# Numerical Study on the Hydrodynamic Characteristics of a Double-Row Floating Breakwater Composed of a Pontoon and an Airbag

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

#### 2.1. Fluid Equations

**g**is the gravitational acceleration; ρ, p, V,

**u**and

**r**are the density, pressure, volume, velocity and position, respectively; ρ

_{0}= 1000 kg/m

^{3}is the reference density; h = 1.5 δ

_{p}is the smoothing length with δ

_{p}being the initial particle spacing; δ = 0.1 is the diffusion coefficient; α = 8 ν/(hc

_{0}) is the artificial viscosity coefficient with ν being the kinematic viscosity of fluid and c

_{0}being the numerical speed of sound. Its value, in this study, varies between 2.7 × 10

^{−7}and 2.4 × 10

^{−5}depending on different cases; W is the Wendland C2 kernel function [84] defined as:

_{ij}has the following expression:

_{ij}has the following expression:

#### 2.2. Floating Body Equations

**V**and

**Ω**are the mass, moment of inertia, linear and angular velocities of the floating body, respectively;

**F**

_{f}and

**T**

_{f}are the fluid force acting on the floating body and its torque to the center of mass, respectively;

**F**

_{m}and

**T**

_{m}are the mooring force and its torque to the center of mass, respectively.

#### 2.2.1. Solid Boundary Treatment

_{p}is the total number of fluid particles i in the kernel support of DBP k; z is the vertical position of the particle. According to Equation (7), we have:

**f**

_{i→k}of all fluid particles in the kernel support, the total fluid force exerted on DBP k is obtained:

**f**

_{k}of all DBPs constituting the floating body:

**r**

_{c}is the position of the center of mass of the floating body.

#### 2.2.2. Mooring System Implementation

_{m}

_{,p}is the mooring force between nodes p and p + 1, and F

_{m}

_{,p−1}is the mooring force between nodes p − 1 and p; θ

_{p}is the angle between segment p and horizontal axis, and θ

_{p}

_{−1}is the angle between segment p − 1 and horizontal axis; ω is the wet weight per unit length of the mooring line; l

_{0}is the initial length of each segment.

_{x}

_{,p}and F

_{z}

_{,p}are the horizontal and vertical mooring force between p and p + 1, respectively; F

_{u}is the uplift force at the anchor point; subscript q is a counter; ω

_{q}= ω/2 when q = 1, and ω

_{q}= ω when 2 ≤ q ≤ p. Thus,

_{q}is:

_{P}

_{+1}is compared with the vertical position of the mooring point z

_{m}. When Z

_{P}

_{+1}< z

_{m}, nodes of the mooring line are lifted successively from No. P + 1 to No. 2. In case all nodes have been lifted, the uplift force F

_{u}begins to increase to reduce the slackness of the mooring line, hence further elevating the mooring end. When Z

_{P}

_{+1}> z

_{m}, F

_{u}decreases, and if Z

_{P}

_{+1}is still greater than z

_{m}when F

_{u}= 0, nodes fall to the seabed successively from No. 2 to No. P + 1. Then, X

_{P}

_{+1}is compared with the horizontal position of the mooring point x

_{m}. When X

_{P}

_{+1}< x

_{m}, the horizontal mooring force F

_{x}

_{,P+1}increases, and when X

_{P}

_{+1}> x

_{m}, F

_{x}

_{,P+1}decreases. The above process is repeated until both the error between Z

_{P}

_{+1}and z

_{m}and the error between X

_{P}

_{+1}and x

_{m}are allowable. Finally, the mooring force F

_{m}

_{,P+1}is calculated according to Equations (21)–(23).

## 3. Model Validation

#### 3.1. Numerical Wave Flume

_{i}is the horizontal position of the fluid particle labeled as i; x

_{s}is the horizontal position of the interface between sponge layer and common fluid domain; L

_{s}= λ is the length of the sponge layer with λ being the wavelength; β is the intensity coefficient of the sponge layer, which can be taken from Reference [91]. In this study, β ranges between 0.8 and 5.7 depending on different cases.

