. 14 compares the numerical and experimental mooring forces on the seaward side F s and leeward side F l of the FB under wave condition of T = 1.4 s and H = 0.10 m. We note that the present numerical model slightly overestimates the mooring line force. This may be because we ignore the hydrodynamic and frictional contributions in the mooring line system. Fig. 15 presents the geometrical configurations of SPFB and DPFB. The width and height of SPFB (B and A) are kept constant with those in the experiment. l is the spacing between two pontoons in DPFB. The position of the centre of gravity and the rotation radius of SPFB and the length and elastic modulus of the mooring lines for both SPFB and DPFB are the same as that in the experiment.

Comparisons of DPFB and SPFB
In this section, we change the draft of FB by changing its mass. For the same draft, the masses of DPFB and SPFB are the same. The rotation radius of DPFB can be directly calculated from that of SPFB. Parameters used in the simulation described in this section are listed in Table 1. Three wave heights (H = 0.02, 0.05, 0.10 m), five wave periods (T =

Reflection, transmission and energy dissipation coefficients
In this subsection, the energy dissipation coefficient K d is defined as K d = 1−K t 2 -K r 2 . The comparisons of the reflection coefficient K r , transmission coefficient K t and energy dissipation coefficient K d under different drafts of DPFB and SPFB are presented. Fig. 16 shows the comparisons of the variations in reflection coefficient K r of DPFB and SPFB at different wave periods T under different drafts w. For DPFB, K r first increases and then decreases with the increasing of wave period T. For different drafts, K r reaches the maximum value at different wave periods T. For w = 0.10 and 0.15 m, it reaches the maximum value when wave period T is 1.2 s. However, for w = 0.20, 0.25 and 0.30 m, that is T = 1.4 s. This is mainly because when w = 0.10 and 0.15 m, the natural period of DPFB is close to 1.2 s and when w = 0.20, 0.25 and 0.30 m, the natural period of DPFB is close to 1.4 s. To explain this, choosing w = 0.15 m and w = 0.25 m of DPFB as examples, a numerical decay test is carried out. In the test, DPFB rotated ten degrees counterclockwise around the center of gravity at the beginning of the calculation. The sketch of the test is depicted in Fig. 17. The time histories of the roll θ for w = 0.15 m and 0.25 m are reported in Figs. 18a and 18b. In those figures, t 1 (t 5 ), t 2 (t 6 ), t 3 (t 7 ), t 4 (t 8 ) are equal to 1.18 s (1.33 s), 2.36 s (2.67 s), 3.52 s (4.04 s), 4.66 s (5.36 s), respectively. The difference between t 1 (t 3 ) and t 2 (t 4 ), (t 7 ) and t 6 (t 8 ) are about 1.16 s and 1.34 s. This reveals that the DPFB's natural period of these two drafts are close to 1.2 s and 1.4 s, respectively. For SPFB, the variations in reflection K r at different wave periods T are also different under different drafts. For w = 0.10, 0.15 and 0.20 m, K r decreases with the increasing of wave period T in general. For w = 0.25 and 0.30 m, K r first decreases and then increases. Comparing DPFB and SPFB, we can find that under the same draft, K r of DPFB is larger than that of SPFB for different wave periods except for T = 1.0 s. The values of K r for both DPFB and SPFB decrease as the wave period T increases from T = 1.4 s to T = 2.0 s. In this process of decreasing, the difference between them is also reduced. When the wave period is T = 2.0 s, the difference in reflection coefficient K r between DPFB and SPFB is very small. Fig. 19 shows the comparisons of the variations in trans-   and SPFB increase from wave period T = 1.2 s to T = 2.0 s. This suggests that the wave attenuation performances of DPFB and SPFB are both better for shorter rather than longer waves. For w = 0.10 and 0.15 m, the difference in K t between DPFB and SPFB is small which can almost be neglected and for other drafts, K t of DPFB is smaller than that of SPFB. This reveals that compared with SPFB, DPFB is more suitable for installation with larger draft. Fig. 20 shows the comparisons of the variations in the energy dissipation coefficients K d of DPFB and SPFB at different wave periods T under different drafts w. It can be found that in general, K d , both for DPFB and SPFB, decreases with the increasing wave period T. Moreover, under the same draft, the difference of K d between SPFB and DPFB is slight. This reveals that with larger draft, compared with SPFB, DPFB reflects more wave incident energy and results in a smaller transmission coefficient K t . The energy dissipation processes directly depend on the formation of vortexes, in which the mechanical energy is transferred from the macroscopic to the microscopic scales and therefore   dissipated into heat. Fig. 21 shows comparisons of vorticity field between DPFB and SPFB, for the case T = 1.4 s, H = 0.05 m. The vorticity of each fluid particle is defined as the curl of the local velocity (Monaghan, 1992): According to the above equation, a clockwise (anticlockwise) circulation corresponds to the negative (positive) value of vorticity. From Fig. 21, it can be seen that there is little difference in the vorticity evolution between DPFB and SPFB and the vortex mainly occurs at the corners of the front end and rear end of two FBs. Moreover, it is surprised for us that there is little vortex at the corners between the two pontoons in DPFB. In the authors' opinion, this may be the main reason why the energy dissipation efficiencies of DPFB and SPFB are not much different.

