1. Introduction
Submerged breakwaters have been applied widely in coastal areas as a coastal disaster prevention method to reduce coastal erosion and protect beaches. Submerged breakwaters do not affect the landscape because their crest lies below the water surface, unlike gravity-type breakwaters. They reduce wave energy by breaking incident waves at the breakwater crest. However, the water level increases behind them owing to wave breaking. This results in a strong flow that attempts to escape to the open sea through the openings between the breakwaters. This rip current around the openings adversely affects the stability of the breakwater through seabed scour, which is the main cause of degradation of the breakwater function. Recent design and construction efforts aimed at improving safety have utilized a construction method that strengthens the ground for scour prevention. Excavation and riprap installation for ground improvement require a long construction period and considerable labor. This reduces the economic feasibility. Furthermore, the planar layout of multiple submerged breakwaters under the application of the area control method in accordance with the length of the beach increases the construction cost. To address these problems, Hur et al. (2019) developed a submerged breakwater with a new function capable of reducing the water level behind the breakwater via drainage channels through hydraulic model experiments. They investigated the effects of the developed breakwater on wave control and water level reduction behind the structure through two-dimensional (2D) experimental analysis. Additionally, they used three-dimensional (3D) numerical analysis to compare the hydraulic characteristics of water level reduction behind the structure by installing the developed breakwater for controlling the rip current around the openings. Recent hydraulic model experiments on submerged breakwaters (Kramer et al., 2005) mostly discussed the energy dissipation and the reduction in transmitted wave heights caused by wave breaking. Empirical formulas (van der Meer et al., 2005; Goda and Ahrens, 2008) were proposed based on the results that involved a wide range of nondimensional factors set in the experiments. In addition, numerical analyses have been conducted on the wave control performance of submerged breakwaters and the flow characteristics of the openings (Johnson et al., 2005; Johnson, 2006; Hur et al., 2012) to examine the mitigation of the aforementioned problems concerning the water level behind the breakwaters using cross-sectional geometry and wave control effects.
Hur et al. (2021) compared the variations in the water level behind the structure for the wave control section and cross-sections with channels through hydraulic model experiments. They performed hydraulic model experiments on an integrated submerged breakwater that utilizes the slit of the crest capable of controlling waves, as well as upper-level and lower-level drainage channels capable of controlling the water level increase. Jeong et al. (2022) compared and examined the hydraulic performance of structures that utilized the presence and absence of slits and channels, as well as wave-dissipating blocks, according to the water level behind the structure and wave control through hydraulic model experiments. They provided quantitative verification data of a submerged breakwater for reducing the water level behind the structure in the numerical wave tank (NWT) used and utilized cross-sectional shapes that reduced the water level behind the structure. Previous studies examined the decrease in water level under the influence of wave control and a submerged breakwater for reducing the water level behind the structure according to the presence or absence of slits and channels for each cross-sectional shape. However, they could not examine the efficiency of wave control and the reduction in the water level behind the structure. This was because they could not consider the specifications of various cross-sectional shapes under the limited conditions of hydraulic model experiments. Therefore, cross-sectional shapes are examined under two conditions for four perspectives to enhance the efficiency of a submerged breakwater for reducing the water level behind the structure. Furthermore, the water level variations and wave control performance owing to variations in specifications (e.g., slits and channels of the structure) are analyzed using NWT to supplement hydraulic model experiments.
In this study, NWT is used to examine the optimal cross-section for enhancing the efficiency of a submerged breakwater to reduce the water level behind the structure. The effectiveness and validity of NWT are examined by comparing the wave control characteristics of the submerged breakwater constructed based on previous hydraulic model experiments and the hydraulic characteristics of the water level behind the structure. In addition, the diverse cross-sectional shapes of the breakwater are analyzed in NWT to numerically investigate the optimal cross-section in reducing the water level behind the structure.
