# Influence of Wave-Absorbing Chamber Width on the Wave Attenuation Performance of Perforated Caisson Sitting on Rubble-Mound Foundation

Authors: Peihong Zhao; Dapeng Sun; Hao Wu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

A Jarlan-type perforated caisson consisted of a perforated front wall, a solid rear wall, and a wave-absorbing chamber between them. The wave-absorbing chamber was the main feature of the perforated caisson, and its width had a great effect on wave attenuation performance. In this study, a larger range of the wave-absorbing chamber width was observed in model experiments to investigate the effect on wave attenuation performance including the reflection coefficients and the horizontal wave forces of a perforated caisson sitting on a rubble-mound foundation. A resistance-type porosity numerical model based on the volume-averaged Reynolds-averaged Navier–Stokes (VARANS) equations was validated by comparing the present results with those of previously reported and present experiments. The validated numerical model was then used for extended research. It was found that the reflection coefficients, the total horizontal wave force, and its components all tended to oscillate in a decrease ⟶ increase ⟶ decrease manner with increasing the wave-absorbing chamber width. The reflection coefficients and wave forces acting on both sides of the perforated front wall were found to be synchronized regardless of perforation ratio or the rubble-mound foundation height.


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## Body

## 1. Introduction

Perforated caissons were first proposed by Jarlan [1] in the 1960s. Since then, many researchers studied this coastal structure in analytical, physical, and numerical ways [2–8]. Some other new types of caisson were also proposed and studied in recent years [9, 10]. A Jarlan-type perforated caisson consisted of a perforated front wall, a solid rear wall, and a wave-absorbing chamber between them. When the incident wave overlapped with the reflected waves from the perforated front wall and the solid rear wall, the wave energy can be greatly reduced. Compared with the traditional caisson, a perforated caisson reduced reflection coefficients and the wave forces. In addition, a perforated caisson had the advantages of easy construction and a wide range of applications, making it a key structure in coastal and port engineering.Chen et al. [11], Ma et al. [12], and Jiang et al. [13, 14] carried out physical model experiments to study the irregular wave interaction with a perforated caisson breakwater on an impermeable and a permeable seabed. In those experiments, the relative wave-absorbing chamber width was between 0.064 and 0.208. It was pointed out that both the reflection coefficients and the horizontal wave forces acting on a perforated caisson were both correlated quadratically with the wave-absorbing chamber width, but this conclusion had not been confirmed for a perforated caisson sitting on a rubble-mound foundation. Xing [15] studied the irregular wave interaction with a perforated caisson breakwater on a rubble-mound foundation of three different heights, with the wave-absorbing chamber width between 0.064 and 0.208. Within this range, both the reflection coefficients and the horizontal wave forces acting on a perforated caisson were also correlated quadratically with the wave-absorbing chamber width.So far, it is generally believed that the wave-absorbing chamber width is one of the key factors affecting the wave attenuation performance of a perforated caisson sitting on a rubble-mound foundation. However, when the wave-absorbing chamber width expanded to a larger range, can the changes in reflection coefficients and horizontal wave forces still be described by a quadratic function? Most experimental studies on perforated caissons limit the wave-absorbing width to a small range, so this problem was not solved. Liu et al. [16] attempted to analytically calculate the reflection coefficients of a perforated caisson based on potential flow theory. In their research, the perforated caisson was bedded on an impermeable step, which differs from a rubble-mound foundation used in engineering practice; therefore, the applicability of their analytical solution had great limitations.Accordingly, in this study, by giving a larger range of the wave-absorbing chamber width, model experiments were carried out to study the effect of the wave-absorbing chamber width on the attenuation performance of a perforated caisson sitting on a rubble-mound foundation. Meanwhile, a resistance-type porosity numerical model based on the volume-averaged Reynolds-averaged Navier–Stokes (VARANS) equations was validated by comparing the present results with those of previously reported and present experiment. The validated numerical model was then used for extended research.


## 2. Numerical Model

In the present research, the resistance-type porosity model proposed by Zhao [17] was adopted. This numerical model used the VARANS equations to uniformly describe the flow inside and outside the porous structure, as well as a volume-averaged k−ε model to simulate the turbulence effect. In addition, the three-step finite element method (FEM) was applied to solve the VARANS equations numerically, and the computational Lagrangian–Eulerian advection remap-volume of fluid (CLEAR-VOF) method was applied to capture the free water surface. In the following sections, the governing equations, calibration of the resistance coefficients, and validation process were explained.


### 2.1. VARANS Equations

Figure1 shows the definition of a porous medium. The surface S for generating the averaging volume V included a solid phase and a fluid phase. S was defined by a circle with the radius r0. The actual volume of the fluid phase Vf varied with the porous medium based on the position of the averaging volume, while the total volume V was constant. The macroscopic length scale L and the pore length scale l were also shown in this figure.Figure 1
Definition of volume average areas and length scales.A volume averaging procedure was applied to convert Reynolds-averaged Navier–Stokes (RANS) equations to VARANS equations, and the length scale constraint was given asl << r0 << L. The superficial volume average denoted by 〈 〉 was defined as the average over the entire volume. The intrinsic average denoted by 〈 〉f was defined as the average over the fluid volume. The superficial and intrinsic volume averages of a scalar, vector, or tensor denoted by B were defined as follows:(1)B=1V∫VfBdV,Bf=1Vf∫VfBdV.The relation between these two averages was as follows:(2)B=nBf,where n was the porosity given by(3)n=VfV.In order to obtain VARANS equations, Navier–Stokes equations were ensemble averaged by applying a time-average operator to obtain the Reynolds-averaged Navier–Stokes (RANS) equations. Then, a volume-average operator was applied to the RANS equations to obtain volume-averaged Reynolds-averaged Navier–Stokes (VARANS) equations:(4)∂u¯i∂xi=0,(5)∂u¯i∂t+u¯j1+cm∂∂xju¯in=−n1+cm∂∂xip¯fρ+n1+cm∂∂xjν+νt∂∂xju¯in+∂∂xiu¯jn+ngi1+cm−fi1+cm,where fi along with the inertia term was the extended Darcy–Forchheimer resistance term Fi defined as follows:(6)nFi=fi+cm∂∂tu¯i=au¯i+bu¯ui+cm∂∂tu¯i.The first term on the right-hand side of equation (6) was the linear resistance term, and the second term was the quadratic resistance term. a and b were experimental coefficients that need to be determined in advance. The last term was the inertia term. cm = 0.34(1-n)/n was the added mass coefficient representing the transient interaction between grain and water. It was obvious that n = 1 and cm = 0 outside the porous media, and then VARANS equations automatically changed back to RANS equations.In equation (5), νt=Cμk2/ε was the volume-averaged eddy viscosity, and the balance equations of k and ε were obtained by taking the volume averaging over the standard k and ε equations which turn to the following:(7)∂∂tk+u¯j∂∂xjkn=2νtnSijfSijf−ε+n∂∂xjν+νtnσk∂∂xjkn+nε∞,(8)∂∂tε+u¯j∂∂xjεn=2νtnCε1εkSijfSijf+Cε2ε2k+n∂∂xjν+νtnσk∂∂xjεn+nCε2ε∞2k∞,where(9)Sijf=12∂∂xju¯in+∂∂xiu¯jn.In equations (7) and (8), σk=1.00, σε=1.30, Cε1=1.44, and Cε2=1.92 as Nakayama and Kuwahara [18] (1999) suggested. k∞ and ε∞ were additional sources of turbulence due to the presence of porous materials. They represented small-scale turbulence that was smaller than the averaging volume. Nakayama and Kuwahara [18] also proposed closure schemes for small-scale turbulence based on the numerical simulation of a fluid flowing through a series of square rods:(10)ε∞=39.01−n5/2nu¯3d50,k∞=3.71−nnu¯2.


### 2.2. Calibration Methods for Experimental Coefficients in VARANS Equations

According to Forchheimer’s research, the Darcy–Forchheimer resistance coefficientsa and b occurred in equation (6) were related to the permeability of the porous medium. They were expressed as a=μ/ρK and b=CF/K, respectively, where μ was the coefficient of kinematic viscosity of the fluid, K was the permeability of the porous medium, which could be written as K=n3d502/α1−n2, and CF was an unknown coefficient, which could be written as CF=β/αn3. Thus, a and b could be expressed as a=αμ1−n2/ρn3d502 and b=β1−n/n3d50 equivalently, where d50 was the mean diameter (or grain diameter if the porous medium was homogeneous) of the porous medium and both α and β were coefficients related to the microscopic geometry of the porous medium. When numerically simulating a specific experimental model, the mean diameter d50 and porosity n could usually be measured. However, α and β were unknown, and they had to be determined through certain calibration procedures.It should be noted that research studies gave different values depending on how the coefficients were determined or calibrated, how the resistance coefficientsa and b were formulated, and even how the numerical model handled turbulence effects. Zhao [17] proposed a calibration method, that was, when using different values of α and β, the wave damping rate of wave motion over porous seabed could be regarded as an index to evaluate the accuracy of numerical simulation. Although the wave motion over the porous seabed was a dynamic process, after a period of time, the entire damping system became stable, and wave heights along the distance remained unchanged. The total error could be expressed as follows:(11)ε=kiexp−kinumkiexp,where kiexp was the wave damping rate measured in experiments; kinum was the wave damping rate calculated by numerical modeling; and ε was the relative error. It was seen that the wave damping rate was the only parameter required in the error estimation procedure, which made this calibration method proposed by Zhao [17] easier to implement. In this paper, the same calibration method was adopted and the coefficients α and β were determined to be 100 and 2.0, respectively.


