1. Introduction
In recent years, coastal erosion has accelerated globally due to rising sea levels caused by climate change and unchecked coastal development. Underwater breakwaters, submerged structures designed to reduce incident wave energy, are sometimes employed to address this issue. Unlike detached breakwaters that extend above the water surface, underwater breakwaters preserve natural landscapes and support ecosystems by allowing some controlled seawater movement. In South Korea, breakwaters are widely used to prevent erosion along the coastline.
Underwater breakwaters typically reduce wave energy by increasing crest height and inducing breaking waves on the crest (Hur and Kim, 2003). However, excessively increasing the height of breakwaters can hinder efficient seawater flow and render ship operations impractical. While the height of breakwaters significantly influences incident wave-energy attenuation (Rahman and Womera, 2013), an alternative approach involves arranging structures in multiple rows rather than increasing their height. This configuration enhances wave reduction and achieves high attenuation performance for specific wavelengths through the Bragg resonance phenomenon (Cho and Jeong, 2003). When properly utilized, this phenomenon allows for effective wave-energy attenuation at the most frequently observed wave frequencies.
Two-dimensional (2D) performance analysis is commonly used for underwater breakwaters, as these structures are typically long, parallel to the shoreline, and have uniform cross sections. The Bragg resonance phenomenon arises from wave reflections caused by such structures. When crest heights are low enough to prevent wave breaking, potential flow-based analysis is suitable because fluid viscosity has minimal impact. Lee et al. (2003) investigated the reflection characteristics of multi-row impermeable breakwaters based on cross-sectional geometry using the eigenfunction expansion method and hydraulic model experiments. Cao et al. (2012) determined optimal incident wave-attenuation conditions by adjusting the spacing between multi-row trapezoidal breakwaters in relation to depth, wavelength, and wave height. To precisely evaluate the wave-attenuation performance of breakwaters, viscous flow simulations based on the Navier–Stokes equations were conducted. Hsu et al. (2004) applied the Reynolds-averaged Navier–Stokes (RANS) model and found that wave energy dissipated in impermeable double breakwaters due to eddy formation. These studies demonstrated the effectiveness of two-row submerged structures for wave attenuation. However, in many previous studies, the shapes of each structure in the two-row breakwater were assumed to be identical.
This study aims to identify the incident wave-energy attenuation effect and its impact, focusing on the Bragg resonance phenomenon caused by a two-row arrangement of rectangular cross-section structures, which is the simplest form, and the height difference between each structure. This can provide important basic data for the optimal design of multi-row underwater breakwaters, and the wave-energy attenuation phenomenon can be better understood.
In this study, numerical analyses and experiments were conducted using linear wave theory to simplify the problem (Fig. 1). Potential flow-based and viscous flow analyses, including computational fluid dynamics (CFD), were performed alongside wave tank experiments for comparative analysis. Through a comparative analysis of each method, this study extends previous research on the wave attenuation effects of two-row breakwaters and examines the flow distribution contributing to wave attenuation. Wave tank experiments were carried out in a two-dimensional wave tank at Inha University. Numerical analyses employed a potential flow-based boundary element method (BEM) using numerical wave tank technique and viscous flow analysis using Star-CCM+ software. Results from the BEM and CFD analyses were compared to evaluate the wave-attenuation effect influenced by fluid viscosity, and the flow changes around the structures were calculated. The accuracy of the numerical results was validated against experimental data. Finally, the study analyzed the wave-attenuation effect and flow characteristics of the two-row submerged structure by examining variations in transmitted and reflected waves relative to the height differences between the front and rear structures.
2. Experimental Setup and Method
Experiments to measure the wave-attenuation effect of the two-row submerged structure were conducted in a 2D wave tank at Inha University (Fig. 2). The wave tank had a length of 6 m, width of 0.3 m, and depth of 0.5 m. It was equipped with a wave maker and wave absorber at each end. The wave absorber, a structure composed of a horizontal plate and an inclined plate, can control reflected waves by attenuating the wave energy of the incident wave. According to the results of research conducted by Jung and Koo (2021) in the same wave tank, the wave absorber had a reflectance of 0.1 or less at a water depth of 0.35 m for waves with an incident wave period of 0.8 to 1.2 s and a wave height of 0.02 m.
