Wave dissipation effect of a new combined breakwater and its protective performance for coastal box girder bridges

Abstract

Breakwaters play an important role in in mitigating wave-induced damage to marine structures. However, conventional submerged breakwaters often exhibit limited wave dissipation capabilities, while floating breakwaters may lack adequate safety performance. Therefore, this study introduces a novel combined breakwater design aimed at addressing the shortcomings of both traditional types. The proposed breakwater integrates a floating structure with a trapezoidal submerged breakwater via an anchor chain connection. To evaluate its efficacy, numerical simulations of wave interactions with structures were conducted using the OpenFOAM computational fluid dynamics (CFD) software in a two-dimensional (2D) numerical flume. Dynamic mesh technology was employed to simulate the motion of the floating body, and the resulting wave loads on a box girder bridge deck positioned behind the breakwater were analyzed to assess the combined breakwater’s protective capabilities and influencing factors. Analysis of wave heights and loads on the bridge deck revealed that the combined breakwater outperformed traditional submerged breakwaters in terms of wave dissipation. Furthermore, it was observed that the protective efficacy of the combined breakwater was more sensitive to variations in the size of the floating body compared to the submerged structure, and more responsive to changes in wave period than wave height. Leveraging the ability of the floating body to attenuate waves near the surface and the enhanced impact resistance provided by the combined floating and submerged structures, the proposed breakwater offers a promising approach to improving wave attenuation performance and enhancing safety for coastal infrastructure.

1 Introduction

As China experiences rapid economic development, its land transportation facilities are confronted with intensified pressure due to the increased demand for cargo transportation. Consequently, there has been a notable rise in the construction of sea-crossing bridges to augment existing infrastructure capabilities (Gao et al. 2019; Zhang 2019). These cross-sea bridges serve not only as essential pathways for human travel but also facilitate economic growth and cultural exchanges between regions. However, they are also susceptible to various hazards, such as ocean waves, which pose a threat to their structural integrity and resilience (Iemura et al. 2005; Mimura et al. 2011).

Minimizing wave-induced damage to coastal structures is a significant concern for infrastructure management stakeholders. In response to this challenge, the concept of a breakwater has emerged as a solution. Typically, two types of breakwaters, submerged and floating, are widely utilized in near-shore ports (Zou 1980; Xie 1982; Wang 1985).Extensive research has been conducted on the interaction of submerged breakwaters with waves. Jiang and Li (2009) and Jiang et al. (2012) observed fluctuating wave-induced forces along the caisson upright dike, leading to serpentine damage and structural safety issues. To enhance resilience, Niu et al. (2003) replaced part of the original upright caisson breakwater with a wing plate, forming a comb-tooth surface on the wave-facing side. Subsequently, Zang et al. (2018), Sun et al. (2020) and Wan et al. (2022) demonstrated that the comb-type breakwater effectively reduces the maximum wave force. Other methods to reduce wave load include placing a permeable pipe in front of the breakwater (Shih 2016) or partially perforating it (Kirca and Kabdaşli 2009). Scholars also aim to improve wave dissipation capacities. Xue et al. (2022) proposed a new triple semicircular breakwater with enhanced wave load reduction capabilities, while Xu et al. (2022) found that joint breakwaters significantly enhance wave dissipation capacity when an aerial barrier is positioned behind the submerged breakwater. However, challenges remain. Structural costs increase with water depth (Deng 2019a), and there is a lack of effective wave force countermeasures during storm surges (He 2009). Moreover, the role of submerged breakwaters is limited (Wang et al. 2018), as most fluctuation energy is concentrated on the free surface and a significant portion of wave energy lies within three times the depth of the wave height below the free surface (Zhan et al. 2017; Dai et al. 2018).

