Experimental Comparisons of the Wave Attenuation Characteristics Among Different Flexible-Membrane Breakwaters

Abstract

In this paper, physical experiments were conducted to analyze the wave attenuation characteristics of a combinational breakwater in a 2D piston-type wave flume. The proposed breakwater consisted of a box and double flexible membranes, one of which was fixed on both sides of the box, while the other was positioned at a specific distance from the box sides. The flexible membranes dropped down naturally and formed a U shape. Optimal configuration parameters of the proposed breakwater were determined separately through a series of comparison experiments among mooring breakwaters in regular waves; the box draft d was determined by a box-type breakwater and the membrane spacings L_u and L_d from the box sides were determined by a U-shaped flexible-membrane breakwater, where the wave attenuation coefficients versus kh and B/λ were provided. Then, using the literature, the present box–membrane breakwater with membrane spacings was compared to two similar flexible-membrane breakwaters. One was the above-mentioned U-shaped flexible-membrane breakwater, the other was a box–membrane breakwater in which the flexible membranes were directly fastened on the box sides. The results indicate that with optimal configuration parameters, the wave attenuation performance of the proposed breakwater had been enhanced due to the increase in both dissipating and reflecting wave energy.

1. Introduction

Free surface breakwaters (FSBs) or floating breakwaters (FBs) are coastal engineering structures that can provide a sheltered region to protect the mariculture zone. FBs such as the box-type, pontoon-type, frame-type, mat-type, tethered-float-type, and horizontal-plate-type [1] generally offer a simpler configuration and lower construction difficulty and cost, as well as higher reusability when comprehensively compared with traditional bottom-mounted breakwaters [2]. The wave attenuation mechanism of FBs mainly comprises reflecting and dissipating waves.

A box-type breakwater is a common FB with a simple box structure, which blocks incident waves dominantly by wave reflection. In addition to acting as a breakwater, it can also serve as a road, warehouse, anchor ship, or fishing dock, so a number of investigations on the hydrodynamic performances of box-type FBs with different improving configurations can be found. As the single-box FBs are relatively ineffective in attenuating long waves, the side-by-side-boxes structure was developed in breakwaters to broaden the breakwater breadth and elevate the wave attenuation performance for relatively long waves. Karmakar et al. [3] experimentally compared the wave attenuation characteristics of single-box-type FB and double box-type FBs. Yang et al. [4] investigated two floating boxes rigidly connected by horizontal plates and found that this reduced the transmission coefficient by 0.4, at most, compared to two floating boxes without rigid plate connection. Diamantoulaki and Angelides [5] numerically studied the wave attenuation performance of an array of boxes flexibly connected by hinges under monochromatic linear waves of different wave headings. Ji et al. [6] experimentally studied the wave attenuation characteristics of double-row detached FBs which outperformed a single-row FB through enhanced energy dissipation caused by eddies and the moon-pool effect. Chen and Wu [7] numerically investigated the wave attenuation characteristics of two surface-piercing fixed boxes without connection by the Desingularized Boundary Integral Equation Method, and found that the transmission coefficient of two boxes reached a valley band where the Bragg resonance occurred. Chen et al. [8] numerically studied the wave attenuation characteristics and dynamic responses of two independent moored floating boxes using the δ-Smoothed Particle Hydrodynamics model, which was validated by experimental tests, and the results demonstrated that the movement of the two floating boxes was detrimental to the Bragg resonance reflection effect.

Beyond approaches that improve wave attenuation performance by leveraging multiple attenuations, researchers have investigated adding appendages to box-type breakwaters to dissipate long-wave energy by disturbing wave particle orbits or utilizing friction between appendage structures and wave particles. He et al. [9] experimentally compared the hydrodynamic performance of a rectangular box-type breakwater with and without pneumatic chambers which could boost wave energy dissipation. Alizadeh [10] studied the performance of a box with attached rigid plates on both its front and rear faces under regular waves. Ji et al. [11] conducted experiments to measure the wave attenuation performance of a hollow box whose upper surface was appended with four perforated-plate compartments which could reduce the motion response and mooring force. He et al. [12] experimentally analyzed the wave protection and motion response performances of a rectangular floating breakwater with upper surface slotting to insert multiple through-penetrating vertical plates extending externally, demonstrating that the primary function of vertical plates as wave-dissipating components is due to water-body confinement. Hu et al. [13] numerically and experimentally researched the interactions between regular waves and a floating box with adjacent porous baffles positioned both upstream and downstream, and corresponding experiments were also conducted to ensure the accuracy of the numerical results. Nevertheless, this kind of breakwater still encounters the challenge of resisting relatively long waves, which leads to enlarging the breakwater dimension accompanied by the loss of economic efficiency.