An ultrasonic wave gauge (WG), shown in Fig. 3(a), was used to measure the wave height, and its main specifications are listed in Table 1. The displacement of the free water surface was measured every 0.05 s using four WGs. The three-point method proposed by Suh et al. (2001) was used to calculate the incident and reflected wave heights from the wave height data measured in front of the structure. Hence, one unit was placed behind the WGs, and the distances of each WG from the wave maker were 2.04 m (WG 1), 2.24 m (WG 2), 2.44 m (WG 3), and 3.92 m (WG 4). WG 3 was installed at a position 0.355 m away from the front of the structure. The structure was installed at a sufficient distance of 2.80 m from the wave maker to minimize the effect of re-reflected waves from the wave maker. Fig. 1 shows the overview of the experimental setup.
The submerged structure model was determined to be rectangular for the convenience of experimental implementation and size modification. The water depth (h) was 0.35 m, and the width of breakwaters (Wb) was 0.15 m. The spacing of the model (Lb) was 0.6 m based on both ends of the two rows. It was approximately half of a wavelength of 1.2 m at a wave period of 0.9 s, a condition that can theoretically cause the Bragg resonance phenomenon. The model was fabricated by combining 0.05 m× 0.05 m zinc rectangular pipes, and it was fixed to the bottom of the tank under its own weight. A horizontal width of 0.15 m was implemented by combining three rectangular pipes, and two heights were created (0.15 and 0.2 m). The length of the pipes was set to 0.28 m, which was close to the tank width. Table 2 shows the cases of the submerged body shape used in the experiment. To analyze the reflection and transmission coefficients by incident wave period under each condition, we assumed the incident wave to be a regular wave, and we used a total of five waves with a period of 0.7 to 1.1 s at an interval of 0.1 s. The wave height was set to 0.02 m (Table 3).
3. Numerical Analysis
3.1 Potential Flow-Based Numerical Analysis
The interaction between the incident wave and submerged structure in the frequency domain was analyzed using a potential flow-based 2D numerical wave tank software (in-house software). The computational fluid domain was assumed to be incompressible, irrotational, and inviscid. Under the conditions of using the velocity potential and the fluid satisfying the continuity equation, the governing equation for the fluid was the Laplace equation.
In the wave diffraction problem for the submerged structure, the velocity potential is composed of the sum of the velocity potentials of the incident and diffracted waves. The incident wave was assumed to be linear, and the incident wave velocity potential can be calculated using Eq. (3).
where ϕI and ϕD are the velocity potentials of the incident and diffracted wave, respectively, and g is the gravitational acceleration. H is the wave height, ω is the incident wave frequency, k is the wave number, and h is the water depth. x is the horizontal coordinate in the incident wave direction, and z is the vertical displacement from the water surface. The Laplace equation can be converted into a boundary integral equation for each boundary of the fluid domain using Green’s second identity (Eq. (4)). α is the solid angle, and has a value of 0.5 on the boundary.G is Green’s function, which discretizes each boundary and can express each node point on the boundary with a value corresponding to the distance to the point by defining the source and field points.
Each boundary of the fluid domain can be divided into free surface, radiation, bottom surface, and fixed body boundary conditions (Fig. 4). Eq. (5) indicates free surface boundary conditions that combine dynamic and kinematic boundary conditions.
Sommerfeld radiation boundary conditions were used on both sides of the fluid domain to implement the radiation boundary. The boundary condition considering the influence of incident waves was applied to ΓRad1 (Koley et al., 2020).
Because the bottom surface and structure boundaries had no fluid velocity in the normal line direction, Eq. (8) was applied as impermeable boundary conditions.
By substituting Eqs. (5) to (8) into Eq. (4) and performing matrix operations through discretization, velocity potentials on all boundaries can be obtained. The reflection and transmission coefficients can be calculated using the velocity potential on the open sea boundary as follows (Vijay et al., 2021).
KR and KT are the reflection and transmission coefficients, respectively. Z0 (z) = cosh k(z + h)/cosh kh is the vertical eigenfunction, and
N 0 2 = ∫ − h 0 Z 0 2 ( z ) d z K R 2 + K T 2 = 1
3.2 Viscous Flow-Based Numerical Analysis (CFD)
To perform CFD analysis based on viscous flow, the commercial software Star-CCM+ (19.02.012) was used to implement a two-dimensional numerical wave tank based on linear wave theory. The governing equations are incompressible unsteady RANS equations. The standard k− ∊ low-Re model, which adequately reflects turbulent flows in ocean and coastal flows with short calculation time, was used as a turbulence model. The two-phase volume of fluid (VOF) technique was applied to the water–air fluid interface. For boundary layers, the bottom surface with a no-slip condition was applied through the all y+ wall treatment setting and the y+ value was set to 1 or less for the walls of the structure. Fig. 5 shows the boundary conditions set for the fluid domain. The symmetry condition was set for the sides of the tank to implement the 2D condition. In addition, the forcing zone was set at the inlet and outlet by the incident wavelength to generate accurate waves inside the domain and prevent reflected waves on boundaries. The time step for time-series calculation was set considering the wave period and mesh conditions as shown in Eq. (11). This is the method recommended by the software, and the reliability of the results can be ensured as the same method was used by Gunawardane et al. (2020).