In response to the limitations of submerged breakwaters in wave dissipation, floating breakwaters have gained increasing attention due to their robust wave attenuation capabilities and simple construction (He 2009; Shen et al. 2016; Deng 2019; Yi 2020). Yin et al. (2021). Numerical simulations conducted by Yin et al. (2021) demonstrated effective attenuation of wave forces on bridge abutments under the protection of rectangular floating breakwaters. Luan (2021) highlighted that more complex float structures yield superior wave dissipation effects compared to regular rectangular floating breakwaters. Innovative designs, such as the inverted Π-type floating breakwater proposed by Tang et al. (2023) and the “factory”-shaped plate breakwater suggested by Jin et al. (2023), have shown enhanced wave dissipation performance compared to traditional rectangular breakwaters. Ji et al. (2015) introduced a novel floating breakwater comprising a rigid cylinder and a flexible mesh cage, which outperformed traditional double pontoon and box-type floating breakwaters in reducing long wave transmission. However, the wave dissipation performance of floating breakwaters is influenced by various factors. For instance, the spacing of tanks in multi-floating tank configurations affects their wave dissipation effectiveness (Williams and Abul-Azm 1997; Williams et al. 2000; Syed and Mani 2006). Wu (2018) observed different wave dissipation properties of upper and lower arc-plate breakwaters in outgoing and incoming water states. Additionally, He et al. (2014) found that increasing the width and draft of rectangular pontoons reduces the transmission coefficient of breakwater to waves. The dynamic nature of sea conditions and severe weather events like tropical cyclones can subject floating breakwaters to damage, potentially rendering them ineffective (Li 2020). Furthermore, in anchor chain-type floating breakwaters, the friction generated by the direct contact between anchor chains and anchorage with the seabed may pose an additional source of damage to the overall system.

To develop a practical breakwater device suitable for greater water depths and storm surge scenarios, this paper proposes a combined breakwater form that incorporates the advantages of both submerged and floating breakwater designs. With this design, there is no need to excessively increase the height of the submerged breakwater to enhance wave dissipation capacity, as the incorporation of the floating structure achieves this goal, thereby reducing project costs. Additionally, the submerged breakwater serves as an anchoring carrier for the floating breakwater, mitigating friction between the anchor chain and the seabed. This arrangement also results in a shortened length of anchor chain, prolonging the safe usage time of the floating breakwater.

The structure of this paper is organized as follows. An introduction to the necessary wave theory and model validation is presented in Sect. 2. This is followed by an investigation into the attenuation performance of the combined breakwater subject to nonlinear waves in Sect. 3. Section 4 provides a test case to demonstrate the protective capability of the combined breakwater in defense of a box girder bridge deck and explores other influencing parameters of interest. Finally, conclusions of the current study are given in Sect. 5.

2 Research theory

2.1 Wave theory

In this study, the wave dissipation effect of the combined breakwater is investigated by simulating Stokes second-order waves and wave-structure interactions. The second-order wave surface equation is given as follows:

$$\eta =\frac{H}{2}\text{c}\text{o}\text{s}(kx-\omega t)+\frac{\pi {H}^{2}}{4 L}(1+\frac{3}{2\text{s}\text{i}\text{n}{\text{h}}^{2 }kh})\text{c}\text{o}\text{s}\text{h}\, kh\, \text{c}\text{o}\text{s}2(kx-\omega t)$$

(1)

where H represents the wave height, k denotes the wave number, h is the water depth, ω stands for the wave angular frequency, and L indicates the wave length.

2.2 Numerical model building

A two-dimensional (2D) numerical flume is constructed using OpenFOAM to investigate the performance of the combined breakwater under the influence of nonlinear waves. The numerical model is established based on the following theory.