Since relying on rigid breakwaters proves insufficient and prohibitively expensive for effective wave attenuation in the face of longer waves, researchers have begun incorporating flexible appendages into FBs to enhance wave attenuation performance and reduce costs. Kee and Kim [14] numerically researched the interaction of water waves with a tensioned vertical flexible membrane attached to a solid cylindrical buoy through the Boundary Element Method (BEM), and showed that the wave attenuation performance of FBs under long waves could be enhanced by adding the underwater appurtenance, the numerical results of which were soon validated by a series of experiments carried by Kim et al. [15] in a two-dimensional regular and irregular wave tank. Cho et al. [16] proposed a dual cylindrical buoy attached with vertical flexible membranes which were then numerically investigated under oblique incident waves, and the results demonstrated that the wave attenuation performance of a dual membrane breakwater can be significantly enhanced compared to the single membrane case. Briggs et al. [17] experimentally tested the Rapidly Installed Breakwater System (RIBS) which attached the fabric membrane to a rigid frame, like a ‘Venetian blind,’ to permit rapid and expedient deployment, reducing Sea State 3 wave conditions to Sea State 2 for safe vessel operations. Hermanson [18] studied the wave attenuation performance of a vertical flexible membrane attached to FBs through experiments and found that the attaching membrane can reduce the average transmission coefficient by about 12–17%. Kee [19], utilizing the BEM, numerically studied the hydrodynamic characteristics of a floating rectangle pontoon attached with triple vertical porous membranes underneath, demonstrating that long vertical porous membranes taking a major fraction of the water column can significantly increase the overall wave attenuation performance in normal and oblique incident waves, including long waves. The above-mentioned flexible-membrane breakwaters had a common feature of using pre-tensioned flexible membranes to reflect waves, which requires huge weights to tauten the flexible membranes especially in larger waves, resulting in difficulty meeting the tension requirements for the weights and the flexible membranes. In view of this, Wu et al. [20] experimentally presented a flexible-membrane appendage without pretension in which two-layer arc membranes naturally hung down from a rectangular box, achieving nearly 50% wave attenuation under longer waves due to introducing a new wave-absorbing mechanism, including the movements of the flexible membranes and of the massive water enveloped in the flexible membranes. Based on the single-box–membrane breakwater of Wu et al. [20], Wu et al. [21] implemented the experiments of a side-by-side box–membrane breakwater, enhancing the wave-blocking ability by additional Bragg reflection and wave energy exhaustion in the gap between two boxes. In addition, Wu et al. [22] removed the box and put forward a bilayer slack U-shaped flexible-membrane breakwater, where the outer membrane was separated from the inner membrane, posing a strong hindering-wave ability mainly through absorbing wave energy.

This study, also based on the single-box–membrane breakwater of Wu et al. [20], and introducing the U-shaped flexible-membrane breakwater presented by Wu et al. [22], proposes a new box–membrane combinational breakwater, combining the advantages of both in terms of reflecting waves and wave-energy dissipating. In this proposed breakwater, the outer membrane is separated from the box sides, unlike that of Wu et al. [20] which was fastened together with the inner membrane on the box sides, according the design of Wu et al. [22]. Compared with the proposed breakwater of Wu et al. [22], this study added a box above the inner membrane. The dynamic motion response modeled by a two-way fluid–structure interaction (FSI) of the U-shaped flexible membranes is comparatively complex; thus we opted to prioritize experimental investigation as the primary research approach. And in this study, the investigated breakwaters were all moored instead of fixed as in Wu et al. [20] and Wu et al. [22]. While acknowledging the broader mooring system influence, this investigation prioritized the evaluation of the wave attenuation characteristics of the proposed breakwater, reserving mooring system analyses for subsequent research. For the proposed breakwater under mooring, the box draft d was determined through varying the draft of a naked mooring box, and the two membrane spacings separated from two box sides were optimized through adjusting the spacings of a mooring U-shaped flexible-membrane breakwater. Then, this proposed breakwater was compared to the box-type breakwater and the U-shaped flexible-membrane breakwater with respect to wave attenuation performance and mechanism, all under mooring. Finally, compared to the box-type flexible-membrane breakwater presented by Wu et al. [20], it was not fixed but was moored. In this paper, the tested breakwaters had the same sizes for box and membranes.