_{s}and leeward mooring force F

_{l}, respectively. On the upstream of the floating body, #1 and #2 wave gauges were arranged, and the two-point method [92] was used to calculate the reflected wave height H

_{r}. On the downstream of the floating body, #3 wave gauge was placed. Since high-order wave components generated during the interaction between wave and floating body were weak, the transmitted wave height H

_{t}was taken as the height difference between the wave crest and wave trough. Thus, the wave reflection coefficient C

_{r}and the wave transmission coefficient C

_{t}of the floating body are defined as

_{s}, heave motion RAO

_{h}and roll motion RAO

_{r}of the floating body are defined as

_{s}, A

_{h}and A

_{r}are the amplitudes of the sway, heave and roll motion of the floating body, respectively. For each wave condition, 30 wave cycles were run and the last 15 wave cycles were taken for analysis. The height and period of the generated wave were checked prior to simulating the interaction between wave and moored floating body.

#### 3.2. Validation Case 1

#### 3.2.1. Setup of Physical and Numerical Models

_{p}= 5, 10 and 20 (i.e., initial particle spacing δ

_{p}= 2 cm, 1 cm and 0.5 cm) were adopted to check the numerical convergence and find the optimal δ

_{p}. Under various H/δ

_{p}and T, a total number of 31,380–799,240 particles were used, resulting in runtimes of 0.4–61.9 h on an Intel Core i9-9900X CPU @ 3.50GHz.

#### 3.2.2. Comparison of Experimental and Numerical Results

_{t}and reflection coefficients C

_{r}of the cuboid floating pontoon. Regardless of the particle resolution H/δ

_{p}, the overall trends are in reasonable agreement. When T ≤ 1.6 s, the numerical C

_{t}and C

_{r}are generally less than the experimental data. This is related to the dissipation nature of the SPH method [94]. Since the physical and numerical wave gauges used to measure the transmitted and reflected waves are at the same distance from the pontoon, more energies have been dissipated before the numerical wave arrives at the wave gauges. The numerical dissipation decreases with the increment of T, and when T > 1.6 s the numerical C

_{t}and C

_{r}become no longer less but even greater than the experimental data. The greater numerical results might be associated with the absence of turbulence model in the present numerical model. On such premise, energy dissipation caused by the numerical turbulent flow is under-predicted. It can be further observed that both numerical C

_{t}and C

_{r}basically converge toward the experimental data with the increase of H/δ

_{p}. Doubling H/δ

_{p}from 5 to 10 significantly enhances the numerical accuracy, while doubling H/δ

_{p}from 10 to 20 has limited effect. Therefore, based on a trade-off between numerical accuracy and computational efficiency, H/δ

_{p}= 10 is a proper particle resolution in terms of C

_{t}and C

_{r}computations. It is worth mentioning that H/δ

_{p}= 10 is also recommended by Altomare et al. [95] as the threshold of accurately and affordably modelling the wave generation, propagation and absorption.

_{s}, heave motion RAO

_{h}and roll motion RAO

_{r}of the cuboid floating pontoon. Intuitively, the experimental and numerical RAO

_{s}are in better agreement. Although the experimental and numerical RAO

_{h}and RAO

_{r}have the same trends, the numerical results are noticeably less than the experimental data. However, if taking account of the different variation ranges of RAO

_{s}, RAO

_{h}and RAO

_{r}, the degrees of agreement shown in Figure 6a–c are comparable and all acceptable. According to Equations (9) and (10), the motion of the pontoon is subjected to fluid force and mooring force, which means that the discrepancies in RAO

_{s}, RAO

_{h}and RAO

_{r}are caused by the computational inaccuracies of fluid force and mooring force.

_{p}= 20 leads to the favorable numerical results compared with the experimental data. H/δ

_{p}= 10 yields the second-best numerical results which are sufficiently close to those of H/δ

_{p}= 20. Therefore, H/δ

_{p}= 10 is the optimal particle resolution in terms of RAO

_{s}, RAO

_{h}and RAO

_{r}computations.