2.2.1 Free surface treatment

The free liquid surface is captured using the Volume of Fluid (VOF) method (Hirt and Nichols 1981). The VOF method is a numerical technique for tracking fluid interfaces in CFD for two immiscible fluids. It ensures mass conservation and demonstrates good convergence and accuracy (Akhtar et al. 2007; Mulbah et al. 2022). The continuity equation for volume fraction is given by:

$$\frac{\partial \alpha }{\partial t}+\nabla \cdot \left(\text{U}\alpha \right)+\nabla \cdot \left({\text{U}}_{r}\alpha \right(1-\alpha \left)\right)=0$$

(2)

where U is the fluid velocity vector in Cartesian coordinates, α denotes the liquid volume fraction in the grid cell. If α = 1, the grid cell is entirely filled with liquid, whereas if α = 0, the grid is entirely occupied with air. When 0 < α < 1, it indicates the presence of the free liquid surface state within the grid cell.

2.2.2 Boundary and initial conditions

Fig. 1

Schematic diagram of the computational domain and boundary conditions of the numerical flume

Table 1 Boundary conditions setting of the numerical flume

In OpenFOAM, the front and back walls of the numerical flume are configured as empty to enable 2D numerical simulation, as depicted in Fig. 1. The overall computational domain encompasses the entire setup. The inlet boundary, responsible for wave generation, is situated at the left end of the flume, with waves propagating along the x-axis. Conversely, the outlet boundary, facilitating wave absorption, is positioned at the right end of the flume. At the bottom of the flume, located at z=-h with a water depth of h, a no-slip boundary condition is set. This implies that the velocity at the bottom is zero, and there is no normal pressure gradient. For a comprehensive overview of the initial boundary conditions, please refer to Table 1.

2.2.3 Wave-generation and wave-absorption technology

The wave generation method employed within the numerical flume is the velocity inlet method. This method facilitates wave simulation by prescribing the water quality point velocity of the target wave at the velocity inlet. For wave dissipation, the active dissipation technique is used (Huang et al. 2019). This technique is based on the linear shallow water theory and no need to expand the computational domain as a dissipation zone, thereby reducing the number of grids and computational costs (Schäffer and Klopman 2000). Additionally, it offers superior wave dissipation capabilities, with the reflection coefficient typically remaining below 10% (Higuera et al. 2013).

Fig. 2

MoorDyn’s Coupling Process within OpenFOAM

2.3 Dynamic grid technology

To facilitate the motion of the anchor chain, a mooring line model (Moordyn) is integrated into OpenFOAM to establish a line constraint. Additionally, within the overInterDyMFoam solver, a new rigid body constraint (sixDoFRigidBodyMotion Restraint) is introduced for the floating body. This flow solver is equipped with the overlapping mesh method, which is particularly suitable for handling large amplitude rigid body motions (Windt et al. 2020).

To compute the motion of the floating body, an open-source dynamic library is employed (Hall and Goupee 2015; Chen and Hall 2022), which is dynamically loaded into the OpenFOAM solver during runtime. The workflow is illustrated in Fig. 2. During the calculation of the float motion, MoorDyn retrieves the position and velocity of the float and forwards them to MoorDyn. Subsequently, MoorDyn computes the node positions, velocities, and segmental tensions of the mooring lines, and transmits the total constraint forces and moments back to the motion solver. This enables the motion solver to update the acceleration of the float.

2.4 Validation of the numerical model

When investigating the interaction between waves and structures, certain scholars assume the fluid to be non-viscous (Ti et al. 2019; Qiu et al. 2022). However, to account for wave dissipation in turbulence modeling during 2D numerical simulations (Deng 2019a), a laminar flow model is adopted in this study to validate the accuracy of the numerical model.

2.4.1 Validation of the submerged breakwater

Figure 3 presents a comparison between the numerical and experimental wave height time histories before and after the breakwater. The experimental data were sourced from a study on the behavior of a submerged trapezoidal breakwater with isolated waves (Zhao et al. 2019). The experimental setup is depicted in Fig. 4.