These preliminary experiments were carried out in a 2D piston-type wave flume under regular waves, where wave elevations were acquired by wave gauges for the hydrodynamic coefficients of transmission coefficient, reflection coefficient, and energy dissipation coefficient. The rest of the paper is organized as follows: Section 2 describes the experimental setup. Section 3 defines three hydrodynamic coefficients. In Section 4, the results are provided and discussed. In Section 5, the conclusions of this study are summarized.

2. Experimental Setup

2.1. The Proposed Breakwater

Herein, a new configuration of a box–membrane breakwater was developed by combining the box-flexible membrane breakwater presented by Wu et al. [20] with the U-shaped flexible-membrane breakwater presented by Wu et al. [22]. The schematic diagram of the proposed new box–membrane combinational breakwater is depicted in Figure 1. It consisted of a wooden box (length L = 0.595 m; box width B = 0.2 m; height D = 0.155 m, as in Wu et al. [21]), U-shaped transparent PVC flexible membranes (1.0 mm thick; length of inner membrane L₁ = 0.45 m; length of outer membrane L₂ = 0.9 m), and six flexible buoys. The U-shaped transparent Poly Vinyl Chloride (PVC) flexible membranes were made of a common tablecloth, whose density of 1220 kg/m³ was slightly weightier than water, with the choice of membrane geometry referring to Wu et al. [22]. The outer membrane was separated from the two box sides in spacings of L_u and L_d, while the inner flexible membrane was fixed directly on both sides of the box. The flexible membranes dropped down freely to form the inner and outer layers of the U-shaped flexible membranes and both layers did not touch the bottom. Such a design had the objective of mitigating waves more effectively than the breakwater of Wu et al. [20], based on retaining the merits of the breakwater in Wu et al. [22] related to wave energy dissipation, and retaining the multiple roles of the box-type breakwater.

2.2. The Tested Breakwaters

To verify the effectiveness of the optimization by incorporating auxiliary buoys on both sides of the box and positioning the flexible outer membrane at a specific distance from the box, preliminary mooring experiments were carried out to separately decide the draft d of the box and the spacings L_u and L_d of bilayer membranes of the novel type of box–membrane combinational breakwater. Specifically, the wave attenuation performance of a moored box breakwater (Model 1, length L = 0.595 m; width B = 0.2 m; height D = 0.155 m), whose size was identical to the box in the proposed breakwater, was experimentally investigated with various drafts d. Meanwhile, the buoy–membrane breakwater (Model 2; middle buoy length, which was equal to the box width, was 0.2 m), consisting of bilayer membranes attached by buoys, was introduced with the same membrane length as in proposed breakwater, which derived from the U-shaped flexible-membrane breakwater proposed by Wu et al. [22], and physical experiments was conducted on the wave attenuation performance of Model 2 under different upstream-side spacings (length of left buoy) L_u = 0.16, 0.18, and 0.2 m and downstream-side spacings (length of right buoy) L_d = 0.16, 0.18, and 0.2 m, with the choice of membranes spacing referring to Wu et al. [22]. Thereby, not only the optimal draft d of Model 1 and the upstream-side and downstream-side spacing (L_u, L_d) of Model 2, but also the reference parameters of the optimal new box–membrane combinational breakwater (Model 3) were obtained through these preliminary experiments. To investigate the effectiveness of the proposed new box–membrane breakwater, we conducted a comparative experiment of wave attenuation performance between the new box–membrane combinational breakwater (Model 3) and the box-type flexible-membrane breakwater (Model 4, L_u = 0 m; L_d = 0 m; B = 0.2 m; L₁ = 0.45 m; L₂ = 0.9 m) proposed by Wu et al. [20]. The models tested are presented in

Table 1. The photographs and sketches of the tested breakwaters.

Photographs in Calm Water	Photographs in Waves	Sketches
Model 1
Model 2
Model 3
Model 4

and the detailed parameters of the models are listed in

Table 2. The parameters of the tested models.