_{s}and leeward side F

_{l}of the cuboid floating pontoon. There were two anchor chains on each side of the 76 cm-wide physical pontoon and there was one anchor chain on each side of the 100 cm-wide numerical pontoon. Thus, the computed mooring force was divided by 2 and multiplied by 0.76 before being compared with the experimental data. From Figure 7, it can be seen that the numerical F

_{s}and F

_{l}follow the same trends as the experimental data, but the numerical results are generally over-predicted. Although the discrepancies seem significant, the relative errors (RE = |numerical value–experimental value|/experimental value) are small. Under H/δ

_{p}= 5, the average REs of F

_{s}and F

_{l}are 8.0% and 5.5%, respectively. Under H/δ

_{p}= 10, the average REs of F

_{s}and F

_{l}decrease to 6.3% and 4.1%, respectively. Under H/δ

_{p}= 20, the average REs of F

_{s}and F

_{l}are as small as 5.1% and 3.4%, respectively. The two aforementioned facts could have accounted for the discrepancies in the mooring force. First, the side-wall effect of the physical experiment slightly interferes with the motion of the pontoon. It is known that the mooring force is largely dependent on the mooring position. Thus, the disturbed motion of the physical pontoon lowers the degree of agreement between the experimental and numerical mooring force. Second, the present mooring model neglects the hydrodynamic, inertial, damping and frictional contributions. As documented by Hall et al. [96] and Davidson and Ringwood [97], the mooring dynamics are quite different, incorporating or not incorporating these contributions.

#### 3.3. Validation Case 2

#### 3.3.1. Setup of Physical and Numerical Models

_{p}= 10 (i.e., δ

_{p}= 1 cm), thus requiring 125,232–199,304 particles in the simulations and resulting in runtimes of 2.8–6.2 h.

#### 3.3.2. Comparison of Experimental and Numerical Results

_{t}and C

_{r}of the dual cylindrical floating pontoon. Again, the overall agreement is satisfactory. In Figure 9a, when T ≤ 1.2 s the numerical C

_{t}are less than the experimental data, and when T > 1.2 s the numerical C

_{t}are greater. This discrepancy has been explained in Section 3.2.2 as the comprehensive result of the dissipation nature of the SPH method and the absence of turbulence model in the present numerical model. It is worth noticing that in Figure 9a the threshold value (less than which the numerical results are under-predicted while greater than which the numerical results are over-predicted) of T is 1.2 s which is less than the 1.6 s in Figure 5a. This is probably because the 2-D numerical model abandons the nine connecting rods of the 3-D physical model. Thus, the turbulent energy dissipation in the process of flow passing through the rods is neglected in the simulation, resulting in over-predicted numerical C

_{t}. Unlike the transmitted wave, the reflected wave mainly depends on the frontal area and structural shape of the floating breakwater. Thus, abandoning the physical rods makes little difference on the numerical C

_{r}. The manifestation is that the numerical C

_{r}in Figure 9b has the same threshold value of T, namely 1.6 s, as Figure 5b.

_{s}, RAO

_{h}and RAO

_{r}of the dual cylindrical floating pontoon. For RAO

_{s}and RAO

_{r}, the maximum relative errors REs are 19.9% and 13.7%, which occur at T = 1.0 s and 2.0 s, respectively. These two REs are remarkable just because the experimental data, as denominators, are small. However, the average REs of RAO

_{s}and RAO

_{r}within the entire T range are only 8.6% and 7.5%, respectively. As for RAO

_{h}, T = 1.2 s corresponds to the maximum RE equaling 9.4%, and the average RE is 5.8%. In view of the insignificant average REs, the RAO

_{s}, RAO

_{h}and RAO

_{r}computations regarding the cylindrical floating pontoon are reliable.

_{s}and F

_{l}of the dual cylindrical floating pontoon. Owing to the side-wall effect of the physical experiment and the lack of hydrodynamic, inertial, damping and frictional contributions in the numerical mooring model, discrepancies between experimental and numerical F

_{s}and F

_{l}can be observed. The degrees of deviation are comparable with those in Figure 7, and quantitatively, the average REs of F

_{s}and F

_{l}are merely 5.4% and 3.6%, respectively. Therefore, the mooring force computation regarding the cylindrical floating pontoon is also reliable.

## 4. Results and Analyses

#### 4.1. Setup of Double-Row Floating Breakwater

_{f}, height H

_{f}, draft d

_{f}

_{1}, mass M

_{1}and moment of inertia I

_{1}of the pontoon were 10 m, 2.5 m, 1.5 m, 15 t, and 136.56 t⋅m

^{2}, respectively, and the height of the center of mass c

_{m}

_{1}was 0.75 m. The diameter Φ

_{c}, draft d

_{f}

_{2}, mass M

_{2}and moment of inertia I

_{2}of the airbag were 5 m, 4 m, 1.5 m, 16.84 t and 76.93 t⋅m

^{2}, respectively, and the height of the center of mass c

_{m}

_{2}was 2.18 m. Both pontoon and airbag were restrained by two mooring lines with mooring angle θ = 30°, tensile stiffness EA = 2.76 MN and wet weight w = 0.19 kN/m. For pontoon, mooring points located at its upper corners. The bending length of the mooring line l