2. Numerical Wave Tank
2.1 Numerical Analysis Method
This study was conducted to examine the optimal cross-section for enhancing the efficiency of a submerged breakwater to reduce the water level behind the structure. Two-dimensional NWT (LES-WASS-2D; Hur and Choi, 2008) was used to analyze the water level variations and wave control performance for various cross-sectional shapes according to the specifications of the structure, such as slits and channels. In addition, the full-nonlinear 2D numerical analysis technique of a 2D Navier–Stokes (N-S) solver based on the volume of fluid (VOF) was used to simulate a complex free surface. The governing equations of the numerical model consist of the continuity equation with wind-wave source terms (Eq. (1)), the modified N-S solver equation (Eq. (2)), and the functional formula of VOF (Hirt and Nichols, 1981) that applies the porous body model (PBM) concept to the continuity equation based on incompressible and viscous fluids (Eq. (3)).
where vi is the flow velocity in the x- and z-directions, q* is the flow density of the source, γv is the volumetric porosity, γi is the areal permeability in the x- and z-directions, t is the time, ρ is the density of water, p is the pressure, νT is the sum of the kinematic viscosity and eddy viscosity of water, Dij is the strain rate tensor, Si is the surface tension term based on the continuum surface force (CSF) (Brackbill et al., 1992), Qi is the wave source term, Ri is the fluid resistance term for the permeable medium, gi is the gravitational acceleration term, Ei is the energy dissipation term of the sponge layer, and F is the volumetric ratio of fluid in each lattice. Details of the numerical model, such as the fluid resistance of the permeable medium (Ergun, 1952, Sakakiyama and Kajima, 1992) and the turbulence model (Smagorinsky, 1963), are presented in Lee et al. (2016).
2.2 Verification of the Numerical Wave Tank (NWT)
The free surface observed in previous 2D hydraulic model experiments (Hur et al., 2021) is shown in Fig. 1. In this study, this free surface was considered to ensure the effectiveness and validity of NWT and, thereby, enhance the efficiency of the submerged breakwater for reducing the water level behind the structure. The experimental and calculated values for the spatial wave distribution and the temporal series of the free surface elevation were compared at each free surface measurement point by nondimensionalizing the variations in water level in front of and behind the structure with the incident wave height under the wave action of CASE-V14 (h = 30 cm, Hi = 5 cm, Ti = 1.5 s). CASE-V14 has intermediate levels of water depth (h), incident wave height (Hi), and incident period (Ti) among the external force conditions. An approximate cross-sectional shape of the submerged breakwater and the configuration of NWT are presented in the middle part of Fig. 1(a). Free surface elevation and mean water level variations measured from two points in front of the structure (WG1–2) and nine points behind it (WG3–11) were nondimensionalized and compared with the incident wave height. Fig. 1(b) –1(l) shows the spatiotemporal variations in free surface elevation nondimensionalized with the incident wave height at each point from WG1 to 11. The red circles (
) in Fig. 1(a)–1(l) represent experimental values, whereas the black solid lines (—) indicate calculated values. From the measured free surface data in Fig. 1, the hydraulic characteristics of the reflection coefficient (KR), transmission coefficient (KT), and dissipation coefficient (KD) (which are calculated using the incident and reflected wave separation method (Goda and Suzuki, 1976)) and the mean water level (η̄) were compared. The cross-sectional verification results of the submerged breakwater in NWT were compared with the results of the hydraulic model experiment performed in the presence of both slits and channels. The experimental and calculated values of hydraulic characteristics under the external force conditions of 27 cases were compared. The results are shown in Fig. 2 and Table 1.
) in Fig. 1(a)–1(l) represent experimental values, whereas the black solid lines (—) indicate calculated values. From the measured free surface data in Fig. 1, the hydraulic characteristics of the reflection coefficient (KR), transmission coefficient (KT), and dissipation coefficient (KD) (which are calculated using the incident and reflected wave separation method (Goda and Suzuki, 1976)) and the mean water level (η̄) were compared. The cross-sectional verification results of the submerged breakwater in NWT were compared with the results of the hydraulic model experiment performed in the presence of both slits and channels. The experimental and calculated values of hydraulic characteristics under the external force conditions of 27 cases were compared. The results are shown in Fig. 2 and Table 1.
Fig. 1(a) effectively reproduces the formation of partially overlapping wave fields under the influence of the reflected waves in front of the breakwater and the wave height distribution characteristics formed owing to the wave height attenuation caused by wave control. This is the phenomenon in which the transmitted wave height decreases owing to the energy dissipation caused by wave breaking on the crest of the breakwater. The spatiotemporal variations in free surface elevation nondimensionalized with the incident wave height at each point are reproduced effectively in Fig. 1(d)–1(l) and the increase in the nonlinearity of waves under the influence of reflected waves in Fig. 1(b) and 1(c). The distribution of the mean water level in front of and behind the structure controlled through the slit, upper-level drainage channel, and low-level drainage channel of the breakwater is also reproduced effectively.
