Wave interactions with multiple semi-immersed Jarlan-type perforated breakwaters

This study examines wave interactions with multiple semi-immersed Jarlan-type perforated breakwaters. A numerical model based on linear wave theory and an eigenfunction expansion method has been developed to study the hydrodynamic characteristics of breakwaters. The numerical results show a good agreement with previous analytical results and experimental data for limiting cases of double partially immersed impermeable walls and double and triple Jarlan-type breakwaters. The wave transmission coefficient CT; reflection coefficient CR, and energy dissipation coefficient CE coefficients and the horizontal wave force exerted on the front and rear walls are examined. The results show that CR reaches the maximum value when B/L = 0.46n while it is smallest when B/L=0.46n+0.24 (n=0, 1, 2,...). An economical triple semi-immersed Jarlan-type perforated breakwater can be designed with B/L = 0.25 and CR and CT ranging from 0.25 to 0.32 by choosing a relative draft d/h of 0.35 and a permeability parameter of the perforated front walls being 0.5 for an incident wave number kh nearly equal to 2.0. The triple semi-immersed Jarlan-type perforated breakwaters with significantly reduced CR, will enhance the structure’s wave absorption ability, and lead to smaller wave forces compared with the double one. The proposed model may be used to predict the response of a structure in the preliminary design stage for practical engineering.


Introduction
Breakwaters are mainly used to protect coasts from erosion, protect harbor facilities, and provide a calm basin for ships by reducing wave-induced disturbances. Rubble mound breakwaters are the oldest type of breakwaters. They have been widely used to shelter harbors. Caisson breakwaters are advantageous over rubble mound breakwaters as they require less space, material, and time for construction. However, caisson breakwaters reflect the incoming wave energy, and thus causing erosion at the breakwater toe.
Researchers have attempted to solve the problem of wave reflection and scouring at the breakwater toe by using perforated walls. Perforated breakwaters were first introduced by Jarlan (1961). He proposed a breakwater with a perforated front wall and a solid rear wall. Since then, several researchers have introduced several modifications to Jarlan-type perforated breakwaters to improve their hydraulic performance. Kondo (1979) presented an analytical approach based on long wave theory to estimate C R and C T of double and triple Jarlan-type perforated breakwaters. Tanimoto and Yoshi-moto (1982) theoretically and experimentally studied the reflection properties of partially perforated caissons. Twu and Lin (1991) examined the reflection of a finite number of porous plates. The authors concluded that the spacing between adjacent porous plates and the alignment of these plates significantly affects wave reflection. Fugazza and Natale (1992) analyzed the wave attenuation produced by the permeable structure and proposed design formulae for an optimized hydraulic design of Jarlan-type breakwaters. Bennet et al. (1992) theoretically and experimentally studied C R of a screen breakwater. Losada et al. (1993) used linear theory for water obliquely impinging on dissipative multilayered media to evaluate C R and C T . The results indicated that C R decreases with the increasing number of absorber units. Suh and Park (1995) developed an analytical model to predict C R of a fully perforated wall breakwater mounted on a rubble mound foundation; later, Suh et al. (2001Suh et al. ( , 2006 extended this model to random waves and a partially perforated wall caisson. Cox et al. (1998) introduced a double semi-immersed Jarlan-type perforated breakwater. Their researches showed that with a suitable design, C R and C T of short waves can be smaller than 0.3. Isaacson et al. (1998) theoretically studied C R , wave run-up, and wave force for a breakwater consisting of a perforated front wall, an impermeable back wall, and a rockfilled core. Brossard et al. (2003) proposed a semi-immersed caisson breakwater with a perforated front wall and demonstrated that both C R and C T of the breakwater could be small. Liu et al. (2007) studied C R of obliquely incident waves for an infinite array of partially perforated Jarlan-type breakwaters. Garrido and Medina (2006) experimentally investigated slotted and perforated double and triple Jarlantype breakwaters under regular and random waves and developed a nonlinear relationship between C R and the structural and incident wave conditions. Bergmann and Omeraci (2008) experimentally studied double-chambered breakwaters. They revealed that for effective wave damping, the relative wave chamber width (B/L) should exceed 0.3. Krishnakumar et al. (2009) experimentally investigated wave interactions with a doublechambered breakwater system consisting of two wave screens placed on the seaward side of an impermeable vertical wall. The authors concluded that at larger and smaller draft to wavelength ratios (d/L), the reflection is reduced by approximately 40% and 60%, respectively. Molin et al. (2009) indicated that C R of semi-immersed caissons can be greatly reduced by adding perforated walls. Liu and Li (2011) developed an analytical solution for describing the hydrodynamic performance of double semi-immersed Jarlan-type perforated breakwaters. They presented the optimal design parameters for the breakwater.
Using wave scattering based on eigenfunction expansions, Liu et al. (2015) studied wave interactions with a semi-immersed breakwater consisting of a perforated front wall, a solid rear wall, and a horizontal perforated plate between the two walls. In addition, they experimentally investigated C R and C T of the breakwater. The new breakwater was found to give a better wave-absorbing performance and smaller wave forces.
By extending Liu and Li's (2011) model, this study developed a mathematical model to investigate the hydrodynamic performance of multiple semi-immersed Jarlantype perforated breakwaters. The studied breakwater consists of a seaward multi-perforated wall and a shoreward impermeable wall. All walls extend from above the seawater level to a distance above the seabed.