_{m}

_{1}was 78.67 m and the horizontal distance between mooring point and anchoring point l

_{x}

_{1}was 74.56 m. 2# and 3# load cells were installed to measure the seaward mooring force F

_{s}

_{1}and leeward mooring force F

_{l}

_{1}, respectively. For airbag, mooring points located at its outermost tips, the bending length of the mooring line l

_{m}

_{2}was 69.04 m and the horizontal distance between mooring point and anchoring point l

_{x}

_{2}was 65.69 m. 4# and 5# load cells were installed to measure the seaward mooring force F

_{s}

_{2}and leeward mooring force F

_{l}

_{2}, respectively. 1# wave gauge was placed 0.75λ downstream from the airbag to obtain the wave transmission coefficient C

_{t}. s varied between λ/4 and λ with an interval of λ/4. Water depth d was fixed at 20 m. Regular waves of four heights, H = 1 m, 2 m, 3 m and 4 m, and four periods, T = 4 s, 5 s, 6 s and 7 s, were tested. Since H/δ

_{p}= 10 has been shown to be suitable regarding numerical accuracy and computational efficiency, δ

_{p}= 10 cm was taken to handle the minimum wave height H = 1 m. Thus, a total numerical of 204,810–641,780 particles were deployed depending on different T and s, and it took 6.3–42.6 h to simulate 30 wave cycles.

#### 4.2. Influence of Separation Distance

_{t}of the double-row floating breakwater with different separation distance s/λ. Meanwhile, the results are compared with C

_{t}of a single pontoon which is identical to the one used in the double-row floating breakwater. When T = 4 s, the average C

_{t}of the pontoon-airbag system is 0.03 less than C

_{t}of the single pontoon, and when T = 6 s, the average C

_{t}of the pontoon-airbag system is 0.14 less than C

_{t}of the single pontoon. It indicates that adding an airbag on the leeward side of the floating pontoon helps to improve the wave attenuation performance, and the improvement effect is more significant in a long-wave regime. In addition, C

_{t}of the pontoon-airbag system decreases with increasing s/λ and turns to increase after s/λ > 0.75. Thus, s/λ = 0.75 is the optimal separation distance of the pontoon-airbag double-row floating breakwater in terms of wave attenuation performance.

_{s}, heave motion RAO

_{h}and roll motion RAO

_{r}of the double-row floating breakwater with different separation distance s/λ. It is known that the wave surface measured at 1# wave gauge in Figure 12 is comprised of the transmitted wave after the incident wave passes through the pontoon-airbag system and the radiated wave generated by the motion of the pontoon-airbag system. Thus, the wave height on the leeward side of the pontoon-airbag system is usually positively correlated with the motion amplitudes of the pontoon and airbag, and the variation trends of RAO

_{s}, RAO

_{h}and RAO

_{r}with s/λ in Figure 14 are consistent with the variation trend of C

_{t}in Figure 13. That is, RAO

_{s}, RAO

_{h}and RAO

_{r}of pontoon and airbag decrease with increasing s/λ and turn to increase after s/λ > 0.75.

_{s}and leeward side F

_{l}of the double-row floating breakwater with different separation distance s/λ. As mooring force is predominantly dependent on mooring position, F

_{s}and F

_{l}in Figure 15 follow the same variation trends as RAO

_{s}, RAO

_{h}and RAO

_{r}in Figure 14. The minimum F

_{s}and F

_{l}are obtained when s/λ = 0.75. However, the maximum F

_{s}and F

_{l}occur when s/λ = 0.25, which is different from the result that the maximum RAO

_{s}, RAO

_{h}and RAO

_{r}appear when s/λ = 1.00. In addition, under both wave periods, F

_{s}and F

_{l}of the airbag are greater than those of the pontoon. This phenomenon is consistent with Figure 14c where RAO

_{r}of the airbag is greater than that of the pontoon, but is different from Figure 14a,b where RAO

_{s}and RAO

_{h}of the airbag are less than those of the pontoon. This implies the roll motion of the floating body has the greatest influence on the mooring force.