### 2.3. Validation Using the Existed Model Experiments

The numerical model proposed by Zhao [17] could be successfully used to simulate the porous media-fluid coupled flow. By comparing the simulation results with the existing model experimental results, the applicability and accuracy of the simulation of the irregular wave interaction with a perforated caisson sitting on a rubble-mound foundation were validated.Xing et al. [15] studied the irregular wave interaction with a perforated caisson breakwater on a rubble-mound foundation. In their model experiments, the widths of the wave chamber bc were set to 0.15, 0.20, and 0.30 m. The front wall of the caisson was partially perforated from 0.2 m below the water surface to the top of the caisson. The porosity μ of the perforated front wall was set to 20% and 40%. The water depth d during the test was maintained at 0.4 m. The heights of the rubble-mound foundation hm were set to 0.10 m, 0.15 m, and 0.20 m. The berm width of the foundation W was 0.25 m. The core of the rubble-mound foundation was the same as the rock-fill core in the wave-absorbing chamber. The grain diameter d50 was about 1.1 cm, and the porosity of the core n was about 0.4. The significant wave height Hs was 0.053 m. Because the water depth d during the test was maintained at 0.4 m, the significant wave lengths Ls were only related to the wave periods. They were 2.35 m, 1.82 m, and 1.44 m, respectively, when the significant wave periods Ts were 1.38 s, 1.15 s, and 0.99 s. After combining all these varied factors, a total of 54 cases were carried out in their work.As shown in Figure2, a 2-D numerical wave flume was developed in this study. The length of the flume is 15 m. The left boundary represents the wave-maker, and the boundary condition is the velocity inlet. The rubble-mound foundation and the perforated caisson sitting on it were located in the middle section of the wave flume. The perforated front wall was 7.7 m away from the left boundary, which made the left side of the perforated wall the seaward side and the right side the leeward side.Figure 2
2-D numerical wave flume and the perforated caisson sitting on the rubble-mound foundation.In order to eliminate the effect of re-reflect waves between the structure and the wave generator, a time-domain method for separating the incident waves and the reflected waves (SIRW method) by Frigaard and Christensen [19, 20] was adopted in the present research. It was proved that this method could be used in investigating the irregular wave interaction with a perforated caisson by Tang [21]. In addition, a sponge layer was located on the right side of the wave flume, and the flow velocity in it decayed exponentially.An irregular wave train could be treated as a stationary stochastic process. It was assumed to be a linear superposition of large numbers of monochromatic wave components with different wave heights, wave angular frequencies, and random initial phases. The incident wave spectrums of experiments and numerical modeling were both the modified JONSWAP spectrum (Goda, 1999 [22]). So, by neglecting the interaction of different component of waves, the reflection coefficient of each component could be calculated by estimation of incident and reflected waves [23]. Once the reflection coefficient of each component was determined, the reflected spectral density was obtained by(12)Srωj=KrjSiωj,where Krj was the reflection coefficient of the jth component and Sr(ωr) and Sr(ωj) were the spectral densities of the reflected and incident waves, respectively. For irregular waves, the average reflection coefficient Kr can be calculated by(13)K¯r=m0,rm0,i,where m0,r and m0,I are the zero-order moments of the reflected and incident wave spectra, respectively, and can be calculated by(14)m0,r=∫0∞Srωdω,m0,i=∫0∞Siωdω.For consistency, the average reflection coefficientKr was considered as the reflection coefficient of irregular waves and just be written as Kr in the following sections.In order to accurately calculate the reflection coefficient of each component of waves, five wave height gauges (numbered 1# to 5#) were arranged 2.8 m, 3.1 m, 3.6 m, 3.91 m, and 4.17 m from the seaward face of the caisson, respectively. Appropriate combination of wave gauges was to be selected according to different wave length of each component of waves.The accuracy of the numerical model could be illustrated by comparing the time series data of wave elevation or the frequency spectrum based on numerical modeling and experimental results. For instance, Figure3 shows a target spectrum of the irregular wave in experiments and numerical modeling for Ls = 1.82 m; Figure 4 shows both the numerical and experimental time series data of wave elevation and the corresponding frequency spectrum measured by the 1# wave height gauge for hm = 0.15 m; bc = 0.2 m; μ = 0.4; and Ls = 1.82 m. It was seen that the numerical results were in good agreement with the experimental results. Only in the high frequency region, some deviation occurred.Figure 3
1# target spectrum of irregular wave and the numerical results.Figure 4
1# time series data of wave elevation and frequency spectrum.In Figure5, the reflection coefficients Kr calculated based on the numerical and experimental results were compared, where the horizontal axis represented the relative wave-absorbing chamber width bc/Ls. It was seen that in Xing’s model experiments, the relative wave-absorbing chamber width was about 0.064–0.208, and within this range, Kr varied nonlinearly. In some cases, as shown in Figure 5(f), the minimum of Kr was found. Therefore, a quadratic function was used to describe the correlation between Kr and the relative wave-absorbing chamber width. The curves for the three hm values were calculated from the formula proposed by Xing et al. [15].Figure 5
Comparison of the reflection coefficientKr: ●, experimental results as hm = 0.10 m; ○, numerical results as hm = 0.10 m; (—), calculation results by Xing as hm = 0.10 m; ■, experimental results as hm = 0.15 m; □, numerical results as hm = 0.15 m; (--), calculation results by Xing as hm = 0.15 m; ▲, experimental results as h hm = 0.20 m; △, numerical results as hm = 0.20 (m) (-.), calculation results by Xing as hm = 0.20 m.
(a)(b)(c)(d)(e)(f)In general, Figure5 indicated a reasonable agreement between the numerical and experimental values of Kr in all 54 cases. The numerical model used in the present research was applicable and sufficiently accurate to simulate the wave interaction with a perforated caisson sitting on a rubble-mound foundation.


## 2.1. VARANS Equations

Figure1 shows the definition of a porous medium. The surface S for generating the averaging volume V included a solid phase and a fluid phase. S was defined by a circle with the radius r0. The actual volume of the fluid phase Vf varied with the porous medium based on the position of the averaging volume, while the total volume V was constant. The macroscopic length scale L and the pore length scale l were also shown in this figure.Figure 1
Definition of volume average areas and length scales.A volume averaging procedure was applied to convert Reynolds-averaged Navier–Stokes (RANS) equations to VARANS equations, and the length scale constraint was given asl << r0 << L. The superficial volume average denoted by 〈 〉 was defined as the average over the entire volume. The intrinsic average denoted by 〈 〉f was defined as the average over the fluid volume. The superficial and intrinsic volume averages of a scalar, vector, or tensor denoted by B were defined as follows:(1)B=1V∫VfBdV,Bf=1Vf∫VfBdV.The relation between these two averages was as follows:(2)B=nBf,where n was the porosity given by(3)n=VfV.In order to obtain VARANS equations, Navier–Stokes equations were ensemble averaged by applying a time-average operator to obtain the Reynolds-averaged Navier–Stokes (RANS) equations. Then, a volume-average operator was applied to the RANS equations to obtain volume-averaged Reynolds-averaged Navier–Stokes (VARANS) equations:(4)∂u¯i∂xi=0,(5)∂u¯i∂t+u¯j1+cm∂∂xju¯in=−n1+cm∂∂xip¯fρ+n1+cm∂∂xjν+νt∂∂xju¯in+∂∂xiu¯jn+ngi1+cm−fi1+cm,where fi along with the inertia term was the extended Darcy–Forchheimer resistance term Fi defined as follows:(6)nFi=fi+cm∂∂tu¯i=au¯i+bu¯ui+cm∂∂tu¯i.The first term on the right-hand side of equation (6) was the linear resistance term, and the second term was the quadratic resistance term. a and b were experimental coefficients that need to be determined in advance. The last term was the inertia term. cm = 0.34(1-n)/n was the added mass coefficient representing the transient interaction between grain and water. It was obvious that n = 1 and cm = 0 outside the porous media, and then VARANS equations automatically changed back to RANS equations.In equation (5), νt=Cμk2/ε was the volume-averaged eddy viscosity, and the balance equations of k and ε were obtained by taking the volume averaging over the standard k and ε equations which turn to the following:(7)∂∂tk+u¯j∂∂xjkn=2νtnSijfSijf−ε+n∂∂xjν+νtnσk∂∂xjkn+nε∞,(8)∂∂tε+u¯j∂∂xjεn=2νtnCε1εkSijfSijf+Cε2ε2k+n∂∂xjν+νtnσk∂∂xjεn+nCε2ε∞2k∞,where(9)Sijf=12∂∂xju¯in+∂∂xiu¯jn.In equations (7) and (8), σk=1.00, σε=1.30, Cε1=1.44, and Cε2=1.92 as Nakayama and Kuwahara [18] (1999) suggested. k∞ and ε∞ were additional sources of turbulence due to the presence of porous materials. They represented small-scale turbulence that was smaller than the averaging volume. Nakayama and Kuwahara [18] also proposed closure schemes for small-scale turbulence based on the numerical simulation of a fluid flowing through a series of square rods:(10)ε∞=39.01−n5/2nu¯3d50,k∞=3.71−nnu¯2.