Fig. 3

Comparison of numerical and experimental wave height time history: a WG1, b WG2

Numerical simulations were conducted using three different minimum grid sizes: MeshI, MeshII, and MeshIII, with minimum grid sizes of 0.0045 m, 0.0065 m, and 0.0085 m, respectively. The numerically simulated wave height time histories at the measured positions exhibit good agreement with the experimental results. However, the error in the peak values of wave height is more pronounced when using MeshIII, while the error is less than 5% when using MeshII. This demonstrates the accuracy of the numerical flume in calculating the wave heights around the fixed breakwater. Although the accuracy of wave height calculation is improved when using MeshI, it has little effect and increases the calculation time. Therefore, MeshII is used throughout the present work.

Fig. 4

Schematic of the experimental setup

2.4.2 Validation of the floating breakwater

In the study by Domínguez et al. (2019) and Wu et al. (2019), the experimental measurements of a floating box moored by four suspension chain lines under the influence of regular waves were validated using a combination of numerical and experimental methods. This section aims to validate the accuracy of MoorDyn’s coupled model within OpenFOAM by utilizing their experimental data. The dimensions of the float tank and the properties of the anchor chain are maintained consistent with the experiment. The schematic arrangement of the numerical model is illustrated in Fig. 5.

Fig. 5

Schematic of the numerical model layout: a front view, b plan view

Figure 6 depicts the wave time histories at WG1 and WG2. The wave time histories generated in the numerical simulation exhibit a good agreement with the experimental data, demonstrating the accuracy of the numerical flume in calculating the wave heights around the floating breakwater.

Fig. 6

Numerical versus experimental wave heights: a WG1, b WG2

Figure 7 illustrates the comparison between the numerical and experimental results for the anchor chain tension. Although there is some deviation in the numerically simulated anchor chain tension, this could be attributed to potential slack in the acceleration of the six-degrees-of-freedom (six-DOF) constrained rigid body when MoorDyn calculates the tension. Nonetheless, the calculation error of peak tension is minimal, indicating the accuracy of numerically estimating the anchor chain tension.

Fig. 7

Comparison of anchor chain tension a H = 0.12 m, T = 1.6s, b H = 0.12 m, T = 1.8s

In summary, the use of the laminar flow model has effectively verified the accuracy of the numerical model in calculating both wave height and anchor chain tension for the floating structure. Therefore, the laminar flow model is employed in this paper to investigate the combined breakwater.

3 Wave dissipation effect of the combined breakwater

3.1 Numerical model setup

The numerical flume is 26 m long in the wave propagation direction, 0.2 m in the vertical wave propagation direction, and has a depth of 1.5 m. Figure 8(a) provides a schematic diagram of the structural arrangement of the combined breakwater, featuring four symmetrically arranged anchor chains with a length of 0.48 m. Meanwhile, Fig. 8(b) illustrates the structural arrangement of the submerged breakwater.

The interaction of Stokes second-order waves with a breakwater is simulated in OpenFOAM and two cases with different wave parameter settings are set up, i.e., case1 (H = 0.15 m, T = 1.00s, L = 1.55 m) and case2 (H = 0.18 m, T = 1.20s, L = 2.17 m).

Measurement points are positioned at the locations of wave gauges in the figure to measure the wave heights. Specifically, WG2 is 3 m from the geometric center of the breakwater, which exceeds a distance of 1L. This distance ensures that the wave height behind the breakwater is fully developed and stabilized.

3.2 Comparison of wave dissipation effect of two types of breakwaters

Figure 9 depicts the time histories of wave height at WG2 over several complete wave cycles under two cases. The figure illustrates that, at the same moment, the wave height behind the combined breakwater is significantly smaller than that behind the submerged breakwater for both cases. This observation suggests that the combined breakwater offers superior wave attenuation performance compared to the conventional submerged breakwater. Furthermore, in Fig. 9(a) and (c), the peak and valley values of wave height at WG2 without a breakwater are nearly equal to the theoretical value of the incident wave under both conditions. This consistency indicates that wave dissipation is primarily caused by the breakwater, thus excluding the influence of numerical calculation errors in wave height on the results.