#### 4.3. Influence of Wave Parameters

_{t}are plotted in Figure 16, and they are compared with C

_{t}of a single pontoon when the leeward airbag is absent. In Figure 16a, C

_{t}of both pontoon-airbag system and single pontoon decrease with the increase of H. The reason may be that as H increases, the interaction between wave and floating body is intensified; thereby, the wave energy loss due to fluid viscosity becomes heavier. In Figure 16b, C

_{t}of both pontoon-airbag system and single pontoon increase with the increase of T, apparently because longer waves have stronger transmission capacity. In fact, the above laws hold for most species of floating breakwaters. Within the concerned wave conditions, the pontoon-airbag system has less C

_{t}than the single pontoon, i.e., presenting better wave attenuation performance. For one thing, the improvement weakens as H increases. When H = 1 m, 1.5 m, 2 m, 3 m and 4 m, C

_{t}of the pontoon-airbag system are 0.23, 0.22, 0.19, 0.19 and 0.13 less than C

_{t}of the single pontoon, respectively. For another, the improvement strengthens as T increases. When T = 4 s, 5 s, 6 s and 7 s, C

_{t}of the pontoon-airbag system are 0.07, 0.13, 0.18 and 0.19 less than C

_{t}of the single pontoon, respectively.

_{s}, RAO

_{h}and RAO

_{r}of the double-row floating breakwater under different H and T. From Figure 17a–c, it can be seen that H has limited influence on RAO

_{s}, RAO

_{h}and RAO

_{r}of the pontoon. As for the airbag, since more wave energy is dissipated during the wave-pontoon interaction with the increase of H, the intensity of the leeward wave field is relatively weakened, leading to the rapid decreases in RAO

_{s}, RAO

_{h}and RAO

_{r}of the airbag. These two trends yield that, when H is relatively small, the motion amplitudes of the airbag are greater than those of the pontoon, while when H is relatively large, the motion amplitudes of the pontoon become greater. For RAO

_{s}, RAO

_{h}and RAO

_{r}, the threshold values of H are 3.6 m, 2.1 m and 4 m, respectively.

_{s}, RAO

_{h}and RAO

_{r}of the pontoon decrease with the increase of T, while RAO

_{s}, RAO

_{h}and RAO

_{r}of the airbag increase with the increase of T. This is evidently because shorter waves have weaker transmission capacity; thus, the wave energy mainly acts on the seaward pontoon, leading to the greater motion amplitudes of pontoon than an airbag. The wave transmission capacity grows with T; thus, more wave energy acts on the leeward airbag, resulting the greater motion amplitudes of the airbag than pontoon. For RAO

_{s}and RAO

_{h}, the threshold values of T are 5.7 and 6.8 s, respectively. However, within the entire T range, RAO

_{r}of the airbag is always greater than that of the pontoon, because the moment of inertia of the airbag, I

_{2}= 76.93 t⋅m

^{2}, is much smaller than that of the pontoon, I

_{1}= 136.56 t⋅m

^{2}.

_{s}and F

_{l}of the double-row floating breakwater under different H and T. Since mooring force is closely related to mooring position, F

_{s}and F

_{l}in Figure 18 follow the same variation trends as RAO

_{s}, RAO

_{h}and RAO

_{r}in Figure 17. That is, under a given T, F

_{s}and F

_{l}of both pontoon and airbag decrease with the increase of H, and F

_{s}and F

_{l}of the airbag decrease more rapidly. Under a given H, F

_{s}and F

_{l}of the pontoon decrease with the increase of T, while F

_{s}and F

_{l}of the airbag increase with the increase of T. In addition, when H ≤ 2.6 m, F

_{s}and F

_{l}of the airbag are greater than those of the pontoon, and when H > 2.6 m, F

_{s}and F

_{l}of the pontoon turn to be greater. When T ≤ 6.2 s, F

_{s}and F

_{l}of the pontoon are greater than those of the airbag, and when T > 6.2 s, F

_{s}and F

_{l}of the airbag become greater.