3. Optimal Cross-Section Analysis
In this study, the verified NWT was used to examine the optimal cross-section for enhancing the efficiency of the submerged breakwater. The hydraulic characteristics and the water level behind the structure were numerically compared under the cross-sectional conditions of 56 cases based on four measures of the cross-sectional shapes of the breakwater. The incident conditions of the intermediate levels (h = 30 cm, Hi = 5 cm, and Ti = 1.5 s) used for verifying NWT were also used as depth and wave conditions. Among the cross-sectional shapes, the first is the crest width (B) and crest depth (R) of the breakwater, and the second is the slit distance (DS) and slit width (BS) that the breakwater has for wave and water level control. The third is the height (hU) and size (SU) of the upper-level drainage channel connected to the slit, and the fourth is the height (hL) and size (SL) of the lower-level drainage channel connected to the back of the structure. Fig. 3 shows a schematic for examining the optimal cross-section to enhance the efficiency of the submerged breakwater.
3.1 Optimal Cross-Section of Low Crest Deign
To examine the optimal cross-section to enhance the efficiency of the submerged breakwater, the hydraulic characteristics and the water level behind the structure according to B and R were numerically compared first (Table 2). The subsequent numerical results are omitted because these are equivalent to the results of the analysis of the hydraulic characteristics and the water level behind the structure conducted to examine the effectiveness and validity of NWT. The subsequent breakwaters are landscape-friendly structures with their crests below the water surface. B and R are important factors in examining the hydraulic performance and optimal cross-section. For each parameter variation, the optimal cross-section is examined based on the verified hydraulic model experiment. The examination results for the verified reference cross-section are expressed in bold letters. These serve as reference values for each factor. Therefore, R values from 0 to 6 cm and B values from 35 to 85 cm were considered, whereas the other specifications were identical to those of the reference cross-section.
From Table 2, an increase in B and a decrease in R are effective for wave control. The water level behind the structure tended to increase as B increased. However, it increased and then decreased as R decreased. This demonstrates the wave-breaking effect caused by the ratio of the crest depth to the incident wave height (R / Hi), which shows clearer correlation than previous study (Jeong et al., 2022) considering the water depth (h) and incident wave height (Hi). These results indicate that it is necessary to set control targets for waves and the water level behind the structure with appropriate B and R values to enhance the efficiency of the submerged breakwater.
3.2 Optimal Cross-Section of Slit Deign
To examine the optimal cross-section for enhancing the efficiency of the submerged breakwater, the hydraulic characteristics and the water level behind the structure were numerically compared according to the slit distance (DS) and slit width (BS) among the cross-sectional shapes of the breakwater. The results are presented in Table 3. For the slit, the optimal cross-section was also examined based on the verified hydraulic model experiment in a manner similar to that for the aforementioned reference cross-section of the crest. Therefore, the examination results for the verified reference cross-section are expressed in bold letters. These serve as reference values for each factor. Thus, the slit widths (BS) from 5 to 50 cm and slit distances (DS) from 2 to 42 cm were considered, whereas the other specifications were identical to those of the reference cross-section.
From Table 3, an increase in BS is effective in reducing the water level behind the structure with no significant difference in the transmission coefficient. With an increase in DS, controlling the mean water level behind the structure becomes more effective than influencing the reflection coefficient that decreases through the control of reflected waves in front of the structure. These results indicate that the slit for controlling the water level (rather than waves behind the structure) is required to enhance the efficiency of the submerged breakwater.
3.3 Optimal Cross-Section of Upper-Level Drainage Channel Design
The optimal cross-section was examined to enhance the efficiency of the submerged breakwater through the upper-level drainage channel height (hU) and size (SU). The hydraulic characteristics and the water level behind the structure were numerically compared according to the conditions. The results are presented in Table 4. In addition, the optimal cross-section was examined based on the verified hydraulic model experiment as mentioned earlier. The reference values are expressed in bold letters. As the upper-level drainage channel size (SU) increased from 3 to 19 cm, the upper-level drainage channel height (hU) decreased from 23 to 8 cm. The upper- and lower-level drainage channels were affected by the size and height, as shown in Table 4.