Fig. 8

Schematic diagram of the structural arrangement of the two types of breakwaters: a the combined breakwater, b the submerged breakwater (Unit: m)

Fig. 9

Time histories of the wave height at WG2: a comparison of theoretical and numerical wave height history (case1), b comparison of the time history of incident wave and wave height behind the breakwater (case1), c comparison of theoretical and numerical wave height history (case2), d comparison of the time history of incident wave and wave height behind the breakwater (case2)

To quantitatively compare the attenuation of wave height values by the two types of breakwaters, the average wave height and attenuation percentage over three stabilization cycles at the markers in Fig. 9(b) and (d) are calculated. This helps mitigate the effect of non-stationarity among cycles on the results. The calculation results are presented in Table 2. The results indicate that in Case 1, the combined breakwater reduces the wave height value by 42.6%, which is twice as much as that of the submerged breakwater. In Case 2, the combined breakwater reduces the wave height by 15%, while the submerged breakwater has a minimal effect on reducing the wave height.

Table 2 The calculation results of average value of wave height and attenuation percentage

3.3 Wave dissipation mechanism of combined breakwater

Figure 10 presents the wave height time history at WG1 behind the two types of breakwaters, marking several time points of wave height change within a wave cycle (t = 12.0s, 12.2s, 12.4s, 12.6s, 12.8s, 13.0s). Positioned less than 1 L away from the breakwater, this location offers a clear view of the wave surface changes induced by the breakwater. Notably, the wave height behind the combined breakwater remains significantly smaller than that of the submerged breakwaters. Furthermore, the wave height time history behind the combined breakwater slightly lags behind that of the submerged breakwater. This delay is attributed to the longer duration of wave–structure interaction for the combined breakwater, consequently reducing the wave propagation speed.

Fig. 10

Comparison of wave height history at WG1 behind the breakwaters (H = 0.15 m, T = 1.0s)

To elucidate the mechanism behind the superior wave reduction effect of the combined breakwater compared to the traditional submerged breakwater, Fig. 11 presents a comparison of the wavefields around the two types of breakwaters at corresponding moments. At t = 12.0s, a wave crest reaches the combined breakwater (Fig. 11a). The presence of the submerged structure causes a degree of wave surface surging. The leeward side of the floating structure moves upward, while the wave-facing side blocks the wave near the wave face, impeding the development of the wave crest. In contrast, at t = 12.0s (Fig. 11b), the wave at the submerged breakwater rises up the slope in front of the structure, with a small degree of wave breaking observed behind the breakwater. However, the wave remains cyclical, and the wave crest is fully developed. This explains why, in Fig. 10, the wave height behind the submerged breakwater is just above the peak, while behind the combined breakwater, it does not reach the peak due to obstruction by the floating structure, resulting in reduced propagation speed. At t = 12.2s, the wave continues to propagate forward above the submerged breakwater, while for the combined breakwater, the wave strikes directly on the top surface of the floating structure, leading to more significant wave energy loss. Subsequently, at t = 12.4s, the wave above the floating structure crosses the top surface and falls to the leeward side, where potential wave energy is converted to mechanical kinetic energy, dissipating wave energy. During this time, the wave height behind the combined breakwater reaches its minimum value, notably smaller than behind the submerged breakwater. Continuing to t = 12.6s, the wave surface deforms behind the combined breakwater as the floating body undergoes fluid–structure interaction, disturbing the motion of the water. By t = 12.8s, the wave surface deformation behind the combined breakwater continues to develop, resulting in a flatter wave surface compared to behind the submerged breakwater, which reaches its peak wave height. Finally, at t = 13s, increased wave breaking is observed behind the submerged breakwater, while behind the combined breakwater, the wave surface height reaches its peak, significantly smaller than that behind the submerged breakwater at t = 12.8s. It’s noteworthy that during the interaction of the combined breakwater with waves, the movement of the anchor chain also contributes to wave energy loss.