## 5. Conclusions and Future Perspectives

- (1)
- The wave transmission coefficient, response amplitude operators and mooring force of the double-row floating breakwater first decrease then increase with the increase of the separation distance between pontoon and airbag. The optimal separation distance is 0.75 times the wavelength.
- (2)
- At the optimal separation distance and within the concerned 1–4 m wave heights and 4–7 s wave periods, the pontoon-airbag system presents better wave attenuation performance than a single pontoon. This improvement weakens as wave height increases while strengthens as the wave period increases.
- (3)
- The wave transmission coefficient, response amplitude operators and mooring force of the double-row floating breakwater all decrease with the increase of incident wave height, which indicates its potential application in a high-wave regime.
- (4)
- The wave transmission coefficient of the pontoon-airbag system, the response amplitude operators and mooring force of the airbag increase with the increase of incident wavelength, while the response amplitude operators and mooring force of the pontoon decrease.

- (1)
- The airbag was temporarily assumed to be rigid and its mass distribution did not change with the airbag motion. To reflect its physical behavior more realistically, the external flexibility and the internal ballast water should be taken into account.
- (2)
- No turbulence model was adopted in the fluid equations, and the lumped-mass mooring model neglected the hydrodynamic, inertial, damping, and frictional contributions. To compute the hydrodynamic characteristics of floating breakwater more accurately, a suitable turbulence model and a sophisticated mooring model should be employed.
- (3)
- Only the influence of separation distance and wave parameters on the hydrodynamic characteristics of double-row floating breakwater was analyzed. To design the floating breakwater more systematically, a parametric study of the pontoon and airbag should be conducted.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Different types of floating breakwaters: (

**a**) Cuboid pontoon [2]; (

**b**) Cylindrical pontoon [3]; (

**c**) Π-type pontoon [4]; (

**d**) Cuboid pontoon-plates [5];

**(e)**Pontoon with an air chamber [6]; (

**f**) Pontoon with two air chambers [7]; (

**g**) Permeable structure [8]; (

**h**) Y-Type pontoon [9]; (

**i**) Trapezoidal pontoon-porous plates [10]; (

**j**) Horizontal plate-nets [11]; (

**k**) Dual cuboid pontoon [12]; (

**l**) Dual cylindrical pontoon [13]; (

**m**) Dual cylindrical pontoon-nets [14]; (

**n**) Triple cuboid pontoons [15]; (

**o**) Double-row cuboid pontoons [16]; (

**p**) Double-row cuboid pontoons-mesh cage [17]; (

**q**) Double-row cylindrical pontoons-mesh cage [18]; (

**r**) F-type pontoon [19]; (

**s**) Trapezoidal porous pontoons [20]; (

**t**) T-type pontoon [24]; (

**u**) Curtain wall [69]; (

**v**) Pontoon with different cross-sections [73].

**Figure 3.**Numerical wave flume used to simulate the interaction between waves and mooring floating bodies.

**Figure 4.**Setup of the cuboid floating pontoon and its mooring system: (

**a**) Physical model; (

**b**) Numerical model.

**Figure 5.**Comparison of the experimental and numerical wave transmission coefficients and reflection coefficients of the cuboid floating pontoon: (

**a**) Wave transmission coefficient C

_{t}; (

**b**) Wave reflection coefficient C

_{r}.

**Figure 6.**Comparison of the experimental and numerical response amplitude operators of the cuboid floating pontoon: (

**a**) Sway motion RAO

_{s}; (

**b**) Heave motion RAO

_{h}; (

**c**) Roll motion RAO

_{r}.

**Figure 7.**Comparison of the experimental and numerical mooring force of the cuboid floating pontoon: (

**a**) Seaward side F

_{s}; (

**b**) Leeward side F

_{l}.

**Figure 8.**Setup of the dual cylindrical floating pontoon and its mooring system: (

**a**) Physical model; (

**b**) Numerical model.

**Figure 9.**Comparison of the experimental and numerical wave transmission coefficients and reflection coefficients of the dual cylindrical floating pontoon: (

**a**) Wave transmission coefficient C

_{t}; (

**b**) Wave reflection coefficient C

_{r}.

**Figure 10.**Comparison of the experimental and numerical response amplitude operators of the dual cylindrical floating pontoon: (

**a**) Sway motion RAO

_{s}; (

**b**) Heave motion RAO

_{h}; (

**c**) Roll motion RAO

_{r}.

**Figure 11.**Comparison of the experimental and numerical mooring force of the dual cylindrical floating pontoon: (

**a**) Seaward side F

_{s}; (

**b**) Leeward side F

_{l}.

**Figure 12.**Numerical wave flume used to simulate the interaction between waves and a pontoon-airbag double-row floating breakwater.