## 2.2. Calibration Methods for Experimental Coefficients in VARANS Equations

According to Forchheimer’s research, the Darcy–Forchheimer resistance coefficientsa and b occurred in equation (6) were related to the permeability of the porous medium. They were expressed as a=μ/ρK and b=CF/K, respectively, where μ was the coefficient of kinematic viscosity of the fluid, K was the permeability of the porous medium, which could be written as K=n3d502/α1−n2, and CF was an unknown coefficient, which could be written as CF=β/αn3. Thus, a and b could be expressed as a=αμ1−n2/ρn3d502 and b=β1−n/n3d50 equivalently, where d50 was the mean diameter (or grain diameter if the porous medium was homogeneous) of the porous medium and both α and β were coefficients related to the microscopic geometry of the porous medium. When numerically simulating a specific experimental model, the mean diameter d50 and porosity n could usually be measured. However, α and β were unknown, and they had to be determined through certain calibration procedures.It should be noted that research studies gave different values depending on how the coefficients were determined or calibrated, how the resistance coefficientsa and b were formulated, and even how the numerical model handled turbulence effects. Zhao [17] proposed a calibration method, that was, when using different values of α and β, the wave damping rate of wave motion over porous seabed could be regarded as an index to evaluate the accuracy of numerical simulation. Although the wave motion over the porous seabed was a dynamic process, after a period of time, the entire damping system became stable, and wave heights along the distance remained unchanged. The total error could be expressed as follows:(11)ε=kiexp−kinumkiexp,where kiexp was the wave damping rate measured in experiments; kinum was the wave damping rate calculated by numerical modeling; and ε was the relative error. It was seen that the wave damping rate was the only parameter required in the error estimation procedure, which made this calibration method proposed by Zhao [17] easier to implement. In this paper, the same calibration method was adopted and the coefficients α and β were determined to be 100 and 2.0, respectively.


## 2.3. Validation Using the Existed Model Experiments

The numerical model proposed by Zhao [17] could be successfully used to simulate the porous media-fluid coupled flow. By comparing the simulation results with the existing model experimental results, the applicability and accuracy of the simulation of the irregular wave interaction with a perforated caisson sitting on a rubble-mound foundation were validated.Xing et al. [15] studied the irregular wave interaction with a perforated caisson breakwater on a rubble-mound foundation. In their model experiments, the widths of the wave chamber bc were set to 0.15, 0.20, and 0.30 m. The front wall of the caisson was partially perforated from 0.2 m below the water surface to the top of the caisson. The porosity μ of the perforated front wall was set to 20% and 40%. The water depth d during the test was maintained at 0.4 m. The heights of the rubble-mound foundation hm were set to 0.10 m, 0.15 m, and 0.20 m. The berm width of the foundation W was 0.25 m. The core of the rubble-mound foundation was the same as the rock-fill core in the wave-absorbing chamber. The grain diameter d50 was about 1.1 cm, and the porosity of the core n was about 0.4. The significant wave height Hs was 0.053 m. Because the water depth d during the test was maintained at 0.4 m, the significant wave lengths Ls were only related to the wave periods. They were 2.35 m, 1.82 m, and 1.44 m, respectively, when the significant wave periods Ts were 1.38 s, 1.15 s, and 0.99 s. After combining all these varied factors, a total of 54 cases were carried out in their work.As shown in Figure2, a 2-D numerical wave flume was developed in this study. The length of the flume is 15 m. The left boundary represents the wave-maker, and the boundary condition is the velocity inlet. The rubble-mound foundation and the perforated caisson sitting on it were located in the middle section of the wave flume. The perforated front wall was 7.7 m away from the left boundary, which made the left side of the perforated wall the seaward side and the right side the leeward side.Figure 2
2-D numerical wave flume and the perforated caisson sitting on the rubble-mound foundation.In order to eliminate the effect of re-reflect waves between the structure and the wave generator, a time-domain method for separating the incident waves and the reflected waves (SIRW method) by Frigaard and Christensen [19, 20] was adopted in the present research. It was proved that this method could be used in investigating the irregular wave interaction with a perforated caisson by Tang [21]. In addition, a sponge layer was located on the right side of the wave flume, and the flow velocity in it decayed exponentially.An irregular wave train could be treated as a stationary stochastic process. It was assumed to be a linear superposition of large numbers of monochromatic wave components with different wave heights, wave angular frequencies, and random initial phases. The incident wave spectrums of experiments and numerical modeling were both the modified JONSWAP spectrum (Goda, 1999 [22]). So, by neglecting the interaction of different component of waves, the reflection coefficient of each component could be calculated by estimation of incident and reflected waves [23]. Once the reflection coefficient of each component was determined, the reflected spectral density was obtained by(12)Srωj=KrjSiωj,where Krj was the reflection coefficient of the jth component and Sr(ωr) and Sr(ωj) were the spectral densities of the reflected and incident waves, respectively. For irregular waves, the average reflection coefficient Kr can be calculated by(13)K¯r=m0,rm0,i,where m0,r and m0,I are the zero-order moments of the reflected and incident wave spectra, respectively, and can be calculated by(14)m0,r=∫0∞Srωdω,m0,i=∫0∞Siωdω.For consistency, the average reflection coefficientKr was considered as the reflection coefficient of irregular waves and just be written as Kr in the following sections.In order to accurately calculate the reflection coefficient of each component of waves, five wave height gauges (numbered 1# to 5#) were arranged 2.8 m, 3.1 m, 3.6 m, 3.91 m, and 4.17 m from the seaward face of the caisson, respectively. Appropriate combination of wave gauges was to be selected according to different wave length of each component of waves.The accuracy of the numerical model could be illustrated by comparing the time series data of wave elevation or the frequency spectrum based on numerical modeling and experimental results. For instance, Figure3 shows a target spectrum of the irregular wave in experiments and numerical modeling for Ls = 1.82 m; Figure 4 shows both the numerical and experimental time series data of wave elevation and the corresponding frequency spectrum measured by the 1# wave height gauge for hm = 0.15 m; bc = 0.2 m; μ = 0.4; and Ls = 1.82 m. It was seen that the numerical results were in good agreement with the experimental results. Only in the high frequency region, some deviation occurred.Figure 3
1# target spectrum of irregular wave and the numerical results.Figure 4
1# time series data of wave elevation and frequency spectrum.In Figure5, the reflection coefficients Kr calculated based on the numerical and experimental results were compared, where the horizontal axis represented the relative wave-absorbing chamber width bc/Ls. It was seen that in Xing’s model experiments, the relative wave-absorbing chamber width was about 0.064–0.208, and within this range, Kr varied nonlinearly. In some cases, as shown in Figure 5(f), the minimum of Kr was found. Therefore, a quadratic function was used to describe the correlation between Kr and the relative wave-absorbing chamber width. The curves for the three hm values were calculated from the formula proposed by Xing et al. [15].Figure 5
Comparison of the reflection coefficientKr: ●, experimental results as hm = 0.10 m; ○, numerical results as hm = 0.10 m; (—), calculation results by Xing as hm = 0.10 m; ■, experimental results as hm = 0.15 m; □, numerical results as hm = 0.15 m; (--), calculation results by Xing as hm = 0.15 m; ▲, experimental results as h hm = 0.20 m; △, numerical results as hm = 0.20 (m) (-.), calculation results by Xing as hm = 0.20 m.
(a)(b)(c)(d)(e)(f)In general, Figure5 indicated a reasonable agreement between the numerical and experimental values of Kr in all 54 cases. The numerical model used in the present research was applicable and sufficiently accurate to simulate the wave interaction with a perforated caisson sitting on a rubble-mound foundation.


## 3. Effect of the Wave-Absorbing Chamber Width on the Wave Attenuation Performance

As mentioned in the previous section, since the range of the wave-absorbing chamber width was very small, within this range, it was reasonable to use a quadratic function to describe the variation of reflection coefficients. However, when the wave-absorbing chamber width continued to increase, what would be the behavior of the reflection coefficient and even the horizontal wave force acting on the perforated caisson sitting on the rubble-mound foundation? In this paper, model experiments and numerical simulations were used to answer this question.