Fig. 11

Comparison of wave surface behind two breakwater breakwaters: a The combined breakwater, b The submerged breakwater

The combined breakwater with additional floating structure inherits the advantages of both traditional submerged and floating breakwaters. In deeper water, waves encounter the fixed structure, causing reflection and breaking. Near the water surface, waves encounter the floating body, which obstructs their path. The resulting relative movement between the wave and the floating body leads to energy dissipation. Consequently, the combined breakwater exhibits superior wave dissipation compared to conventional submerged breakwaters.

3.4 Comparison of forces on box girder bridges behind two breakwaters

In order to study the protective effect of the combined breakwater on a bridge, a box girder bridge model is placed at WG2 in the numerical model, as described in Sect. 3.1 to calculate its wave loads under different wave conditions. In the numerical wave flume, the dimensions of the box girder bridge model (model solid and scaled ratio of length are from Huang et al. (2019) and the refined mesh of the bridge perimeter are given in Fig. 12. A refined mesh around the structure is used to more accurately capture wave features around the structure and to calculate wave loads.

Fig. 12

Model of box girder bridge: a size of the model, b refined grid around the bridge

Fig. 13

Time histories of wave loads on box girder bridge behind breakwaters: a the horizontal wave force (case1), b the vertical wave force (case1), c the horizontal wave force(case2), d the vertical wave force (case2)

A comparison of the wave load time histories of the box girder bridge behind the two types of breakwaters is given in Fig. 13. The figure shows that the combined breakwater can significantly reduce the wave loads on the box girder bridge behind the breakwater in two cases. In addition, the temporal lag of the wave load time histories of the box girder bridge after the combined breakwater is observed in Fig. 13, which indicates that the breakwater can delay the time of wave action on the bridge.

Since the two types of breakwaters show certain different rules of load reduction effects on bridges behind the breakwaters at different cycles, the average peak values of wave load as well as the percentage of reduction will be calculated for the three stabilization cycles labeled in Fig. 13(a-d). Calculation results in Table 3 show that the wave force reduction percentage of the combined breakwater is at least three times as much as that of the submerged breakwater. This illustrates the combined breakwater can provide a better protection effect for the box girder bridge.

Table 3 Calculation results of average peak values of wave load(unit: N) and reduction percentage of load(unit: %)

4 Factors affecting the protective performance of the combined breakwater

The combined breakwater has a protective effect on the box girder bridge behind it, which is manifested in the reduction of wave loads on the bridge. However, the magnitude of this protective effect depends on the influence of many factors. In this section, the wave loads of box girder bridges under different setting conditions are calculated to investigate the changes in protective performance of the breakwater under the influence of different factors.

4.1 Influence of the size of the submerged structure

The submerged structure in the combined breakwater provides the role of wave dissipation, hence the influence of its height is studied. The variations of the box girder bridge force behind the combined breakwater under the conditions of different submerged structure heights (0.18 m, 0.22 m, 0.26 m, 0.30 m, 0.34 m) are given in Fig. 14. In addition, When the height of the submersible structure changes, the anchoring form of the anchor cable remains unchanged, as shown in Fig. 8(a). The cable diameter, stiffness and other parameters are kept unchanged, and the length relaxation coefficient of the cable is given the same in the tension state. This can keep the tension stability of the anchor cable unchanged, and reduce the influence of the change of the length of the anchor cable on the load reduction performance of the breakwater to a certain extent. Note that it is only explored here for assessing the variation in the performance of the combined breakwater, and the wave forces are shown as peak values for each calculation, taking into account the role of ultimate wave load control. The figure shows that when the height of the submerged structure (h_s) changes, the bridge wave load (including horizontal and vertical forces) only fluctuates, indicating that increasing the height of the submerged structure does not correspondingly increase the protective effect of the combined breakwater on the bridge. It is not the same as the conclusion that the wave dissipation capacity of a conventional submerged box breakwater increases with its height (Dai et al. 2018). This is because when the height of the submerged structure increases, even though the incident wave blocking height increases, the upwelling height of the wave increases after the wave passes through the slope, the wave cannot interact with the floating structure, and the wave energy dissipation decreases, so that the force on the box girder bridge does not decrease significantly.

