Meshless Method with Domain Decomposition for Submerged Porous Breakwaters in Waves

CHIOUKH Nadji, and YÜKSEL Yalçın

Meshless Method with Domain Decomposition for Submerged Porous Breakwaters in Waves

CHIOUKH Nadji1), 2), *, and YÜKSEL Yalçın3)

1) Department of Hydraulics, University of Sidi Bel-Abbes, Sidi Bel-Abbes 22000, Algeria 2) Laboratory of Civil Engineering and Environment, University of Sidi Bel-Abbes, Sidi Bel-Abbes 22000, Algeria 3) Laboratory of Hydraulics and Coastal Engineering, Yildiz Technical University, Esenler-Istanbul 34220, Turkey

Based on the improved version of the meshless singular boundary method (ISBM) in multi domain (MD), a numerical method is proposed in this paper to study the interaction of submerged permeable breakwaters and regular waves at normal incidence. To account for fluid flow inside the porous breakwaters, the conventional model of Sollitt and Cross for porous media is adopted. Both single and dual trapezoidal breakwaters are examined. The physical problem is formulated in the context of the linear potential wave theory. The domain decomposition method (DDM) is employed, in which the full computational domain is decomposed into separate domains, that is, the fluid domain and the domains of the breakwaters. Respectively, appropriate mixed type boundary and continuity conditions are applied for each subdomain and at the interfaces between domains. The solution is approximated in each subdomain by the ISBM. The discretized algebraic equations are combined, resulting in an overdetermined full system that is solved using a least-square solution procedure. The numerical results are presented in terms of the hydrodynamic quantities of reflection, transmission, and wave-energy dissipation. The relevance of the results of the present numerical procedure is first validated against data of previous studies, and then selected computations are discussed for various structural conditions.The proposed method is demonstrated to be highly accurate and computationally efficient.

meshless method; domain decomposition; regular waves; breakwaters; porosity; reflection; transmission; dissipation; coastal environment

1 Introduction

Coastal regions around the world are constantly subjected to violent aggressions of waves, currents, and storms,resulting in the permanent transformation of our coastlines. Thus, to reduce some of these effects, breakwaters have been constructed along coasts and entrances of harbors and marinas, creating calmer areas at the back of these structures. In places where the aesthetics are a primary concern, submerged porous breakwaters are constructed, which can offer potentially cost-effective solutions to solve many problems, if designed well. They can attenuate the action of currents and waves while keeping a low environmental and visual impact. Because these structures are placed below the water surface, they permit wave overtopping and allow water to circulate, leading to a restriction in coastal erosion, redistribution in the movement of the sediment transport, and creation of a dynamic habitat for marine life.

In the last few decades, a significant amount of experimental and theoretical research has been carried out aim-ing to investigate the hydrodynamic performance of sub- merged breakwaters interacting with waves, including the studies of Dattatri(1978), Sulisz (1985), Hsu(2002), Cho(2004), Lokesha(2015), and Sri- neash(2020), to name a few. Analytical solutions based on the eigenfunction expansion method (EFEM) and in the context of linear potential theory were also pro- vided by a number of investigators, including Abul-Azm (1994), Cho(2004), Twu and Liu (2004), Liu and Li (2014), Zhao(2017), and Venkateswarlu and Kar- makar (2019). Numerical studies based on the boundary element method (BEM) in the context of the linear wave theory have also been developed successfully by a num- ber of investigators, including Sulisz (1985), Koley(2015a, 2015b), and Koley (2019). With the advent of high-speed computers and availability of more computa- tional power, numerical modeling with the use of compu- tational fluid dynamics has become very popular. To study flows in porous media and particularly the interaction of water waves and porous coastal structures, the volume-averagedReynolds-averagedNavier-Stokes(VRANS)equa-tions are resolvedusing efficient numerical methods (Sasi-kumar, 2020; Srineash, 2020).

In recent years, new emerging numerical methods, that is, meshless methods, have also been exploited to study the transmission of waves over submerged breakwaters. These include the method of fundamental solutions (MFS) (Fairweather and Karageorghis, 1998), the regularized meshless method (RMM) of Young(2005), and the improved version of the singular boundary method (ISBM) of Gu(2012b) and Gu and Chen (2013, 2014). These methods are advantageous since they require no meshing and no special numerical integration over the elements but require only scattered nodes to determine simple internodal algebraic relations. For further information on these methods and their advantages over the mesh-type methods, reference is made to Senouci(2019) and Chi- oukh(2019). Tsai and Young (2011) employed the MFS to solve the waterwave diffraction problem through a thin porous vertical breakwater of semi-infinite extent. Chen(2011) successfully applied the RMM to solve the problem of obliquely incident water waves past a single submerged breakwater with rigid and absorbing bound-ary conditions. Ouyang(2016) developed a numerical solution based on the RMM to study the Bragg reflections for a train of surface water waves from a series of impermeable submerged bottom breakwaters. Chen(2014) was the first to attempt to test the feasibility of the ISBM to a 2D problem of obliquely incident water waves past a submerged breakwater. Fu(2015) applied the ISBM to various exterior wave problems, and the efficiency of the method was confirmed by several numerical tests. Li(2016) applied the ISBM to investigate the interaction of obliquely incident water waves past single and dual submerged breakwaters with rigid and absorbing boundary conditions. Results indicate that a dual breakwater systemtraps more water wave energy compared with a single breakwater system. Recently, Senouci(2019) appliedthe ISBM to investigate the hydrodynamic performance of submerged impermeable prismatic breakwaters in regular waves. Chioukh(2019) extended their study to include fully absorbing permeability effects and seawall prox-imity.