### 3.1. Model Experiments and Numerical Simulations

The model experiments were carried out at the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology. The wave flume was 56 m in length and 0.7 m in width. A hydraulic servo push-plate type wave generator was installed on one side of the flume. An antislope device was installed on the other side to dissipate wave energy. The setup of the wave height gauges was the same as in Xing et al. [15].The roofless caisson was made of PMMA with a thickness of 1 cm. Four rows of uniformly distributed rectangular holes were cut out from 0.2 m below the water surface to the top of the caisson, which led to a perforation ratioμ of 40%. The height of the rubble-mound foundation hm was set to 0.20 m. In these experiments, the range of wave-absorbing chamber width bc was very large, measuring 0.15, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, and 0.80 m. The perforated caisson model is shown in Figure 6.Figure 6
The perforated caisson used in the experiments.In order to calculate the wave forces on the caisson, pressure gauges were installed on both sides of the perforated front wall and the solid rear wall, as shown in Figure7. Once the time series data of pressure were collected, the horizontal wave forces acting on each wall and the total horizontal wave forces acting on the entire structure could be calculated by a simple integration algorithm.Figure 7
Perforated front wall and solid rear wall of the caisson: ●, pressure gauges on the seaward side of the front perforated wall and the solid rear wall; ○, pressure gauges on the leeward side of the perforated front wall.In the whole experiment, the significant wave heightHs was 0.053 m, and other geometric parameters and wave experimental conditions are shown in Table 1. The numerical model validated in the previous section was adopted here to simulate the model experiment cases.Table 1
Experimental conditions and geometric parameters.
Significant wave periodTs (s)1.38, 1.15, 0.99Chamber widthbc (m)bc0.15, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80Foundation heighthm (m)0.20Relative foundation heighthm/Ls hm/L1/30.043∼0.139Relative chamber widthbc/Ls bc/L1/30.064∼0.590Relative water depthd/Ls d/L1/30.17∼0.278


#### 3.1.1. Reflection Coefficients

Figure8 shows the obtained numerical and experimental reflection coefficients as the wave-absorbing chamber width varied from 0.15 m to 0.80 m. In Figures 8(a)–8(c), three different groups of cases with the significant wave lengths Ls = 2.35 m, 1.82 m, and 1.44 m were shown, respectively. It was seen that the numerical results were in good agreement with the experimental results. Results showed that resonant conditions occurred for various bc/Ls values introducing an oscillation in a decrease ⟶ increase ⟶ decrease manner of the reflection coefficients.Figure 8
Experimental and numerical reflection coefficients.
(a)(b)(c)This indicated that for incident waves with the same periods, as the wave-absorbing chamber width increased, the phase differences between the incident wave, the wave reflected from the perforated front wall, and the solid rear wall oscillated. This oscillation behavior is most clearly seen in Figure8(c) in which the significant wave length Ls was the shortest, and the range of the relative wave-absorbing chamber width bc/Ls was therefore largest. According to Figure 8, for a greater range of the wave-absorbing chamber width changed, a quadratic function could no longer describe the correlation between the relative wave-absorbing chamber width and reflection coefficients; therefore, the formula of Xing [15] did not apply in such cases.


#### 3.1.2. Horizontal Wave Forces

Another important subject for studying the wave attenuation performance of a perforated caisson sitting on a rubble-mound foundation was the horizontal wave force. This subject included the horizontal wave forces acting on each wall of a perforated caisson and the total horizontal wave force acting on the entire structure. For a typical Jarlan-type perforated caisson, the total horizontal wave forceFh could be decomposed into three different components, as shown in Figure 9(a). The total horizontal wave force could be expressed as Fh = Fh1 − Fh2 + Fh3, where Fh1 represented the wave force acting on the seaward side of the perforated front wall, Fh2 was the force acting on the leeward side of the perforated front wall, and Fh3 was the force acting on the solid rear wall.Figure 9
Wave forces acting on (a) perforated and (b) traditional caisson sitting on a rubble-mound foundation.
(a)(b)Under the same conditions,Fh0 was the horizontal wave force acting on a traditional caisson, as shown in Figure 9(b).To study the horizontal wave forces acting on each part of the perforated caisson, dimensionless ratio parametersFh1/Fh0, Fh2/Fh0, and Fh3/Fh0 were introduced. The dimensionless total horizontal wave force could be expressed by Fh/Fh0. Since Fh2 was opposite in direction to Fh0, Fh2/Fh0 was always negative. Figure 10 shows the numerical and experimental horizontal wave forces as the wave-absorbing chamber width varied from 0.15 m to 0.80 m. In Figures 10(a)–10(c), three different groups of cases with the significant wave lengths Ls = 2.35 m, 1.82 m, and 1.44 m were shown, respectively. It could be seen that the numerical results were in good agreement with the experimental results. Based on the numerical results, spline curves are used to smooth the curves in Figure 10 and in other graphs in the following sections.Figure 10
Horizontal wave forces acting on each part of the perforated caisson with irregular waves: black ○, numerical results ofFh1/Fh0; black ●, experimental results of Fh1/Fh0; black □, numerical results of Fh2/Fh0; black ■, experimental results of Fh2/Fh0; black △, numerical results of Fh3/Fh0; black ▲, experimental results of Fh3/Fh0.
(a)(b)(c)As the wave-absorbing chamber width increased over a large range, results showed that resonant conditions occurred for variousbc/Ls values introducing an oscillation in a decrease ⟶ increase ⟶ decrease manner of all Fh1/Fh0, Fh2/Fh0, and Fh3/Fh0. The oscillation of curves of Fh1/Fh0 and Fh2/Fh0 indicated that for incident waves with the same periods, as the wave-absorbing chamber width increased, the phase differences between the incident wave, the wave reflected from the perforated front wall, and the solid rear wall oscillated. The oscillation of Fh3/Fh0 curve indicated that for incident waves with the same periods, as the wave-absorbing chamber width increased, the phase differences between the incident wave, the wave reflected from the solid rear wall, and the re-reflected wave from the leeward side of the perforated front wall oscillated. The curve of Fh2/Fh0 showed the most obvious oscillation, and the oscillating amplitude of the Fh1/Fh0 curve was smaller than that of Fh2/Fh0. This indicated that the wave forces acting on the seaward and the leeward side of the perforated front wall of the caisson varied asymmetrically with the increase in the wave-absorbing chamber width.It must be noted that synchronization could be found when investigating the curves ofFh1/Fh0, Fh2/Fh0, and Kr. This meant that when Kr reached its maximum or minimum value, Fh1/Fh0 and Fh2/Fh0 also simultaneously reached their maximum or minimum values. On the other hand, the Fh3/Fh0 curve showed some hysteresis.Figure11 shows the numerical and experimental values of the total horizontal wave force as the wave-absorbing chamber width varied from 0.15 m to 0.80 m. In Figures 11(a)–11(c), three different groups of cases with the significant wave lengths Ls = 2.35 m, 1.82 m, and 1.44 m were shown, respectively. Some differences can be seen but generally the numerical results were consistent with the experimental results.Figure 11
Total horizontal wave force acting on the perforated caisson with irregular waves: black ○, numerical results ofFh/Fh0.; black ●, experimental results of Fh/Fh0.
(a)(b)(c)Results showed that resonant conditions occurred for variousbc/Ls values introducing an oscillation in a decrease ⟶ increase ⟶ decrease manner of Fh/Fh0. Compared with the curves of Fh1/Fh0 and Fh2/Fh0, the Fh/Fh0 curve was more similar to the Fh3/Fh0 curve. This was because, on the one hand, the horizontal wave force acting on the solid rear wall was always larger than that acting on the perforated front wall under experimental conditions. On the other hand, the wave force acting on the seaward side of the perforated front wall was offset by that acting on the leeward side, so the contribution to the total horizontal force was rather small.In general, for all values of reflection coefficientKr, the total horizontal wave force or its components, results showed that resonant conditions occurred for various bc/Ls values introducing an oscillation in a decrease ⟶ increase ⟶ decrease manner. It was clear that their relationships could not be described by a quadratic function.