The mesh size of the computational domain was set to vary depending on the wavelength and wave height of the incident wave to implement accurate waves. To determine the appropriate mesh size, we conducted a convergence test on the ratio of the measured wave height to the input wave height according to cells per wave height (CPH), and the results are shown in Fig. 6. CPH refers to the number of meshes in the z direction set in the incident wave area to implement accurate waves on the free surface (Katsidoniotaki and Göteman, 2022). Based on the test results and software recommendations, the mesh size in the x-axis direction was set to 1/100 of the wavelength and that in the z-axis direction to 1/20 of the wave height. The y-axis direction was set to be very thin assuming a 2D calculation. Meshes were constructed by applying the prismatic layer around the submerged structure and on the bottom surface to represent the viscosity effect in the boundary layers. Fig. 7 shows the mesh setup of Case 3 at a wave period of 0.8 s.
For the incident wave, CFLI of 50.0, CFLu of 100.0, sharpening factor of 0.1, and angle factor of 0.15 were entered into the fifth Stokes wave, and a linear wave corresponding to a wave height of 0.02 m and a period of 0.7 to 1.1 s was generated. To verify the wave generation reproducibility and the effectiveness of the wave absorber in preventing the reflection of waves at the end of the computational domain, time-series results for waves with a period of 1.1 s in the absence of any structures in the fluid domain are presented in Fig. 8. The water surface displacement was calculated at the same position as WG 1 in the experiment. The generated wave remained constant in the steady state, and the reflections at the end of the boundary were insignificant.
Wave height measurements were conducted at the same locations as in the experiments, and the wave height time-series data recorded in front of the structure were separated into incident and reflected waves using the three-point method. The data after approximately nine periods were used to extract the wave height in the steady state after the propagation of the incident wave. Fig. 9 shows the wave height time-series data of Case 4 at a period of 0.9 s and a wave height of 0.02 m. The steady-state time-series data were measured at each wave gauge.
4. Result Analysis
To compare the results of the experiment and numerical analysis conducted in this study, Fig. 10 shows all of the reflection coefficients (KR) and transmission coefficients (KT) of Cases 1 to 4 in black and red lines, respectively. BEM represents the results of analyzing potential flow through the boundary element method, and CFD shows the results of the viscous flow-based analysis. They were compared with the experimental results (EXP). For all cases, the reflection coefficient was highest, and the transmission coefficient was lowest around kh = 1.54 (T = 1.0 s). The experiment and CFD analysis results were relatively in good agreement for the transmission coefficient, and they were lower than the potential flow analysis (BEM). This difference is attributed to the effects of fluid viscosity, which are not accounted for in BEM. Notably, the discrepancy between the CFD and BEM results is more pronounced for the transmission coefficients than for the reflection coefficients. This is because fluid viscosity has minimal influence on wave reflection from the structure, whereas the passage of incident waves through the submerged structure is affected by vortex generation at the edges of the structure, which contributes to wave energy dissipation.
For the reflection and transmission coefficients according to the height of the submerged structure, the reflection coefficient increased and the transmission coefficient decreased as the structure height increased. In particular, the reflection coefficient was highest in Case 4 (20 × 20), where both the two-row submerged bodies were high, and lowest in Case 1 (15 × 15), where both of them were low. The transmission coefficient was similar in Case 3 (20 × 15), where the front body was higher, and Case 2 (15 × 20), where the rear body was higher.
When the reflection coefficient was compared between the CFD and experimental results, significant differences occurred, unlike for the transmission coefficient. In particular, the experimental values were significantly lower than for the CFD or BEM near kh = 1.54, where the reflection coefficient was high. This indicated that the tank used in the experiment was not long and the reflected waves by the submerged structure may have not been perfectly controlled. Even though the wave maker can partially control reflected waves, it is believed that complete control was not achieved due to the short propagation distance. This indicates the need for further research, including more precise measurements and verification of the wave maker’s performance.
Fig. 11 shows the transmission coefficient, reflection coefficient, and dissipation coefficient (KD) by case at kh = 1.54 (T = 1.0 s), where Bragg resonance occurred with the highest reflection coefficient. The relationship among the coefficients can be calculated using Eq. (12).