Model	Rectangular Box Size (m)				Membrane Length (m)		Lu (m)	Ld (m)
	L	B	D	d	Inner	Outer
Model 1	0.595	0.2	0.155	0.05, 0.075, 0.01	-	-	-	-
Model 2	0.595	0.2	-	-	0.45	0.9	0.16, 0.18, 0.2	0.16, 0.18, 0.2
Model 3	0.595	0.2	0.155	0.075	0.45	0.9	0.16	0.2
Model 4	0.595	0.2	0.155	0.075	0.45	0.9	0	0

.

All the tested breakwaters were moored by a four-point symmetric mooring system on the bottom of the breakwater using wire ropes, with a mooring angle θ = 45°. Four wire ropes serving as the mooring lines were symmetrically attached to the corner nodes of the bottom of the breakwater; each wire was connected to a tension spring after threading through four pulleys fixed to the flume bed. The mooring method moored the structure on both sides by mooring lines with a certain mooring angle referred to in Jonkman’s research [23]. The mooring lines extended to the flume top, where their free ends were secured via screw-fastened clamps, enabling precise tension adjustment and the spring was used to simulate the stiffness of the mooring lines with a stiffness coefficient of 3.13 kN/m which was assumed to be constant. Applying a modest pretension of 250 N to each mooring line helped reduce horizontal drift while allowing vertical heave motion, which is a common laboratory practice when adjusting model restoring and the similar uses of springs to simulate mooring line stiffness at model scale, and controlled pretension to control the horizontal drift of the main structure under waves can also be found in recent experimental mooring setups such as in Zhai et al. [24]. The sketch of the experimental facilities arrangement is shown in Figure 2.

2.3. Experimental Facilities

The arrangement of the wave gauges and tested breakwater (Models 1–4) in the wave flume is sketched in Figure 3. The dimension of the wave flume in the Laboratory of Fluid Mechanics at Wuhan University of Technology (Figure 4) is 18 m × 0.6 m × 0.8 m, where the wave height and period exhibit repeatability and stability errors of less than 5% and 0.5%, respectively, and the decrease in the degree of wave height per unit distance in the flume is less than 2%, mainly due to fluid viscosity [25]. A piston-type wavemaker controls the push plate to carry on the reciprocating motion to create the wave with a maximum stroke of ±300 mm and a set of porous triangle boards is set up at the tail end of the flume as a wave absorber, as shown in Figure 4. The wave elevations are acquired by the capacitance-type digital wave gauges, and the absolute error is less than ±1 mm.

The leading edge of the tested breakwater was fixed at 6.9 m from the wavemaker. Wave gauges WG1 and WG2 fixed in front of the breakwater were used for the reflected waves, while gauges WG3 and WG4 placed behind the breakwater were intended for the transmitted waves. The distances between the wave gauges are shown in Figure 3. It should be noted that WG1 and WG2 (WG3 and WG4) act as substitutes for each other in case the measurement fails.

2.4. Wave Conditions

The water depth was h = 0.4 m and was kept constant. The wave periods T varied within 0.8 s to 1.76 s, and corresponding wavelengths λ varied between 0.986 m and 3.182 m according to the dispersion relation. Wave steepness H/λ was selected as 0.02 and water depth h was chosen as 0.4 m. As can be seen in

Table 3. Wave parameters.

T (s)	λ (m)	H/λ	H (mm)	h (m)
0.8–1.76	0.986–3.182	0.02	19.74–63.66	0.4

, the wave parameters were designed based on the water depth and length of the flume, as well as the wave making conditions of the flume. The actual experimental wave heights were slightly smaller than the wave heights H of Table 3 due to the fluid viscosity and the slits between the wavemaker and the flume walls.

Before each test, the flexible membrane part of the models (except for Model 1) was restored, and the free surface was stationary.

3. The Hydrodynamic Coefficients

3.1. Transmission Coefficient

The transmission coefficient K_t is defined as the ratio of the transmitted wave height H_t to the incident wave height H_i:

$$K_{\mathrm{t}}={\displaystyle \frac{H_{\mathrm{t}}}{H_{\mathrm{i}}}}.$$

(1)

where both H_t and H_i are obtained directly from the wave elevations measured by WG3 (see Figure 2). H_i is an average value of several wave heights without the model, and H_t is that with the model (breakwater).