### 3.2. Extended Research Using the Numerical Model

The experimental cases mentioned in the previous section were limited. In fact, this experiment only included the cases of rubble-mound foundation heighthm = 0.2 m and the perforation ratio μ = 0.4. In other cases, how would the reflection coefficient Kr, the total horizontal wave force, and its components vary as the wave-absorbing chamber width increased? Would there be any similarities or differences? To answering these questions, the numerical model was used for extended research to extrapolate the above results beyond the parameters of the model experiments.Different cases with three rubble-mound foundation heights, two perforation ratios, eight wave-absorbing chamber widths, and three significant wave lengths were considered. It totally included 144 cases after combing all these factors. Table2 showed all the modeling conditions and geometric parameters.Table 2
Numerical modeling conditions and geometric parameters.
Significant wave periodTs (s)1.38, 1.15, 0.99Significant wave lengthLs (m)2.35, 1.82, 1.44Chamber widthbc (m)0.15, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8Foundation heighthm (m)0.10, 0.15, 0.20Perforation ratioμ0.2～0.4Relative foundation heighthm/L1/30.043～0.139Relative chamber widthbc/L1/30.064～0.556Relative water depthd/L1/30.17～0.278


#### 3.2.1. Effect of Wave-Absorbing Chamber Width on Reflection Coefficients

Figure12 shows the obtained numerical and experimental reflection coefficients as the wave-absorbing chamber width varied from 0.15 m to 0.80 m. Three different groups of cases with the significant wave lengths Ls = 2.35 m, 1.82 m, and 1.44 m were shown, respectively. In Figures 12(a)–12(c), the perforation ratio was 0.4. In Figures 12(d)–12(f), the perforated ratio was 0.2.Figure 12
Reflection coefficients for irregular waves: ○, numerical results forhm = 0.10 m; □, numerical results for hm = 0.15 m; △, numerical results for hm = 0.20 m.
(a)(b)(c)(d)(e)(f)Generally, as the wave-absorbing chamber widened, the reflection coefficients tended to oscillate in a decrease ⟶ increase ⟶ decrease manner. This was a common feature regardless of the rubble-mound foundation height or the perforation ratio.From Figures12(a)–12(c), it was found that when the perforated ratio μ = 0.4, the oscillating amplitude became smaller for higher foundations. However, in Figures 12(d)–12(f), this phenomenon was not evident.Comparison between Figures12(a), 12(b), and 12(c) with 12(d), 12(e), and 12(f) in that order showed that as the wave-absorbing chamber width increased, different perforation ratios also resulted in different amplitudes of the oscillating curves.


#### 3.2.2. Effect of Wave-Absorbing Chamber Width on the Horizontal Wave Forces

Figure13 shows the numerical and experimental horizontal wave forces acting on each part of the perforated caisson as the wave-absorbing chamber width varied from 0.15 m to 0.80 m. Three different groups of cases with the significant wave lengths Ls = 2.35 m, 1.82 m, and 1.44 m were shown, respectively. In Figures 13(a)–13(c), the perforated ratio was 0.4, and in Figures 13(d)–13(f), the perforated ratio was 0.2.Figure 13
Horizontal wave forces acting on each part of a perforated caisson for irregular waves: red ○,Fh1/Fh0 for hm = 0.10 m; blue ○, Fh1/Fh0 for hm = 0.15 m; black ○, Fh1/Fh0 for hm = 0.20 m; red □, Fh2/Fh0 for hm = 0.10 m; blue □, Fh2/Fh0 for hm = 0.15 m; black □, Fh2/Fh0 for hm = 0.20 m; red △, Fh3/Fh0 for hm = 0.10 m; blue △, Fh3/Fh0 for hm = 0.15 m; black △, Fh3/Fh0 for hm = 0.20 m.
(a)(b)(c)(d)(e)(f)Generally, as the wave-absorbing chamber width increased,Fh1/Fh0, Fh2/Fh0, and Fh3/Fh0 tended to oscillate in a decrease ⟶ increase ⟶ decrease manner. The Fh2/Fh0 curve oscillated most obviously, and the oscillating amplitude of Fh1/Fh0 was smaller than that of Fh2/Fh0. Synchronization occurred between Fh1/Fh0, Fh2/Fh0, and Kr to some extent that they reached their maxima and minima simultaneously. However, Fh3/Fh0 showed some hysteresis. These features were seen regardless of rubble-mound foundation height or the perforation ratio.For different perforation ratios, it could be found that whenμ = 0.4, the horizontal wave force acting on the seaward side of the perforated front wall was smaller than that for μ = 0.2. Conversely, the horizontal wave force acting on the solid rear wall was greater. This was mainly because the perforation ratio of the front wall determined how much incident wave energy passed into the wave-absorbing chamber and how much was reflected from the front wall. When the perforation ratio was small, less wave energy entered, which meant that the perforated front wall was subject to greater horizontal wave forces. In addition, compared with the case of μ = 0.4, the curves of Fh1/Fh0 and Fh3/Fh0 were flatter near their minima and change more sharply when they began to climb or fall. When μ = 0.2, the curves were less symmetrical than for μ = 0.4.By comparing figures with different rubble-mound foundation heights, it could be found that the rubble-mound foundation height had limited influence on the horizontal wave forces. Whenhm became higher, Fh1/Fh0 gets smaller only on the seaward side of the perforated front wall. This indicated that the horizontal wave forces acting on the area close to the bottom of the caisson were very small. In addition, some deviation was found in the Fh2/Fh0, curves when hm = 0.2 m especially near their maxima.Figure14 shows the numerical and experimental total horizontal wave forces as the wave-absorbing chamber width varied from 0.15 m to 0.80 m. Three different groups of cases with the significant wave lengths Ls = 2.35 m, 1.82 m, and 1.44 m were shown, respectively. In Figures 14(a)–14(c), the perforation ratio was 0.4, and in Figures 14(d)–14(f), the perforation ratio was 0.2.Figure 14
Total horizontal wave force acting on the perforated caisson with irregular waves: black ○,Fh/Fh0 for hm = 0.10 m; black □, Fh/Fh0 for hm = 0.15 m; black △, Fh/Fh0 for hm = 0.20 m.
(a)(b)(c)(d)(e)(f)In Figure14, as the wave-absorbing chamber width increased in a larger range, Fh/Fh0 tended to oscillate in a decrease ⟶ increase ⟶ decrease manner, but the oscillation was weaker than for either Fh1/Fh0 or Fh2/Fh0. The Fh/Fh0 curve showed a certain amount of hysteresis compared with that of Kr. These were the common features regardless of the rubble-mound foundation height or the perforation ratio.In addition, the variation of perforation ratio had almost no effect on theFh/Fh0 curve although the Fh/Fh0 curve appeared to resemble the Fh3/Fh0 curve more closely as the perforation ratio increased. This was mainly due to the greater force on the solid rear wall of the perforated caisson with greater perforation ratio.Comparing the figures for different rubble-mound foundation heights, it could be found that the rubble-mound foundation height had limited influence on the total horizontal wave force. As in the figures forKr, when the perforation ratio μ = 0.4, the oscillation amplitudes became smaller with greater foundation heights, but this phenomenon was no longer evident when μ = 0.2.


## 3.1. Model Experiments and Numerical Simulations

The model experiments were carried out at the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology. The wave flume was 56 m in length and 0.7 m in width. A hydraulic servo push-plate type wave generator was installed on one side of the flume. An antislope device was installed on the other side to dissipate wave energy. The setup of the wave height gauges was the same as in Xing et al. [15].The roofless caisson was made of PMMA with a thickness of 1 cm. Four rows of uniformly distributed rectangular holes were cut out from 0.2 m below the water surface to the top of the caisson, which led to a perforation ratioμ of 40%. The height of the rubble-mound foundation hm was set to 0.20 m. In these experiments, the range of wave-absorbing chamber width bc was very large, measuring 0.15, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, and 0.80 m. The perforated caisson model is shown in Figure 6.Figure 6
The perforated caisson used in the experiments.In order to calculate the wave forces on the caisson, pressure gauges were installed on both sides of the perforated front wall and the solid rear wall, as shown in Figure7. Once the time series data of pressure were collected, the horizontal wave forces acting on each wall and the total horizontal wave forces acting on the entire structure could be calculated by a simple integration algorithm.Figure 7
Perforated front wall and solid rear wall of the caisson: ●, pressure gauges on the seaward side of the front perforated wall and the solid rear wall; ○, pressure gauges on the leeward side of the perforated front wall.In the whole experiment, the significant wave heightHs was 0.053 m, and other geometric parameters and wave experimental conditions are shown in Table 1. The numerical model validated in the previous section was adopted here to simulate the model experiment cases.Table 1
Experimental conditions and geometric parameters.
Significant wave periodTs (s)1.38, 1.15, 0.99Chamber widthbc (m)bc0.15, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80Foundation heighthm (m)0.20Relative foundation heighthm/Ls hm/L1/30.043∼0.139Relative chamber widthbc/Ls bc/L1/30.064∼0.590Relative water depthd/Ls d/L1/30.17∼0.278


### 3.1.1. Reflection Coefficients

Figure8 shows the obtained numerical and experimental reflection coefficients as the wave-absorbing chamber width varied from 0.15 m to 0.80 m. In Figures 8(a)–8(c), three different groups of cases with the significant wave lengths Ls = 2.35 m, 1.82 m, and 1.44 m were shown, respectively. It was seen that the numerical results were in good agreement with the experimental results. Results showed that resonant conditions occurred for various bc/Ls values introducing an oscillation in a decrease ⟶ increase ⟶ decrease manner of the reflection coefficients.Figure 8
Experimental and numerical reflection coefficients.
(a)(b)(c)This indicated that for incident waves with the same periods, as the wave-absorbing chamber width increased, the phase differences between the incident wave, the wave reflected from the perforated front wall, and the solid rear wall oscillated. This oscillation behavior is most clearly seen in Figure8(c) in which the significant wave length Ls was the shortest, and the range of the relative wave-absorbing chamber width bc/Ls was therefore largest. According to Figure 8, for a greater range of the wave-absorbing chamber width changed, a quadratic function could no longer describe the correlation between the relative wave-absorbing chamber width and reflection coefficients; therefore, the formula of Xing [15] did not apply in such cases.