From Table 4, hU decreased as SU increased. However, waves were controlled in conjunction with the reduced water level behind the structure. When SU exceeded a specific threshold, the impact on the water level reduction behind the structure became insignificant. This suggests that as the incident wave height (Hi) propagates over the structure, the backwater level increased by wave breaking is affected by the ratio of the upper-level drainage channel size to the incident wave height (SU / Hi) notwithstanding variations in the size (SU). In addition, the impact of hU was not significant. The location of the lower-level drainage channel determined by hU increased the water level behind the structure owing to the elongated flow path. For the specifications in which SU ≥ 1.8Hi, the water level behind the structure decreased by over 72%, and the transmission coefficient (KT) decreased by over 10%.
3.4 Optimal Cross-Section of Lower-Level Drainage Channel Deign
Finally, the optimal cross-section was examined to enhance the efficiency of the submerged breakwater through the lower-level drainage channel height (hL) and size (SL). The hydraulic characteristics and the water level behind the structure were numerically compared by applying the specifications that can be considered in NWT from the cross-section of the breakwater. The results are listed in Table 5. In addition, the optimal cross-section was examined based on the cross-section of the verified hydraulic model experiment. The reference values are expressed in bold letters. As the lower-level drainage channel size (SL) increased from 3 to 19 cm, the lower-level drainage channel height (hL) varied from 18 to 3 cm. Here, because the lower-level drainage channel has no impact according to the size and height, unlike the upper-level drainage channel, the other specifications were identical to those of the reference cross-section.
From Table 5, SL had no significant impact on wave control effects, although variations in cross-sectional specifications similar to those for the aforementioned upper-level drainage channel were considered. However, it effectively reduced the water level behind the structure. hL had no significant impact on wave control and the water level behind the structure. When SL = Hi, the water level behind the structure decreased by approximately 36%. As SL increased, the water level behind the structure decreased by up to approximately 70%.
4. Conclusions
In this study, a numerical model experiment was performed to enhance the efficiency of a submerged breakwater for reducing the water level behind the structure. The results were compared with those of previous hydraulic model experiments to ensure the effectiveness and validity of the constructed numerical wave tank (NWT). To examine the optimal cross-section for strengthening the function of the submerged breakwater, the hydraulic characteristics and the water level behind the structure were compared considering various cross-sectional shapes. The hydraulic characteristics were calculated from the free surface data using the incident and reflected wave separation method. Meanwhile, the water level behind the structure was compared by nondimensionalizing the averaged free surface elevation with the incident wave height. The key findings regarding cross-sectional shapes examined in this study are as follows:
The overall hydraulic characteristics were reproduced effectively. This is based on the fact that the differences between experimental and calculated values for KR, KT, and KD were less than ±10%. The agreement between these in terms of the water level behind the structure was approximately 90%. This verifies the effectiveness and validity of the NWT used.
With regard to the first factor in enhancing the efficiency of the submerged breakwater, an increase in crest width (B) and a decrease in crest depth (R) were effective for wave control. Owing to the wave-breaking effect caused by the ratio of the crest depth to the incident wave height (R / Hi), the water level behind the structure increased as the crest depth (R) decreased. However, it tended to decrease gradually again under the influence of the slit and drainage channels.
Second, an increase in slit width (BS) was highly effective in reducing the water level behind the structure. As the slit distance (DS) decreased, it effectively reduced the water level behind the structure. However, the wave control efficiency was similar or marginally lower.
Third, an increase in upper-level drainage channel size (SU) tended to improve wave control and the efficiency of reducing the water level behind the structure. Meanwhile, the impact of the upper-level drainage channel height (hU) on these was insignificant.
Fourth, an increase in lower-level drainage channel size (SL) was effective in reducing the water level behind the structure. However, it had no significant impact on the wave control effects.
The optimal cross-section of the submerged breakwater was determined to be for a case in which SU ≥ 1.8Hi. In this case, the water level behind the structure decreased by over 72%, and wave control improved by over 10%.
The above numerical analysis results indicate that the submerged breakwater cross-sectional shape examined in this study can appropriately regulate the increase in water level behind the structure. Thus, it is likely to be efficient in controlling the rip current around openings that may occur in sea areas where multiple submerged breakwaters are installed. In the future, the correlation between the rip current around openings and a reduction in the water level behind the structure would be examined quantitatively using three-dimensional (3D) numerical models for coastal areas where multiple submerged breakwaters for reducing the water level behind the structure are installed. In addition, numerical examinations (e.g., a reduction in the run-up of waves behind the structure) considering submerged breakwaters capable of controlling the mean water level would be required for protection against coastal disasters that occur frequently owing to the climate crisis.
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