Mathematical formulation
Consider the multiple rows of the two-dimensional multiple semi-immersed Jarlan-type perforated breakwater sketched in Fig. 1, it consists of "J-1" seaward perforated walls and a shoreward impermeable wall "J" at a constant water depth h. The draft of the j-th breakwater is d j , in which b j is the thickness of the j-th wall. A Cartesian coordinate system (x, z) is defined with the positive x-axis in the direction of wave propagation from a point in front of the first wall and the z-axis in the upward vertical direction. The center of the j-th wall is located at x=x j .

Φ
The fluid is assumed incompressible and inviscid and the flow being irrotational. A velocity potential (x, z, t) for the monochromatic wave propagating with the angular frequency ω over the water depth h can be expressed as: where Re represents the real part of the complex expression in [ ], is the spatial velocity potential, , ω is the angular frequency, t is time, and g is the gravitational acceleration. In different regions, the spatial potentials satisfy the Laplace equation: where the subscript j represents variables with respect to the region j. The potentials must satisfy the following boundary conditions: ∂ϕ j ∂z − ω 2 g ϕ j = 0, at z = 0, j = 0, 1, 2, ..., J; (3) In addition, the boundary conditions at the breakwater walls are ∂ϕ where G j is the permeability parameter of the j-th permeable wall (Chwang and Li, 1983;Yu, 1995).
In this study, the formula proposed by Yu (1995) is followed. G j is given by where is the argument of the complex G j , ε j is the geometrical porosity of the j-th wall, b j is its thickness, f j is the friction coefficient, and is the inertial coefficient given by where C m is the added mass coefficient, and f j is the linearized resistance coefficient through the permeable wall. f j in Eq. (13) can be calculated by the following formula (Li et al., 2006):