### 3.1.2. Horizontal Wave Forces

Another important subject for studying the wave attenuation performance of a perforated caisson sitting on a rubble-mound foundation was the horizontal wave force. This subject included the horizontal wave forces acting on each wall of a perforated caisson and the total horizontal wave force acting on the entire structure. For a typical Jarlan-type perforated caisson, the total horizontal wave forceFh could be decomposed into three different components, as shown in Figure 9(a). The total horizontal wave force could be expressed as Fh = Fh1 − Fh2 + Fh3, where Fh1 represented the wave force acting on the seaward side of the perforated front wall, Fh2 was the force acting on the leeward side of the perforated front wall, and Fh3 was the force acting on the solid rear wall.Figure 9
Wave forces acting on (a) perforated and (b) traditional caisson sitting on a rubble-mound foundation.
(a)(b)Under the same conditions,Fh0 was the horizontal wave force acting on a traditional caisson, as shown in Figure 9(b).To study the horizontal wave forces acting on each part of the perforated caisson, dimensionless ratio parametersFh1/Fh0, Fh2/Fh0, and Fh3/Fh0 were introduced. The dimensionless total horizontal wave force could be expressed by Fh/Fh0. Since Fh2 was opposite in direction to Fh0, Fh2/Fh0 was always negative. Figure 10 shows the numerical and experimental horizontal wave forces as the wave-absorbing chamber width varied from 0.15 m to 0.80 m. In Figures 10(a)–10(c), three different groups of cases with the significant wave lengths Ls = 2.35 m, 1.82 m, and 1.44 m were shown, respectively. It could be seen that the numerical results were in good agreement with the experimental results. Based on the numerical results, spline curves are used to smooth the curves in Figure 10 and in other graphs in the following sections.Figure 10
Horizontal wave forces acting on each part of the perforated caisson with irregular waves: black ○, numerical results ofFh1/Fh0; black ●, experimental results of Fh1/Fh0; black □, numerical results of Fh2/Fh0; black ■, experimental results of Fh2/Fh0; black △, numerical results of Fh3/Fh0; black ▲, experimental results of Fh3/Fh0.
(a)(b)(c)As the wave-absorbing chamber width increased over a large range, results showed that resonant conditions occurred for variousbc/Ls values introducing an oscillation in a decrease ⟶ increase ⟶ decrease manner of all Fh1/Fh0, Fh2/Fh0, and Fh3/Fh0. The oscillation of curves of Fh1/Fh0 and Fh2/Fh0 indicated that for incident waves with the same periods, as the wave-absorbing chamber width increased, the phase differences between the incident wave, the wave reflected from the perforated front wall, and the solid rear wall oscillated. The oscillation of Fh3/Fh0 curve indicated that for incident waves with the same periods, as the wave-absorbing chamber width increased, the phase differences between the incident wave, the wave reflected from the solid rear wall, and the re-reflected wave from the leeward side of the perforated front wall oscillated. The curve of Fh2/Fh0 showed the most obvious oscillation, and the oscillating amplitude of the Fh1/Fh0 curve was smaller than that of Fh2/Fh0. This indicated that the wave forces acting on the seaward and the leeward side of the perforated front wall of the caisson varied asymmetrically with the increase in the wave-absorbing chamber width.It must be noted that synchronization could be found when investigating the curves ofFh1/Fh0, Fh2/Fh0, and Kr. This meant that when Kr reached its maximum or minimum value, Fh1/Fh0 and Fh2/Fh0 also simultaneously reached their maximum or minimum values. On the other hand, the Fh3/Fh0 curve showed some hysteresis.Figure11 shows the numerical and experimental values of the total horizontal wave force as the wave-absorbing chamber width varied from 0.15 m to 0.80 m. In Figures 11(a)–11(c), three different groups of cases with the significant wave lengths Ls = 2.35 m, 1.82 m, and 1.44 m were shown, respectively. Some differences can be seen but generally the numerical results were consistent with the experimental results.Figure 11
Total horizontal wave force acting on the perforated caisson with irregular waves: black ○, numerical results ofFh/Fh0.; black ●, experimental results of Fh/Fh0.
(a)(b)(c)Results showed that resonant conditions occurred for variousbc/Ls values introducing an oscillation in a decrease ⟶ increase ⟶ decrease manner of Fh/Fh0. Compared with the curves of Fh1/Fh0 and Fh2/Fh0, the Fh/Fh0 curve was more similar to the Fh3/Fh0 curve. This was because, on the one hand, the horizontal wave force acting on the solid rear wall was always larger than that acting on the perforated front wall under experimental conditions. On the other hand, the wave force acting on the seaward side of the perforated front wall was offset by that acting on the leeward side, so the contribution to the total horizontal force was rather small.In general, for all values of reflection coefficientKr, the total horizontal wave force or its components, results showed that resonant conditions occurred for various bc/Ls values introducing an oscillation in a decrease ⟶ increase ⟶ decrease manner. It was clear that their relationships could not be described by a quadratic function.


## 3.1.1. Reflection Coefficients

Figure8 shows the obtained numerical and experimental reflection coefficients as the wave-absorbing chamber width varied from 0.15 m to 0.80 m. In Figures 8(a)–8(c), three different groups of cases with the significant wave lengths Ls = 2.35 m, 1.82 m, and 1.44 m were shown, respectively. It was seen that the numerical results were in good agreement with the experimental results. Results showed that resonant conditions occurred for various bc/Ls values introducing an oscillation in a decrease ⟶ increase ⟶ decrease manner of the reflection coefficients.Figure 8
Experimental and numerical reflection coefficients.
(a)(b)(c)This indicated that for incident waves with the same periods, as the wave-absorbing chamber width increased, the phase differences between the incident wave, the wave reflected from the perforated front wall, and the solid rear wall oscillated. This oscillation behavior is most clearly seen in Figure8(c) in which the significant wave length Ls was the shortest, and the range of the relative wave-absorbing chamber width bc/Ls was therefore largest. According to Figure 8, for a greater range of the wave-absorbing chamber width changed, a quadratic function could no longer describe the correlation between the relative wave-absorbing chamber width and reflection coefficients; therefore, the formula of Xing [15] did not apply in such cases.