**Figure 13.**Comparison of the wave transmission coefficients C

_{t}of the double-row floating breakwater with different separation distance s/λ.

**Figure 14.**Comparison of the response amplitude operators of the double-row floating breakwater with different separation distance s/λ: (

**a**) Sway motion RAO

_{s}; (

**b**) Heave motion RAO

_{h}; (

**c**) Roll motion RAO

_{r}.

**Figure 15.**Comparison of the mooring force of the double-row floating breakwater with different separation distance s/λ: (

**a**) Seaward side F

_{s}; (

**b**) Leeward side F

_{l}.

**Figure 16.**Comparison of the wave transmission coefficients C

_{t}of the double-row floating breakwater under different wave parameters: (

**a**) Wave height H = 1–4 m and wave period T = 6 s; (

**b**) Wave height H = 3 m and wave period T = 4–7 s.

**Figure 17.**Comparison of the response amplitude operators of the double-row floating breakwater under different wave parameters: (

**a**) Sway motion RAO

_{s}, H = 1–4 m and T = 6 s; (

**b**) Heave motion RAO

_{h}, H = 1–4 m and T = 6 s; (

**c**) Roll motion RAO

_{r}, H = 1–4 m, T = 6 s; (

**d**) Sway motion RAO

_{s}, H = 3 m, T = 4–7 s; (

**e**) Heave motion RAO

_{h}, H = 3 m, T = 4–7 s; (

**f**) Roll motion RAO

_{r}, H = 3 m, T = 4–7 s.

**Figure 18.**Comparison of the mooring force of the double-row floating breakwater under different wave parameters: (

**a**) Seaward side F

_{s}, H = 1–4 m and T = 6 s; (

**b**) Leeward side F

_{l}, H = 1–4 m and T = 6 s; (

**c**) Seaward side F

_{s}, H = 3 m and T = 4–7 s; (

**d**) Leeward side F

_{l}, H = 3 m and T = 4–7 s.

Floating Pontoon | Length L_{f} | Width W_{f} | Height H_{f} | Draft d_{f} | Mass M | Moment of Inertia I |

Exp. values | 50 cm | 76 cm | 20 cm | 10 cm | 28.6 kg | 0.669 kg·m^{2} |

Num. values | same | 100 cm | same | same | 28.6/76 × 100 = 37.63 kg | 0.669/76 × 100 = 0.88 kg·m^{2} |

Mooring System | Tensile Stiffness EA | Wet Weight w | Bending Length l_{m} | Horizontal Length l_{x} | ||

Exp. values | 3.15 kN | 6.18 N/m | 1.6 m | 1.15 m | ||

Num. values | 3.15 × 2/76 × 100 = 8.29 kN | 6.18 × 2/76 × 100 = 16.26 N/m | same | same |

Floating Pontoon | Diameter F_{c} | Width W_{f} | Interspacing L_{r} | Draft d_{f} | Total Mass M | Total Moment of Inertia I |
---|---|---|---|---|---|---|

Exp. values | 20 cm | 76 cm | 10 cm | 10 cm | 19.1 kg | 0.474 kg·m^{2} |

Num. values | same | 100 cm | same | same | 19.1/76 × 100 = 25.14 kg | 0.474/76 × 100 = 0.624 kg·m^{2} |

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**MDPI and ACS Style**

Cheng, X.; Liu, C.; Zhang, Q.; He, M.; Gao, X.
Numerical Study on the Hydrodynamic Characteristics of a Double-Row Floating Breakwater Composed of a Pontoon and an Airbag. *J. Mar. Sci. Eng.* **2021**, *9*, 983.
https://doi.org/10.3390/jmse9090983

**AMA Style**

Cheng X, Liu C, Zhang Q, He M, Gao X.
Numerical Study on the Hydrodynamic Characteristics of a Double-Row Floating Breakwater Composed of a Pontoon and an Airbag. *Journal of Marine Science and Engineering*. 2021; 9(9):983.
https://doi.org/10.3390/jmse9090983

**Chicago/Turabian Style**

Cheng, Xiaofei, Chang Liu, Qilong Zhang, Ming He, and Xifeng Gao.
2021. "Numerical Study on the Hydrodynamic Characteristics of a Double-Row Floating Breakwater Composed of a Pontoon and an Airbag" *Journal of Marine Science and Engineering* 9, no. 9: 983.
https://doi.org/10.3390/jmse9090983