To assess reproducibility, each test condition was conducted twice and the corresponding incident and transmitted wave elevations were compared directly, with an example shown in Figure 5. As illustrated by the wave elevations time history curves of H_t1 and H_t2 obtained from WG3 with Model 1 under draft d = 0.05 m and the corresponding H_i1 and H_i2 measured by WG3 without the breakwater, the two runs exhibited only minor differences in both the incident and transmitted wave elevations. Here, H_t1 and H_t2 denote the transmitted wave heights obtained from the first and second experimental runs with Model 1, respectively, while H_i1 and H_i2 represent the corresponding incident wave heights measured without the breakwater.

More importantly, despite the minor differences in the raw wave elevation records, the resulting transmission coefficients obtained from the two runs remained highly consistent (e.g., K_t1 = 0.710 and K_t2 = 0.713 for the example in Figure 5), yielding an averaged value of K_t = 0.71. It can be seen that the difference between K_t1 and K_t2 calculated from the two measurements was not significant. This demonstrates that the transmission coefficient, as a main non-dimensional coefficient, is insensitive to small run-to-run variations in wave elevations and therefore provides repeatability assessments and a relatively robust and reproducible characterization of the experiment. Accordingly, the reported K_t values in this study were obtained by averaging the two independent runs for each test condition. For repeatability assessments of other hydrodynamic parameters, the same procedure was applied to their calculation following the same statistical analysis framework.

Here, wave elevations measurements reveal relatively flat wave crests during winter months, though the underlying mechanisms remain undetermined.

3.2. Reflection Coefficient

The reflection coefficient K_r can be determined from the ratio of the reflected wave height H_r to the incident wave height H_i:

$$K_{\mathrm{r}}={\displaystyle \frac{H_{\mathrm{r}}}{H_{\mathrm{i}}}}.$$

(2)

The incident wave height H_i is the averaged value of several wave heights measured by WG2 (see Figure 2) without the model. A composite wave, which is the superposition of the incident wave with the reflected wave by the breakwater, is obtained by WG2 with the breakwater. By subtracting the incident wave from the composite wave, the reflected wave height H_r is obtained by the average value of the wave heights in the steady section of the subtraction result [26,27], as in the case of Model 3, which is shown in the box frame of Figure 6.

In this paper, the number j of the steady wave contour for incident waves, reflected waves and transmitted waves is chosen as N = 3 due to the limitation of both the shorter wave-flume length and the longer wave length. The H_i and H_r can be further written as

$$\left\{\begin{array}{c}{H}_{\mathrm{i}}={\displaystyle \frac{1}{N}}{\sum }_{j=1}^N{H_{\mathrm{i}}}_j\\ H_{\mathrm{r}}={\displaystyle \frac{1}{N}}{\sum }_{j=1}^N{H_{\mathrm{r}}}_j\end{array}\right..$$

(3)

3.3. Energy Dissipation Coefficient

The transmission coefficient K_t, reflection coefficient K_r, and energy dissipation coefficient K_e satisfy the following relationship according to the principle of energy conservation [26]:

$$K_{\mathrm{t}}^2+K_{\mathrm{r}}^2+K_{\mathrm{e}}^2=1.$$

(4)

4. Discussion

4.1. The Influence of Draft d on the Breakwater of Model 1

In this section, except for investigating the influence of draft d on the breakwater of Model 1, the moored box-type breakwater (Model 1) is compared with the moored box-type breakwater of Liang et al. [28] with respect of wave attenuation performance to verify the present experiments. The photograph of the breakwater of Liang et al. [28] is shown in Figure 7, and its geography parameters and tested wave parameters are listed in

Table 4. The parameters of the compared breakwaters.

Breakwater	Box Size (m)		Mooring System		h (m)	T (m)	H (m)	H/λ
	B	d	Stiffness (kN/m)	Pretension Force (N)
Liang (2022) [28]	0.5	0.16	2.36	0	0.6	1.0–1.8	0.1	0.026–0.065
Model 1	0.2	0.05 0.075 0.1	3.13	250	0.4	0.80–1.76	0.020–0.064	0.02

, as well as those of Model 1. Figure 8 shows the variations in hydrodynamic coefficients with dimensionless relative water depth kh for Model 1 and the breakwater of Liang et al., where k = 2π/λ is wave number. Model 1 was tested with three drafts d = 0.05, 0.075, and 0.1 m, under water depth h = 0.4 m; the Liang et al. [28] breakwater had a draft of d = 0.16 m, under h = 0.6 m. Their wave conditions are referred to in Table 4.