Motions of FB
After comparing the reflection coefficient K r , transmission coefficient K t and energy dissipation coefficient K d of DPFB and SPFB under different drafts, the motions of the DPFB and SPFB are compared here. Figs. 22-24 show the comparisons of the variations in amplitudes of motions including heave motion R z , sway motion R x and roll motion R r of DPFB and SPFB at different wave periods T under different drafts. We can find that under the same draft, the trends of R z , R x and R r of DPFB with the wave period T are  the same as those of SPFB. In addition, R z , R x and R r of DPFB are generally smaller than those of SPFB. Based on this finding, we can conclude that DPFB is somewhat safer than SPFB.

Influence of pontoon spacing for DPFB
In this section, we analyze the influence of the spacing between two pontoons l on the hydrodynamic characteristics of DPFB. Parameters used in the simulation described in this section are listed in Table 2. There are three wave heights (H = 0.02, 0.05, 0.10 m), five wave periods (T = 1.0, 1.2, 1.4, 1.7, 2.0 s), six spacing (l = 0.3, 0.5, 0.7, 0.9, 1.1, 1.3 m) and one draft (w = 0.15 m), which are the same as those in the experiment.
The variations in the reflection coefficient K r , transmission coefficient K t and energy dissipation coefficient K d at different normalized spacing l/B under different wave periods T for DPFB are presented in Fig. 25. We can find that, except for the wave period T = 1.0 s, K r increases with the increasing of spacing. Also, it can be seen that, in general, K t decreases with the increasing of l/B, but for the wave periods of T = 1.7 s and T = 2.0 s, the spacing l has a nonsignificant influence on K t . Generally speaking, the energy dissipation coefficient K d increases with the increasing spacing. In conclusion, increasing the spacing is beneficial to improving the wave attenuation performance of DPFB, but we should point out that this improvement effect is very slight.
The variations in the amplitudes of sway, R x , heave, R z , and roll, R r , motions at different normalized spacing l/B under different wave periods T for DPFB are presented in Fig. 26. From this figure, we can find that except for the wave period T = 1.0 s, the amplitudes of the motions of DPFB (including roll, heave and sway) decrease with the increasing of the spacing. Comparing with wave period T = 1.2 s and 1.4 s, this trend at T = 1.7 s and 2.0 s is weaker, and even for T = 2.0 s, this trend can be ignored. Based on the above description we can conclude that increasing the spacing between two pontoons can make DPFB safer.

Conclusions
The hydrodynamic characteristics of the DPFB and SPFB were numerically investigated using an in house δ-SPH model. An experiment was conducted to validate the present numerical model, and the numerical and experimental results were noted to be in good agreement. Based on the present numerical model, the hydrodynamic characteristics of the DPFB and SPFB were compared, and then the influence of the spacing between two pontoons on the hydrodynamic characteristics of DPFB was investigated. Numerical results showed that in comparison with SPFB, DPFB possessed a larger reflection coefficient and after with a larger draft, it has a smaller transmission coefficient. So, we suggest that DPFB should be more suitable for installation with a larger draft. Numerical results also showed that there was little difference in vorticity evolution between DPFB and SPFB, and thus their energy dissipation coefficients were nearly identical. As for motions, the amplitudes of motions including  CHEN Yong-kun et al. China Ocean Eng., 2022, Vol. 36, No. 6, P. 894-910 sway, heave and roll of DPFB were smaller than those of SPFB. In addition, increasing the spacing between two pontoons of DPFB could further reduce the amplitudes of FB motions and improve the wave attenuation ability.

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