## 3.1.2. Horizontal Wave Forces

Another important subject for studying the wave attenuation performance of a perforated caisson sitting on a rubble-mound foundation was the horizontal wave force. This subject included the horizontal wave forces acting on each wall of a perforated caisson and the total horizontal wave force acting on the entire structure. For a typical Jarlan-type perforated caisson, the total horizontal wave forceFh could be decomposed into three different components, as shown in Figure 9(a). The total horizontal wave force could be expressed as Fh = Fh1 − Fh2 + Fh3, where Fh1 represented the wave force acting on the seaward side of the perforated front wall, Fh2 was the force acting on the leeward side of the perforated front wall, and Fh3 was the force acting on the solid rear wall.Figure 9
Wave forces acting on (a) perforated and (b) traditional caisson sitting on a rubble-mound foundation.
(a)(b)Under the same conditions,Fh0 was the horizontal wave force acting on a traditional caisson, as shown in Figure 9(b).To study the horizontal wave forces acting on each part of the perforated caisson, dimensionless ratio parametersFh1/Fh0, Fh2/Fh0, and Fh3/Fh0 were introduced. The dimensionless total horizontal wave force could be expressed by Fh/Fh0. Since Fh2 was opposite in direction to Fh0, Fh2/Fh0 was always negative. Figure 10 shows the numerical and experimental horizontal wave forces as the wave-absorbing chamber width varied from 0.15 m to 0.80 m. In Figures 10(a)–10(c), three different groups of cases with the significant wave lengths Ls = 2.35 m, 1.82 m, and 1.44 m were shown, respectively. It could be seen that the numerical results were in good agreement with the experimental results. Based on the numerical results, spline curves are used to smooth the curves in Figure 10 and in other graphs in the following sections.Figure 10
Horizontal wave forces acting on each part of the perforated caisson with irregular waves: black ○, numerical results ofFh1/Fh0; black ●, experimental results of Fh1/Fh0; black □, numerical results of Fh2/Fh0; black ■, experimental results of Fh2/Fh0; black △, numerical results of Fh3/Fh0; black ▲, experimental results of Fh3/Fh0.
(a)(b)(c)As the wave-absorbing chamber width increased over a large range, results showed that resonant conditions occurred for variousbc/Ls values introducing an oscillation in a decrease ⟶ increase ⟶ decrease manner of all Fh1/Fh0, Fh2/Fh0, and Fh3/Fh0. The oscillation of curves of Fh1/Fh0 and Fh2/Fh0 indicated that for incident waves with the same periods, as the wave-absorbing chamber width increased, the phase differences between the incident wave, the wave reflected from the perforated front wall, and the solid rear wall oscillated. The oscillation of Fh3/Fh0 curve indicated that for incident waves with the same periods, as the wave-absorbing chamber width increased, the phase differences between the incident wave, the wave reflected from the solid rear wall, and the re-reflected wave from the leeward side of the perforated front wall oscillated. The curve of Fh2/Fh0 showed the most obvious oscillation, and the oscillating amplitude of the Fh1/Fh0 curve was smaller than that of Fh2/Fh0. This indicated that the wave forces acting on the seaward and the leeward side of the perforated front wall of the caisson varied asymmetrically with the increase in the wave-absorbing chamber width.It must be noted that synchronization could be found when investigating the curves ofFh1/Fh0, Fh2/Fh0, and Kr. This meant that when Kr reached its maximum or minimum value, Fh1/Fh0 and Fh2/Fh0 also simultaneously reached their maximum or minimum values. On the other hand, the Fh3/Fh0 curve showed some hysteresis.Figure11 shows the numerical and experimental values of the total horizontal wave force as the wave-absorbing chamber width varied from 0.15 m to 0.80 m. In Figures 11(a)–11(c), three different groups of cases with the significant wave lengths Ls = 2.35 m, 1.82 m, and 1.44 m were shown, respectively. Some differences can be seen but generally the numerical results were consistent with the experimental results.Figure 11
Total horizontal wave force acting on the perforated caisson with irregular waves: black ○, numerical results ofFh/Fh0.; black ●, experimental results of Fh/Fh0.
(a)(b)(c)Results showed that resonant conditions occurred for variousbc/Ls values introducing an oscillation in a decrease ⟶ increase ⟶ decrease manner of Fh/Fh0. Compared with the curves of Fh1/Fh0 and Fh2/Fh0, the Fh/Fh0 curve was more similar to the Fh3/Fh0 curve. This was because, on the one hand, the horizontal wave force acting on the solid rear wall was always larger than that acting on the perforated front wall under experimental conditions. On the other hand, the wave force acting on the seaward side of the perforated front wall was offset by that acting on the leeward side, so the contribution to the total horizontal force was rather small.In general, for all values of reflection coefficientKr, the total horizontal wave force or its components, results showed that resonant conditions occurred for various bc/Ls values introducing an oscillation in a decrease ⟶ increase ⟶ decrease manner. It was clear that their relationships could not be described by a quadratic function.


## 3.2. Extended Research Using the Numerical Model

The experimental cases mentioned in the previous section were limited. In fact, this experiment only included the cases of rubble-mound foundation heighthm = 0.2 m and the perforation ratio μ = 0.4. In other cases, how would the reflection coefficient Kr, the total horizontal wave force, and its components vary as the wave-absorbing chamber width increased? Would there be any similarities or differences? To answering these questions, the numerical model was used for extended research to extrapolate the above results beyond the parameters of the model experiments.Different cases with three rubble-mound foundation heights, two perforation ratios, eight wave-absorbing chamber widths, and three significant wave lengths were considered. It totally included 144 cases after combing all these factors. Table2 showed all the modeling conditions and geometric parameters.Table 2
Numerical modeling conditions and geometric parameters.
Significant wave periodTs (s)1.38, 1.15, 0.99Significant wave lengthLs (m)2.35, 1.82, 1.44Chamber widthbc (m)0.15, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8Foundation heighthm (m)0.10, 0.15, 0.20Perforation ratioμ0.2～0.4Relative foundation heighthm/L1/30.043～0.139Relative chamber widthbc/L1/30.064～0.556Relative water depthd/L1/30.17～0.278


### 3.2.1. Effect of Wave-Absorbing Chamber Width on Reflection Coefficients

Figure12 shows the obtained numerical and experimental reflection coefficients as the wave-absorbing chamber width varied from 0.15 m to 0.80 m. Three different groups of cases with the significant wave lengths Ls = 2.35 m, 1.82 m, and 1.44 m were shown, respectively. In Figures 12(a)–12(c), the perforation ratio was 0.4. In Figures 12(d)–12(f), the perforated ratio was 0.2.Figure 12
Reflection coefficients for irregular waves: ○, numerical results forhm = 0.10 m; □, numerical results for hm = 0.15 m; △, numerical results for hm = 0.20 m.
(a)(b)(c)(d)(e)(f)Generally, as the wave-absorbing chamber widened, the reflection coefficients tended to oscillate in a decrease ⟶ increase ⟶ decrease manner. This was a common feature regardless of the rubble-mound foundation height or the perforation ratio.From Figures12(a)–12(c), it was found that when the perforated ratio μ = 0.4, the oscillating amplitude became smaller for higher foundations. However, in Figures 12(d)–12(f), this phenomenon was not evident.Comparison between Figures12(a), 12(b), and 12(c) with 12(d), 12(e), and 12(f) in that order showed that as the wave-absorbing chamber width increased, different perforation ratios also resulted in different amplitudes of the oscillating curves.


### 3.2.2. Effect of Wave-Absorbing Chamber Width on the Horizontal Wave Forces

Figure13 shows the numerical and experimental horizontal wave forces acting on each part of the perforated caisson as the wave-absorbing chamber width varied from 0.15 m to 0.80 m. Three different groups of cases with the significant wave lengths Ls = 2.35 m, 1.82 m, and 1.44 m were shown, respectively. In Figures 13(a)–13(c), the perforated ratio was 0.4, and in Figures 13(d)–13(f), the perforated ratio was 0.2.Figure 13
Horizontal wave forces acting on each part of a perforated caisson for irregular waves: red ○,Fh1/Fh0 for hm = 0.10 m; blue ○, Fh1/Fh0 for hm = 0.15 m; black ○, Fh1/Fh0 for hm = 0.20 m; red □, Fh2/Fh0 for hm = 0.10 m; blue □, Fh2/Fh0 for hm = 0.15 m; black □, Fh2/Fh0 for hm = 0.20 m; red △, Fh3/Fh0 for hm = 0.10 m; blue △, Fh3/Fh0 for hm = 0.15 m; black △, Fh3/Fh0 for hm = 0.20 m.
(a)(b)(c)(d)(e)(f)Generally, as the wave-absorbing chamber width increased,Fh1/Fh0, Fh2/Fh0, and Fh3/Fh0 tended to oscillate in a decrease ⟶ increase ⟶ decrease manner. The Fh2/Fh0 curve oscillated most obviously, and the oscillating amplitude of Fh1/Fh0 was smaller than that of Fh2/Fh0. Synchronization occurred between Fh1/Fh0, Fh2/Fh0, and Kr to some extent that they reached their maxima and minima simultaneously. However, Fh3/Fh0 showed some hysteresis. These features were seen regardless of rubble-mound foundation height or the perforation ratio.For different perforation ratios, it could be found that whenμ = 0.4, the horizontal wave force acting on the seaward side of the perforated front wall was smaller than that for μ = 0.2. Conversely, the horizontal wave force acting on the solid rear wall was greater. This was mainly because the perforation ratio of the front wall determined how much incident wave energy passed into the wave-absorbing chamber and how much was reflected from the front wall. When the perforation ratio was small, less wave energy entered, which meant that the perforated front wall was subject to greater horizontal wave forces. In addition, compared with the case of μ = 0.4, the curves of Fh1/Fh0 and Fh3/Fh0 were flatter near their minima and change more sharply when they began to climb or fall. When μ = 0.2, the curves were less symmetrical than for μ = 0.4.By comparing figures with different rubble-mound foundation heights, it could be found that the rubble-mound foundation height had limited influence on the horizontal wave forces. Whenhm became higher, Fh1/Fh0 gets smaller only on the seaward side of the perforated front wall. This indicated that the horizontal wave forces acting on the area close to the bottom of the caisson were very small. In addition, some deviation was found in the Fh2/Fh0, curves when hm = 0.2 m especially near their maxima.Figure14 shows the numerical and experimental total horizontal wave forces as the wave-absorbing chamber width varied from 0.15 m to 0.80 m. Three different groups of cases with the significant wave lengths Ls = 2.35 m, 1.82 m, and 1.44 m were shown, respectively. In Figures 14(a)–14(c), the perforation ratio was 0.4, and in Figures 14(d)–14(f), the perforation ratio was 0.2.Figure 14
Total horizontal wave force acting on the perforated caisson with irregular waves: black ○,Fh/Fh0 for hm = 0.10 m; black □, Fh/Fh0 for hm = 0.15 m; black △, Fh/Fh0 for hm = 0.20 m.
(a)(b)(c)(d)(e)(f)In Figure14, as the wave-absorbing chamber width increased in a larger range, Fh/Fh0 tended to oscillate in a decrease ⟶ increase ⟶ decrease manner, but the oscillation was weaker than for either Fh1/Fh0 or Fh2/Fh0. The Fh/Fh0 curve showed a certain amount of hysteresis compared with that of Kr. These were the common features regardless of the rubble-mound foundation height or the perforation ratio.In addition, the variation of perforation ratio had almost no effect on theFh/Fh0 curve although the Fh/Fh0 curve appeared to resemble the Fh3/Fh0 curve more closely as the perforation ratio increased. This was mainly due to the greater force on the solid rear wall of the perforated caisson with greater perforation ratio.Comparing the figures for different rubble-mound foundation heights, it could be found that the rubble-mound foundation height had limited influence on the total horizontal wave force. As in the figures forKr, when the perforation ratio μ = 0.4, the oscillation amplitudes became smaller with greater foundation heights, but this phenomenon was no longer evident when μ = 0.2.