It can be seen from Figure 8 that the hydrodynamic coefficients of Model 1 and the breakwater of Liang et al. [28] are both at the same level, and the transmission coefficients K_t of Model 1 under three drafts and the moored box-type breakwater of Liang et al. [28] all exhibit an increasing-then-decreasing trend with increasing kh, which demonstrates the validity of the present experiments. What is more, Model 1 with a middle draft d = 0.075 m achieves better attenuation performance than Model 1 with a shallower draft d = 0.05 m through higher reflection and comparable wave attenuation performance to Model 1 with the deeper draft d = 0.1 m, for shorter waves. In the meantime, the moored Model 1 with the deeper draft d = 0.1 m exhibits a relatively high transmission coefficient K_t under longer waves, which corresponds to the observation of comparatively more intense motion response of the moored Model 1 and thus generates more radiation waves during the test, deteriorating the wave attenuation performance [29]. Here, the deeper the box is immersed, the more the incident waves can be reflected through the wetted surface, which can be validated by the curves of reflection coefficient K_r in Figure 8.

Due to the comparatively good wave attenuation performance of the moored Model 1 with middle draft d = 0.075 m under regular waves in various frequency bands in the experiments, for the new box–membrane combinational breakwater (Model 3) proposed in this study we selected middle draft d = 0.075 m in subsequent mooring tests.

4.2. The Influence of Upstream-Side Spacing L_u on the Breakwater of Model 2

Figure 9 reveals the variation in hydrodynamic coefficients of different L_u (0.16, 0.18, and 0.2 m) with relative widths B/λ for the buoy–membrane breakwater (Model 2) consisting of bilayer membranes attached by buoys under constant L_d = 0.16, 0.18, and 0.2 m, respectively, where water depth h = 0.4 m and wave steepness H/λ = 0.02.

As can be seen in the transmission coefficient K_t curves in Figure 9a,b as well as Figure 9c, under different L_u and L_d, the transmission coefficient K_t of Model 2 generally decreases with the increase in B/λ. What is more, when L_u = 0.16 m and L_d = 0.2 m, the transmission coefficient K_t of the Model 2 is relatively low, and the model with this structural dimension has the best wave attenuation performance among the different structural dimensions, which provided a reference for the “short-front long-back” design for the proposed new box–membrane combinational breakwater in subsequent experiments. As shown in Figure 9a, when L_d = 0.2 m (i.e., L_u < L_d), the smaller the distance L_u, the better the wave attenuation performance of Model 2, which indicates that increasing the width of the model does not necessarily enhance its wave attenuation performance.

In the reflection coefficient K_r curves in Figure 9, the reflection coefficient K_r of Model 2 with different structural dimensions shows an overall upward trend with the increase in B/λ, indicating that the model has higher reflection for shorter waves. When the downstream-side spacings L_d are kept constant, the model with shorter upstream-side spacing L_u = 0.16 m corresponds to a relatively larger reflection coefficient than those of the models with other L_u, which shows that strategic shortening of L_u has the function of simultaneously reducing transmission and increasing reflection of Model 2 under relatively smaller wavelengths, meanwhile providing a reference of the short-front design for the Model 3.

By comprehensively comparing the K_t and K_e curves of Model 2, it can be seen that the model has relatively low K_t with high K_e, indicating that the wave attenuation mechanism of the flexible membrane system primarily could be directed to the dissipation of wave energy. The water column trapped by the bilayer flexible membranes can absorb wave energy, and the deformation motion of the flexible membranes oscillates in a manner of phase cancellation [22]. The dissipation coefficient K_e curves in Figure 9 indicate that the dissipation coefficient of Model 2 generally increases with the growth in B/λ, validating that the model has a stronger dissipation effect on waves with smaller wavelengths. Moreover, the impact of models with different structural dimensions on wave energy dissipation capacity is not significantly different. Nevertheless, when B/λ < 0.103, the energy dissipation coefficient of the model with L_u = 0.16 m and L_d = 0.2 m is higher than that of models with other L_u and L_d, which indicates that the model with these structural dimensions can maintain a relatively stronger ability to dissipate waves under long waves compared to models with other structural dimensions. Based on the comparatively stronger energy consumption capabilities of the model with L_u = 0.16 m and L_d = 0.2 m across the selected wave frequency bands, it can be concluded that drawing lessons from this “short-front long-back” design for the Model 3 can lead to its better energy consumption performance.