Overall, the transmission coefficient is inversely proportional to the reflection coefficient. In the comparison between Case 3, where the front structure of the two-row submerged breakwater is taller, and Case 2, where the rear structure is taller, both the transmission and reflection coefficients were smaller in Case 2. This indicates that when the height of the rear structure is greater, wave transmission decreases, resulting in a more effective breakwater performance. Additionally, the reflection coefficient was higher in Case 3, where the front structure is taller. This is attributed to the fact that incident waves are primarily reflected at the front of the structure.
Fig. 11 shows that Case 2 exhibited the highest dissipation coefficient. This appeared to be owing to the incident wave entering relatively easily because the front body was low, and its energy loss was large owing to the height of the rear body. However, the transmission coefficient of the incident wave was the lowest in Case 4, where both bodies were high, indicating the highest wave-breaking effect.
To examine the effects of fluid viscosity on the reflection and transmission coefficients, Table 4 shows the difference between the BEM and CFD results under Bragg reflection conditions (kh = 1.54). The difference in transmission coefficient was largest in Case 2, where the rear body was higher under the influence of fluid viscosity (approximately 11%), whereas the difference in reflection coefficient was largest in Case 3, where the front body was higher (approximately 12%). This agreed with the comparison results of Fig. 9. In other words, fluid viscosity directly affects changes in the reflection and transmission coefficients of the submerged structure.
Fig. 12 shows the distribution of the sum of velocity vectors inside the flow field under Bragg reflection conditions (kh = 1.54). The flow velocity was more pronounced in Case 4 compared to Case 1, especially above the front body, where the sum of velocity vectors was greater. This suggests, as indicated in the previous analysis, that the flow variation increases due to the front structure. In addition, the velocity rapidly changed and weak eddies occurred at both edges of the structure. This was expected to attenuate the incident wave energy.
The analysis of the results revealed that the transmission coefficient of the incident wave for the two-row submerged structure was most significantly affected by the height of the rear body, whereas the reflection coefficient was affected by the height of the front body. In addition, the energy loss of the incident wave increased when the front body was low and the rear body was high.
5. Conclusions
This study compared the wave-attenuation performance of an impermeable two-row submerged structure and the changes in surrounding flow using potential flow-based numerical analysis, viscous flow-based numerical analysis, and two-dimensional (2D) wave tank experiments. A numerical wave tank, developed and validated by the research team, was employed for potential flow analysis, while StarCCM+ was utilized for viscous flow analysis. The key conclusions and implications of this study are as follows:
(1) The results of the potential flow-based numerical analysis, viscous fluid analysis, and experiments exhibited similar trends, demonstrating that potential flow-based analysis is effective for roughly estimating the wave reduction effects of underwater breakwaters.
(2) The highest reflection coefficient and the lowest transmission coefficient were observed under the Bragg resonance condition, where the spacing of the two-row submerged structure was 1/2 of the incident wavelength.
(3) Under the Bragg resonance condition, potential flow analysis showed differences from the computational fluid dynamics (CFD) and experimental results under the influence of fluid viscosity.
(4) As the height of the two-row submerged structure increased, the transmission coefficient of the incident wave decreased, whereas the reflection coefficient increased.
(5) The transmission coefficient was affected by the height of the rear body of the two-row submerged structure, whereas the reflection coefficient was affected by the height of the front body.
(6) The greatest wave-energy loss occurred when the front structure was low, and the rear structure was high. This was likely because the low front structure allowed the incident wave to enter easily, while the high rear structure caused significant energy dissipation between the two structures.
The results of this study can provide an engineering concept of decreasing the height of structures using two-row (or multi-row) submerged structures and controlling the transmission of the incident wave using the Bragg resonance effect. This is a promising alternative that can overcome the limitation of conventional underwater breakwaters in which the construction of structures near the water surface makes seawater flow and ship operation impossible. Additionally, it is valuable as a basic research tool to determine the height and arrangement of breakwaters. However, the numerical model used in this study is a 2D linear model that did not consider the impact of nonlinear waves on the coast. Thus, a nonlinear numerical model calculation that can address this problem is required in the future. Additionally, for simplicity, the structures were arranged in only two rows, resulting in relatively high transmission coefficients. However, optimizing the geometry in multi-row structures could lead to a reduction in the transmission coefficient. Therefore, further research is required to determine conditions that can maximize the incident wave-attenuation effect by adjusting the size and arrangement of submerged structures.
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