Fig. 14

Different submerged structure heights versus peak wave loads on box girder bridges: a the peak horizontal force, b the peak vertical force

The streamwise width of the submerged structure (w) is varied through: 0.2 m, 0.4 m, 0.6 m, 0.8 m, 1.0 m. Figure 15 gives the variation of the wave load of the box girder bridge under different streamwise width conditions. It is seen in the figure that the force of the box girder bridge behind the breakwater is more obviously influenced by the width of the submerged structure than the height in Fig. 14, and the wave load does not decrease linearly with the increase in the submerged structure width, but is significantly smaller than that in the low width scenario when the width is larger. This suggests that increasing the width of the sunken structure can lead to a reduction in bridge forces, which could be a measure to improve the performance of the combined breakwater for bridge protection. This is due to the fact that more waves fall and break at the top surface of the submerged structure, resulting in more energy dissipation.

Fig. 15

Different submerged structure widths versus peak wave loads on box girder bridges: a the peak horizontal force, b the peak vertical force

4.2 Influence of floating body structural dimensions

The relationship between the peak wave load of the box girder bridge and the height of the floating body (${h}^{{\prime }}$) in the combined breakwater is given in Fig. 16, which shows that the height of the floating body has a significant effect. When the height is 0.06 m, at which time the floating body structure size is too small, the breakwater can no longer reduce the bridge vertical force as shown in Fig. 16(b). Then, the box girder bridge force decreases with the increase of the height of the floating body, when the height of the floating body is 0.22 m, the horizontal force of the bridge has been reduced to nearly zero, as shown in Fig. 16(a). This is because when the height of the floating body increases, most of the waves are blocked, and very few of them pass over the floating structure and propagate backward, so the protective effect of the breakwater is enhanced.

Fig. 16

Different floating structure heights versus peak wave loads on box girder bridges: a the peak horizontal force, b the peak vertical force

As shown in Fig. 17, the streamwise width (w’) of the floating structure in the combined breakwater ranges from 0.1 m, 0.2 m, 0.3 m, 0.4 m, and 0.5 m. The figure indicates that the wave loads of the box girder bridge roughly decrease with the increase of the width of the floating structure, which indicates that increasing the width of the floating component is favorable for the reduction of the loads of the box girder bridge. Because when the w’ increases, the interaction time between the wave and the floating structure is longer, therefore more kinetic energy can be generated.

Fig. 17

Different floating structure widths versus peak wave loads on box girder bridges: a the peak horizontal force, b the peak vertical force

4.3 Influence of wave parameters

The propagation process of incoming waves in complex marine environment changes with weather and wave topography. therefore, waves arriving at breakwaters are different. To study the effect of different wave conditions on the load reduction effect of breakwaters, Table 4 gives the wave parameter settings for different cases.

Table 4 Wave parameter settings for different cases

Due to the different forces on the box girder bridge under different wave conditions, the peak load change cannot be used to characterize the load reduction effect of the breakwater. Defining the load reduction rate η as.

$$\eta=\frac{{a-b}}{{a}}\times100\%$$

(3)

where a denotes the peak wave load on box girder bridge without breakwater, b denotes the peak wave load on box girder bridge behind the combined breakwater, ${\eta }_{x}$ denotes the horizontal load reduction rate and ${\eta }_{z}$ denotes the vertical load reduction rate. The larger the value of η, the better the protection effect of the combined breakwater on the box girder bridge.