4.3. The Influence of Downstream-Side Spacing L_d on the Breakwater of Model 2

Figure 10 reveals the variation in hydrodynamic coefficients of different L_d = 0.16, 0.18, and 0.2 m with relative widths B/λ for the buoy–membrane breakwater (Model 2) consisting of bilayer membranes attached by buoys under constant L_u = 0.16, 0.18, and 0.2 m, where water depth h = 0.4 m and wave steepness H/λ = 0.02.

It can be observed from the transmission coefficient curves in Figure 10a,b as well as Figure 10c, that the transmission coefficient K_t generally shows a decreasing trend with the increase in the B/λ. As demonstrated in Figure 10a for the Model 2 with L_u = 0.16 m, the models with longer L_d can achieve better wave attenuation performance and dissipate more wave energy compared to those with shorter L_d, the reason for which lies in the fact that longer L_d can increase the relative width of the model and the distance of wave energy dissipation, and increasing the width of the breakwater could bring to optimization the wave attenuation performance. As can be seen in the reflection coefficient K_r curves in Figure 10, when the downstream-side spacing L_u is kept constant, the model with shorter downstream-side spacing L_d = 0.16 m generally corresponds to a comparatively larger reflection coefficient under B/λ < 0.103. Based on the above analysis, strategic shortening of L_u has the function of simultaneously increasing reflection and reducing transmission of Model 2 under long waves, meanwhile providing a reference for the short-front design for the Model 3.

4.4. The Comparisons of Model 3 to Model 1 and Model 2

This section aims to investigate the influence of the bilayer membranes attached by buoys on wave attenuation characteristics of the proposed breakwater through a comparison of hydrodynamic coefficients between Model 3 and Model 1, while the effect of the box is examined by comparing Model 3 with Model 2. Figure 11 presents the variation tendency of hydrodynamic coefficients varying with relative width B/λ for single moored box-type breakwater (Model 1, d = 0.075 m), the buoy–membrane breakwater (Model 2, L_u = 0.16 m; L_d = 0.2 m; B = 0.2 m; L₁ = 0.45 m; L₂ = 0.9 m) consisting of bilayer membranes attached by buoys and the newly proposed box–membrane combinational breakwater (Model 3, L_u = 0.16 m; L_d = 0.2 m; B = 0.2 m; L₁ = 0.45 m; L₂ = 0.9 m), when water depth h = 0.4 m, wave steepness H/λ = 0.02, and draft d = 0.075 m. For comparative analysis, Model 1 is developed by removing the bilayer membranes from Model 3 and equipped with the same box and draft d = 0.075 m as Model 3. Similarly, Model 2 is introduced by eliminating the main box from Model 3 and has the same bilayer membranes and upstream-side and downstream-side spacing (L_u, L_d) as Model 3. The detailed parameters of models are listed in Table 2.

As seen in the transmission coefficient curves in Figure 11, the transmission coefficient K_t of the three models generally decreases with increasing B/λ as the wavelength increases. And the new box–membrane combinational breakwater (Model 3) consistently exhibits the lowest transmission coefficients across the designed wave periods and has improved wave attenuation performance by 56.8%, at most, compared to Model 1 under B/λ = 0.203 and by 21.2%, at most, compared to Model 2 under B/λ = 0.118, outperforming Model 1 and Model 2 as the wave period increases, because Model 3 exhibits the highest reflection under B/λ > 0.084 and the most severe wave energy dissipation, while B/λ is less than 0.137 in three models. This indicates that combining the box and the bilayer membranes attached by buoys is effective in optimizing the wave attenuation performance by providing more wave reflection for short waves and more wave energy dissipation for long waves.

What is more, as revealed in Figure 11, the wave attenuation performance and wave energy dissipation of Model 1 are significantly and stably weaker than those of Model 2 and Model 3, which manifests that applying flexible membranes to the box-type breakwater contributes to higher wave energy dissipation of the breakwater. The reason lies in the fact that the water column enclosed by the bilayer membranes can actually dissipate a significant amount of wave energy, given that its mass is relatively large compared to that of the breakwater itself, which also demonstrates that the part of the bilayer membranes attached to the buoys can primarily serve as an energy dissipation component to improve the wave attenuation performance.