Analytical solutions
Expressions for that satisfy the seabed, free surface, and boundary conditions, as well as the abovementioned boundary conditions along x=x i , may be developed in terms of coefficients A jm and B jm , which are the component waves propagating forward and backward, respectively. The first subscript (j) indicates the row of the wall, whereas the second one (m) indicates the wave component. The reduced velocity potentials are obtained using the eigenfunction expansion method, as in Isaacson et al. (1998) and Suh et al. (2006). The velocity potentials are expressed as a series of an infinite number of solutions. The solutions to Eq. (2) satisfying the boundary conditions, Eq.
(3) to Eq. (6), are given by where the pair of imaginary roots μ 0 =±ik for propagating waves are the solution to the dispersion relation, i.e. ω 2 =gμ 0 tanh(μ 0 h). Let μ 0 =-ik such that the propagating waves in Eqs. (15)- (17) correspond to the reflected and transmitted waves. The wave numbers μ m are the solution to the dispersion relation, i.e., ω 2 =-gμ m tan(μ m h), which has an infinite discrete set of real roots ±μ m (m ≥ 1) for nonpropagating evanescent waves. In addition, let the positive roots for m ≥ 1 be such that the nonpropagating waves die out exponentially with the increasing distance from the wall.
In Eqs. (15)- (17), the depth-dependent functions and are given by and To solve the unknown coefficients of Eqs. (15)- (17), they must automatically satisfy all relevant boundary conditions, i.e. Eqs. (7)-(12); for convenience, these matching boundary conditions are rewritten as: The least squares technique, suggested by Dalrymple and Martin (1990), can be used to determine the coefficients , which requires the value of to be minimized. Minimizing these integrals with respect to coefficient leads to where is the complex conjugate of , and Finally, for x=x J , Once the wave potentials are calculated, the various engineering wave properties can be obtained. The reflection and transmission coefficients are given by and The wave energy dissipation coefficient C E is expressed as: The wave force on each wall can be calculated by integrating the wave pressure acting on both the upwave and downwave sides of the wall. The magnitude of the horizontal wave force on the unit width of the front wall, F f , is expressed as: The magnitude of the horizontal wave force on the unit width of the rear wall, F r , is expressed as: The dimensionless wave forces and are expressed as:

Validation of the mathematical model
The effectiveness of the present model is validated by comparing the calculated results with previous analytical results by Horiguchi (1976), Natale (1983), Fugazza and Natale (1992), and Das et al. (1997) as well as previous experimental data of Kondo (1979) and Cox et al. (1998).

Comparisons with other theoretical models
For double partially immersed solid walls (|G| for the perforated front wall is zero), the present calculated reflection coefficient was compared with the corresponding result of Das et al. (1997), as shown in Fig. 2. The calculated conditions are d/h=0.6 and B=2.0h. As shown in Fig. 2, the results obtained by the present model agree well with the analytical results of Das et al. (1997).
For a double Jarlan-type perforated breakwater, data from the theoretical results of Horiguchi (1976) and Natale (1983) are considered, in which the water depth h was 3.0 m, the wave height H was 1.0 m, the chamber width B was 4.0 m, d/h was 1.0, and the porosity of the perforated wall was ε=0.3. As shown in Fig. 3, the results of the present model are close to the theoretical results of Horiguchi (1976) and Natale (1983).
In addition, for a double Jarlan-type perforated breakwater, data from the theoretical results of Fugazza and Natale   Horiguchi (1976) and Natale (1983) for a double Jarlan-type perforated breakwater.
(1992) are considered, in which the wave height H was 4.0 cm, the chamber width B was 0.5 m, d/h was 1.0, and the porosity of the perforated wall ε was 0.2. Fig. 4 shows that the present results agree reasonably well with the theoretical results of Fugazza and Natale (1992).

Comparisons with experimental Data
The results of the present model for C R of double semiimmersed Jarlan-type perforated breakwaters were validated by comparison with the results of Cox et al. (1998). In tests by Cox et al., ε was 20%, s was 1, b/h was 0.02, and d/h were 0.3 and 0.5. As shown in Fig. 5, the results obtained by the present model agree well with the experimental results of Cox et al. (1998).
Next, the present model results for C R of double and triple Jarlan-type perforated breakwaters were validated by comparison with the experimental results of Kondo (1997). The porosity of the perforated wall, ε, was 0.2. As shown in Figs. 6 and 7, the present results agree well with the experimental results of Kondo (1979).