The curves of values of η versus wave period for different wave height conditions are given in Fig. 18. Under the same wave height condition, η decreases with increasing period. Taking H = 0.12 m as an example, when T = 0.8s, the horizontal load reduction rate reaches 65% as shown in Fig. 18(a), and the vertical load reduction rate reaches 98% as shown in Fig. 18(b), and the energy dissipation effect of the combined breakwater on the short-period wave is significant. As the period increases to 1.4 s, the horizontal load reduction rate is -0.5% as shown in Fig. 18(a), and the vertical load reduction rate is -10% as shown in Fig. 18(b), and the box girder bridge load can no longer be reduced, which indicates that the load reduction effect of the breakwater is severely constrained by the long-period waves. This is because long-period waves have larger wavelengths, concentrated energy and strong penetrating power, and when they reach the breakwater, they are easy to bypass the breakwater and continue to propagate, and the energy dissipation is small. The figure also shows that under the same period, there is no obvious change rule of the load reduction rate to the wave height, this may be due to the fact that as the wave height increases, fewer waves are blocked by the floating structure, but at this point the wave breaks and consumes more energy, which results in the rate of load shedding not varying linearly with the increase of wave height.

Fig. 18

Variation of η-values with wave period and wave height: a ${\eta }_{x}$, b ${\eta }_{z}$

Figure 19 provides the relationship between the anchor chain tension of the combined breakwater and the wave period under different wave height conditions. Since the anchor chain tension shows a steady cyclical variation under the action of regular waves, and considering the controlling effect of the tension limit value, the tension is taken as the peak value in the figure. It can be seen in the figure that the value of the front anchor chain tension in Fig. 19(a) is slightly larger than that of the rear anchor chain in Fig. 19(b), because the wave-facing surface is subjected to the direct impact of the waves and the wave impact is larger. It is also seen in the figure that under the same wave height condition, the anchor chain tension is not significantly affected by the wave period, which is reflected in the small fluctuation of the tension with the period. However, under the same period, except T = 0.8s, the pulling force of the front anchor chain and the rear anchor chain increases with the increase of wave height. This is because when the wave height increases, the wave propagates to the floating breakwater, which produces a stronger impact on the wave front, and the pulling force of the anchor chain will increase accordingly. Anchor chain design should consider this point, and take measures to enhance the tensile capacity of the anchor chain under the condition of a large force, so as to ensure the safety of the breakwater system.

Fig. 19

Variation of peak tensions with wave period and wave height: a the front anchor chain, b the rear anchor chain

5 Conclusions

In this paper, aimed at enhancing wave dissipation and safety performance of conventional breakwaters to inform disaster prevention and mitigation engineering design, a new combined breakwater is proposed. By numerically simulating the interaction between the breakwater and waves, the wave dissipation effect of the traditional submerged breakwater and the combined breakwater are investigated. The study focuses on the influence of structural parameters and wave parameters on the load reduction performance of a bridge deck, leading to the following conclusions:

• (1) The combined breakwater integrates a small-sized floating body onto the traditional submerged breakwater. This design significantly increases the wave height reduction percentage and the load reduction percentage for the box girder bridge without excessively increasing the size of the traditional breakwater. This approach offers good economic efficiency and feasibility. Additionally, the combined breakwater can collectively withstand greater wave impact, thereby ensuring the safety of the floating structure.

• (2) The enhanced wave dissipation capacity of combined breakwaters compared to conventional submerged breakwaters primarily stems from the presence of the floating body. This component effectively absorbs wave energy by obstructing wave propagation near the water surface, leading to wave breakage and deformation.

• (3) The load reduction effectiveness of the combined breakwater on box girder bridges behind it is significantly influenced by several factors, including the streamwise width of the submerged structure and the floating structure, as well as the height of the floating structure. Generally, as these dimensions increase, so does the protective effect of the combined breakwater on the bridge. However, variations in the height of the submerged structure have a negligible impact on the load reduction effect.

• (4) The load reduction rate of the combined breakwater on bridges decreases as the wave period increases, suggesting that its wave dissipation performance is limited by the wave period itself. Additionally, as the wave height increases, the tension in the anchor chain also increases, highlighting the necessity to enhance the tensile performance of the anchor chain, particularly in scenarios with larger wave heights, to ensure the breakwater’s stability and resilience.