4.5. The Comparisons of Model 3 and Model 4 for the Influence of the Spacings

Figure 12 presents the variations in variation tendency of transmission coefficient K_t, reflection coefficient K_r, and energy dissipation coefficient K_e with relative width B/λ for the new box–membrane combinational breakwater (Model 3, L_u = 0.16 m; L_d = 0.2 m; B = 0.2 m; L₁ = 0.45 m; L₂ = 0.9 m) and box-type flexible-membrane breakwater (Model 4, B = 0.2 m; L₁ = 0.45 m; L₂ = 0.9 m) proposed by Wu et al. [20] with no spacings (i.e., L_u = 0 m; L_d = 0 m), water depth h = 0.4 m, wave steepness H/λ = 0.02, and draft d = 0.075 m. The detailed parameters of the models are listed in Table 2.

As shown in the transmission coefficient curves in Figure 12, the K_t of both Model 3 and Model 4 generally decrease with increasing B/λ as the wavelength increases. And the curves of transmission and energy dissipation coefficients in Figure 12 demonstrate that, for short wavelength waves with B/λ > 0.103, the wave attenuation performance and the wave energy dissipation of Model 3 is significantly and stably stronger than that of Model 4 and has improved wave attenuation performance by 47.29% compared to Model 4 at most under B/λ = 0.164, because attaching buoys can increase the relative width of the model and the distance of wave energy dissipation to a certain extent, as increasing the width of the breakwater is effective in optimizing the wave attenuation performance for short-period waves. Another reason for the enhanced wave attenuation performance to the short wave domain is that the oscillation of flexible membranes and water column enclosed when waves pass through a structure can actually dissipate a large amount of energy [21], and due to the longer top distance of the flexible membranes in Model 3 compared to Model 4, the bilayer flexible membranes in Model 3 are able to enclose a larger mass of the water column than those in Model 4, therefore resulting in a stronger wave energy dissipation ability, which can be validated by the energy dissipation coefficient curves in Figure 12.

It is also interesting to see, in the reflection coefficient K_r curves in Figure 12, that the Model 3 attached buoys, with the outer membrane that is positioned at a specific distance from the box, have lower reflection but better attenuation performance, in contrast to Model 4 with bilayer membranes directly fixed to the box. We attribute the reasons to the fact that when the incident waves approach Model 3, they first encounter the upstream-side buoy and the water column enclosed by the membranes, which dissipate a portion of the incident wave energy before the waves reach the box. Consequently, the waves that eventually impinge upon the box in Model 3 carry less energy compared to those directly acting on the box in Model 4, which results in weaker reflected waves. Moreover, the reflected waves from the box in Model 3 must pass through the upstream-side buoy again before propagating to the wave gauge for obtaining the reflected waves, therefore dissipating the wave energy. This indicates that the boxes in two models are the primary source of wave reflection in the breakwaters, and meanwhile, the addition of the buoys modifies the configuration, especially enabling the flexible membranes to trap a greater mass of oscillating water column to dissipate wave energy, resulting in enhanced wave energy dissipation and wave attenuation performance.

5. Conclusions

In this paper, a new box–membrane combinational breakwater is proposed. Preliminary experiments were conducted to explore potential structural optimizations for the proposed breakwater and to assess its wave attenuation performance. Preliminary mooring experiments were carried out on the box and the bilayer membranes attached separately by the buoys of the novel type of box–membrane combinational breakwater, based on which a middle draft was selected for the box and a “short-front long-back” arrangement was chosen for the bilayer membranes. The new box–membrane combinational breakwater proposed in this paper can achieve relatively good wave attenuation performance.

Increasing the model’s width and adding buoys could enhance wave attenuation stability, but the wave energy dissipation mechanisms of flexible membranes require further investigation. Furthermore, due to the effectiveness of addition of the U-shaped flexible membrane, testing models with more flexible materials and similar criteria for dynamic response of flexible membranes are necessarily needed to evaluate their wave dissipation performance for practical engineering applications. Another limitation of our study lies in insufficient statistical analysis and research into the influence of the factors that real-world conditions typically involve, such as the mooring system and the dynamic response of the breakwater; irregular, multidirectional waves; and three-dimensional effects, which are not addressed in the present work. In addition, the new design of box–membrane combinational breakwater requires further numerical investigation of its motion state and force in the flow field, and flexible membranes need to be modeled by a two-way fluid–structure interaction (FSI) method.