Numerical examples
The relationships among C R , C T , and C E of double semiimmersed Jarlan-type perforated breakwaters and the relative wave chamber width B/L for different values of the relative draft d/h (d/h=0.35, 0.5, 0.75, and 1.0) with kh=1.6, G=0.5, and H/h=0.1 are shown in Fig. 8. Both C R and C T oscillate with changing B/L with the same trend. The mentioned effect is due to the dissipation of the wave's energy by the perforated front wall and the C E oscillation with changing B/L. The maximum value of C R appears when B/L = 0.46n and its minimum value occurs when B/L = 0.46n+0.24 (n=0, 1, 2, ...). The maxima and minima of C R curve are shifted to the left compared with that of the original bottom-standing Jarlan-type perforated breakwaters presented in Chwang and Dong (1984). Fig. 8 also represents the change in C E against B/L with opposite maximum and minimum values compared with C R and C T . Furthermore, C T decreases gradually with the increasing d/h. Fig. 9 shows C R , C T , and C E of double semi-immersed Jarlan-type perforated breakwaters against d/h for varied values of B/L (B/L=0, 0.25, 0.75, and 1.0) with kh=1.6, G=0.5, and H/h=0.1. From Fig. 9, it is obvious that C T decreases with the increasing d/h. However, the variation in C R with increasing d/h is related to B/L. That is, an increasing d/h does not necessarily lead to a larger C R . It agrees with the results of Brossard et al. (2003) and Liu and Li (2011). It is also obvious that C E increases with the increasing d/h until reaching a peak; after that, it becomes fixed, except in the case of B/L=0.5, where it decreases. Comparing the different curves in Fig. 9 shows that the values of B/L=0. 25, d/h=0.35, B/L=0.75 and d/h=0.35 should be suitable for simultaneously obtaining smaller C R and C T values.   Cox et al. (1998) for double semi-immersed Jarlan-type perforated breakwaters. Fig. 6. Comparison between the results of the present model and those of Kondo (1979) for a double Jarlan-type perforated breakwater. Fig. 7. Comparison between the results of the present model and those of Kondo (1979) for a triple Jarlan-type perforated breakwater.
The relationships among C R , C T , and C E and B/L for different values of G (G=0.5, 1.0, and 2.0) at kh=1.6, H/h=0.1, and d/h=0.5 for double semi-immersed Jarlan-type perforated breakwaters are shown in Fig. 10. It can be observed that the variations in C R and C T with B/L are somewhat similar. The variation in C E with B/L is just the opposite of that in C R and C T with B/L. Furthermore, it can be observed that the positions of the maximum and minimum values of C R against B/L are not affected by G, which is similar to the results of Liu and Li (2011).
The variations in the horizontal wave force exerted on the front wall ( ) and rear wall ( ) against B/L for different values of G (0.5 and 2.0) with kh=1.6, d/h=0.5, and H/h=0.1 for double semi-immersed Jarlan-type perforated  Fig. 9. Variation in C R , C T , and C E of double semi-immersed Jarlan-type perforated breakwaters against d/h for different values of B/L with kh=1.6, G=0.5, and H/h=0.1. Fig. 11. It can be seen that decreases as G increases. This might be because the increase in G allows more wave energy to pass through the porous front wall, and thus, less force is exerted on the front wall. The comparison between Fig. 10 and Fig. 11 shows that the maximum C R corresponds with the minimum and vice versa. This agrees with the results of Yip and Chwang (2000) and Liu and Li (2011). C R and can be related to the difference in the water level between the two sides of the perforated wall as follows: when C R is maximum, the difference in the water level is very small, and hence, is minimum. The opposite is also applicable: when C R is minimum, the difference in the water level is very large, and hence, is maximum. Moreover, the relationship between G and can be observed from Fig. 11: increases. The variations in both and C T with B/L are similar. This can be observed in Figs. 10 and 11. This result agrees with the results of Liu and Li (2011).
The location of the middle wall between the front and rear walls is varied three times such that for the three trials, Δx 1 =0.5h and Δx 2 =1.5h, Δx 1 =1.0h and Δx 2 =1.0h, and Δx 1 =1.5h and Δx 2 =0.5h. Fig. 12 shows that the curves of C R , C T , and C E against B/L are periodic, which is similar to that of a double semi-immersed Jarlan-type perforated breakwater. It is also clear that C T and C R follow the same trend while C E shows an opposite trend. From the results, it can be concluded that the location of the middle wall has little effect on C R , C T , and C E .
The relationships among the C R , C T , and C E of triple semi-immersed Jarlan-type perforated breakwaters and kh for different values of B/L (B/L=0.25, 0.5, and 0.75) with kh=1.6, H/h=0.1, d//h=0.25, and Δx 1 =Δx 2 =1.0h are shown in Fig. 13. It indicates that C R increases with the increasing kh at a fixed B/L. Furthermore, C T decreases with the increasing kh; that is, with increasing kh, the sheltering ability of Fig. 10. Variation in C R , C T , and C E of double semi-immersed Jarlan-type perforated breakwaters against B/L for different values of G with kh=1.6, d/h=0.5, and H/h=0.1. Fig. 11. Variation in and of double semi-immersed Jarlan-type perforated breakwaters against B/L for different values of G with kh=1.6, d/h=0.5, and H/h= 0.1. Fig. 12. Variation in C R , C T , and C E of triple semi-immersed Jarlan-type perforated breakwaters against B/L for different values of the relative distance between the walls with kh=1.6, d//h=0.5, G=0.5, and H/h=0.1. the breakwater increases. This agrees with the results of Liu and Li (2011) and Neelamani and Vedagiri (2002). Fig. 13 also shows that C E increases with the increasing kh until reaching the peak point; after that, it decreases and the values of C R and C T range from 0.25 to 0.32 for kh=2:2.2 and B/L=0.25 and 0.75.