## 3.2.1. Effect of Wave-Absorbing Chamber Width on Reflection Coefficients

Figure12 shows the obtained numerical and experimental reflection coefficients as the wave-absorbing chamber width varied from 0.15 m to 0.80 m. Three different groups of cases with the significant wave lengths Ls = 2.35 m, 1.82 m, and 1.44 m were shown, respectively. In Figures 12(a)–12(c), the perforation ratio was 0.4. In Figures 12(d)–12(f), the perforated ratio was 0.2.Figure 12
Reflection coefficients for irregular waves: ○, numerical results forhm = 0.10 m; □, numerical results for hm = 0.15 m; △, numerical results for hm = 0.20 m.
(a)(b)(c)(d)(e)(f)Generally, as the wave-absorbing chamber widened, the reflection coefficients tended to oscillate in a decrease ⟶ increase ⟶ decrease manner. This was a common feature regardless of the rubble-mound foundation height or the perforation ratio.From Figures12(a)–12(c), it was found that when the perforated ratio μ = 0.4, the oscillating amplitude became smaller for higher foundations. However, in Figures 12(d)–12(f), this phenomenon was not evident.Comparison between Figures12(a), 12(b), and 12(c) with 12(d), 12(e), and 12(f) in that order showed that as the wave-absorbing chamber width increased, different perforation ratios also resulted in different amplitudes of the oscillating curves.


## 3.2.2. Effect of Wave-Absorbing Chamber Width on the Horizontal Wave Forces

Figure13 shows the numerical and experimental horizontal wave forces acting on each part of the perforated caisson as the wave-absorbing chamber width varied from 0.15 m to 0.80 m. Three different groups of cases with the significant wave lengths Ls = 2.35 m, 1.82 m, and 1.44 m were shown, respectively. In Figures 13(a)–13(c), the perforated ratio was 0.4, and in Figures 13(d)–13(f), the perforated ratio was 0.2.Figure 13
Horizontal wave forces acting on each part of a perforated caisson for irregular waves: red ○,Fh1/Fh0 for hm = 0.10 m; blue ○, Fh1/Fh0 for hm = 0.15 m; black ○, Fh1/Fh0 for hm = 0.20 m; red □, Fh2/Fh0 for hm = 0.10 m; blue □, Fh2/Fh0 for hm = 0.15 m; black □, Fh2/Fh0 for hm = 0.20 m; red △, Fh3/Fh0 for hm = 0.10 m; blue △, Fh3/Fh0 for hm = 0.15 m; black △, Fh3/Fh0 for hm = 0.20 m.
(a)(b)(c)(d)(e)(f)Generally, as the wave-absorbing chamber width increased,Fh1/Fh0, Fh2/Fh0, and Fh3/Fh0 tended to oscillate in a decrease ⟶ increase ⟶ decrease manner. The Fh2/Fh0 curve oscillated most obviously, and the oscillating amplitude of Fh1/Fh0 was smaller than that of Fh2/Fh0. Synchronization occurred between Fh1/Fh0, Fh2/Fh0, and Kr to some extent that they reached their maxima and minima simultaneously. However, Fh3/Fh0 showed some hysteresis. These features were seen regardless of rubble-mound foundation height or the perforation ratio.For different perforation ratios, it could be found that whenμ = 0.4, the horizontal wave force acting on the seaward side of the perforated front wall was smaller than that for μ = 0.2. Conversely, the horizontal wave force acting on the solid rear wall was greater. This was mainly because the perforation ratio of the front wall determined how much incident wave energy passed into the wave-absorbing chamber and how much was reflected from the front wall. When the perforation ratio was small, less wave energy entered, which meant that the perforated front wall was subject to greater horizontal wave forces. In addition, compared with the case of μ = 0.4, the curves of Fh1/Fh0 and Fh3/Fh0 were flatter near their minima and change more sharply when they began to climb or fall. When μ = 0.2, the curves were less symmetrical than for μ = 0.4.By comparing figures with different rubble-mound foundation heights, it could be found that the rubble-mound foundation height had limited influence on the horizontal wave forces. Whenhm became higher, Fh1/Fh0 gets smaller only on the seaward side of the perforated front wall. This indicated that the horizontal wave forces acting on the area close to the bottom of the caisson were very small. In addition, some deviation was found in the Fh2/Fh0, curves when hm = 0.2 m especially near their maxima.Figure14 shows the numerical and experimental total horizontal wave forces as the wave-absorbing chamber width varied from 0.15 m to 0.80 m. Three different groups of cases with the significant wave lengths Ls = 2.35 m, 1.82 m, and 1.44 m were shown, respectively. In Figures 14(a)–14(c), the perforation ratio was 0.4, and in Figures 14(d)–14(f), the perforation ratio was 0.2.Figure 14
Total horizontal wave force acting on the perforated caisson with irregular waves: black ○,Fh/Fh0 for hm = 0.10 m; black □, Fh/Fh0 for hm = 0.15 m; black △, Fh/Fh0 for hm = 0.20 m.
(a)(b)(c)(d)(e)(f)In Figure14, as the wave-absorbing chamber width increased in a larger range, Fh/Fh0 tended to oscillate in a decrease ⟶ increase ⟶ decrease manner, but the oscillation was weaker than for either Fh1/Fh0 or Fh2/Fh0. The Fh/Fh0 curve showed a certain amount of hysteresis compared with that of Kr. These were the common features regardless of the rubble-mound foundation height or the perforation ratio.In addition, the variation of perforation ratio had almost no effect on theFh/Fh0 curve although the Fh/Fh0 curve appeared to resemble the Fh3/Fh0 curve more closely as the perforation ratio increased. This was mainly due to the greater force on the solid rear wall of the perforated caisson with greater perforation ratio.Comparing the figures for different rubble-mound foundation heights, it could be found that the rubble-mound foundation height had limited influence on the total horizontal wave force. As in the figures forKr, when the perforation ratio μ = 0.4, the oscillation amplitudes became smaller with greater foundation heights, but this phenomenon was no longer evident when μ = 0.2.


## 4. Conclusions

Model experiments were carried out to study the effects of a large range of the wave-absorbing chamber width on the wave attenuation performance of a perforated caisson sitting on a rubble-mound foundation. Meanwhile, a porosity-type numerical model based on VARANS equations was validated by comparing with the existing and present model experiments. The validated numerical model was then used for extended research. The conclusions are given as follows:(i)
The resistance-type porosity model used in the present research took into account the porosity of the rubble-mound foundation and the fluid viscosity. The numerical results were in good agreement with the existing and present model experiments. Therefore, the numerical model could be used to further study the wave interaction with a perforated caisson sitting on a rubble-mound foundation.(ii)
Resonant conditions occurred for various relative wave-absorbing chamber width introducing an oscillation in a decrease ⟶ increase ⟶ decrease manner of the reflection coefficients. This was a common feature regardless of the rubble-mound foundation height or the perforated ratio.(iii)
Resonant conditions occurred for various relative wave-absorbing chamber width introducing an oscillation in a decrease ⟶ increase ⟶ decrease manner of the horizontal wave forces acting on each part of the caisson. The horizontal wave force acting on the perforated front wall varied synchronously with the reflection coefficients, but the horizontal wave force acting on the solid rear back wall showed a certain amount of hysteresis.(iv)
Resonant conditions occurred for various relative wave-absorbing chamber width introducing an oscillation in a decrease ⟶ increase ⟶ decrease manner of the total horizontal wave force acting on the caisson. A certain amount of hysteresis was evident in the variation of the total horizontal wave force compared to that of the reflection coefficient.(v)
With the variation of the wave-absorbing chamber width, the reflection coefficient and horizontal wave force acting on a perforated caisson sitting on a rubble-mound foundation had minimum and maximum values. Considering the attenuation performance and wave force conditions, the wave-absorbing chamber should be neither too narrow nor too wide.



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*Source: 8845331-2021-02-09.xml*