C Ff C Fr
14 presents the relationships among and and B/L for a triple semi-immersed Jarlan-type perforated breakwater with the following parameters: G=0.5 and 2.0, kh=1.6, d/h=0.5, Δx 1 =Δx 2 = 1.0h, and H/h= 0.1. Comparison between Fig. 14 and Fig. 12b shows that the variations in both and C R against B/L are opposite. The triple semiimmersed Jarlan-type perforated breakwaters were found to be very helpful in enhancing the structure's wave-absorbing ability compared with the double one. Moreover, Figs. 12b and 14 show that the variations in both and C T against B/L are similar.

Conclusions
A mathematical model based on an eigenfunction expansion method and a least squares technique for linear waves has been developed to study the hydrodynamic performance of multiple semi-immersed Jarlan-type perforated breakwaters. The model is validated by comparing the predicted results with the analytical results of Horiguchi (1976), Natal (1983, Fugazza and Natale (1992), and Das et al. (1997) and the experimental data of Kondo (1979) and Cox et al. (1998). The comparisons showed that the results of the presented mathematical model agree reasonably well with the previous analytical results and experimental data. Thus, the model can be used to analyze the performance of multiple semi-immersed Jarlan-type perforated breakwaters. In general, it is possible to construct a perforated breakwater with a smaller wave-absorbing chamber width and low C R value by choosing suitable d/h and G. A small value of C R can be obtained in a narrow range of B/L and with a small d/h. In addition, the sheltering ability of the breakwater increases with the increasing kh.
For triple breakwaters, the location of the middle wall between the front and rear walls has little effect on C R , C T , and C E . The triple semi-immersed Jarlan-type perforated breakwaters significantly reduced C R values compared with the double one. Moreover, the triple type was found to be very helpful in enhancing the structure's wave-absorbing ability compared with the double one.
In future work, more investigations should be carried out on multiple semi-immersed Jarlan-type perforated breakwaters with obliquely incident waves in cases of both absence and presence of sea currents. Fig. 13. Variation in C R , C T , and C E of triple semi-immersed Jarlan-type perforated breakwaters against kh for different values of B/L with H/h=0.1, d//h=0.25, and Δx 1 =Δx 2 = 1.0h. Fig. 14. Variations in and of triple semi-immersed Jarlan-type perforated breakwaters against B/L for different values of G with kh=1.6, d/h=0.5, Δx 1 =Δx 2 =1.0h, and H/h=0.1.