Furthermore, some investigations have successfully com-bined several meshless methods with domain decomposition methods (DDM). Chen(2005) analyzed the ei- genproblem of thin circular membranes with degenerate boundary conditions by means of the DDM together with the MFS. Young(2006) used the DDM along with MFS to solve degenerate boundary problems in ground water flows. Tsai and Young (2011) successfully combined the DDM and the MFS to simulate the waterwave diffraction by thin porous breakwaters. Htike(2011) applied the DDM to investigate the application of the SBM to 2D problems of steady-state heat conduction in isotropic bimaterials. Gu(2012a) used the DDM andthe ISBM to analyze the stress of multilayered elastic materials. Recently, Qiu(2019) developed a numerical method using the DDM and the SBM to study the transient heat conduction problems in layered materials.

The present paper aims to develop a numerical model based on the ISBM and the linear potential wave theory to analyze single and dual submerged porous trapezoidal breakwaters in regular normally incident waves. This study uses the traditional model of Sollitt and Cross (1972) describing flows in porous media to represent continuity of mass flow at the interfaces of the porous breakwaters. In the previous study of Chioukh(2019), simple relations represented the permeability of the breakwaters; hence, the solution procedure did not require a DDM. In this study, the DDM is required to solve the problem due to the more complicated nature of the porous model of Sollitt and Cross (1972). The mathematical model is applied in each subdomain, and the resulting algebraic equations are recombined leading to an overdetermined full system of algebraic equations that is solved by a least-square method. The wave reflection, transmission, and energy dissipation are assessed for several wave conditions and a number of influencing parameters, such as the relative breakwater’s height and width, porosity, friction coefficient, and interspacing between breakwaters.

2 Formulation of the Problem

In this study, we consider single and dual submerged porous trapezoidal breakwaters, shown in Figs.1a and b, respectively. For the sake of generality, the method is developed for a system of dual trapezoidal breakwaters.

Fig.1 Breakwater systems considered in this study.

Fig.2 shows the idealized geometry of a 2D problem in the Cartesian coordinate system (), with originlocated on the seabed and midpoint of the two breakwaters. The structures are subjected to regular normal waves of small amplitude, wavelength,periodand propagate in water with an undisturbed depth.

Fig.2 Problem definitions for the breakwater system.

The breakwaters are assumed similar and have a height, bottom widthw, and top widthw. They are also separated by a distanceXmeasured from their centers. In the proposed work, a DDM is employed where the total fluid domain is divided in five regions (Fig.2). Region (1) at (−∞) denotes the region with incoming waves (inflow), and region (5) at (+∞) is where the waves are transmitted (outflow). Region (2) lies between regions (1) and (5) and is delimited by the porous walls of the breakwaters (Γ(3)bandΓ(4)b), the free surface boundaryΓ, the seabed boundaryΓ(2)s, and the radiation boundariesΓ−and Γ+of the inflow and outflow regions, respectively. Region (3) rep- resents the interior domain of the front porous breakwater delimited by the porous boundaries (Γ(3)b=Γ1+Γ2+Γ3) and the solid boundaryΓ(3)s. Similarly, region (4) represents the interior domain of the back porous breakwater delimited by the porous boundaries (Γ(4)b=Γ4+Γ5+Γ6) and the solid boundaryΓ(4)s. The spatial velocity potentialsatisfies the following conditions:

in the fluid and breakwater regions (=2, 3, 4), (1)

free surface boundary Γ(=2), (2)

seabed boundary Γ=Γ(2)s+Γ(3)s+Γ(4)s(=2, 3, 4), (3)

whereis the normal to the boundary pointing out of the flow region.

At the left and right auxiliary vertical boundaries (Γ−and rightΓ+), continuity of pressure and mass flux yield the following relations:

In the infinite strip problem, the vertical left (Γ−) and right (Γ+) boundaries are located at finite distances, respectively,=−xand=+x. Due to the continuity of pressure and mass flow through these boundaries, the ra- diation conditions can be written as follows:

radiation condition at=−x(boundary Γ−), (5a)

radiation condition at=+x(boundary Γ+), (5b)

whereincis the incident velocity potential, which is defined as follows:

In addition, at the interfaces between the porous breakwaters and the surrounding fluid,, common boundaries between regions (2)–(3) and (2)–(4), the solution must be continuous. The conventional model given by Sollitt and Cross (1972) for porous media, expressing continuity of pressure and mass flux, is adopted:

where,, andare the inertial coefficient, the linearized friction coefficient, and the porosity of the porous material of the breakwaters, respectively. The subscripts (3) and (4) denote the values for the breakwater in regions (3) and (4), respectively. The inertial coefficientdepends on the added massCand the porosity, such that=1+C(1−)/. On the other hand, the linearized friction coefficient should be determined implicitly by employing the Lorentz principle of equivalent work (see Appendix). Further details can also be found in Sollitt and Cross (1972), Sulisz (1985), Koley (2019), and Venkateswarlu and Karmakar (2019).

The analytical series of the potentials at the vertical boundaries (Γ−and Γ+) are given by the following expressions:

where−and+are unknown complex coefficients to be determined. The disturbances are guaranteed to be outgoing waves only (Chioukh, 2016, 2017, 2019;Bakhti, 2017; Senouci, 2019).

where

Respectively, the reflection and transmission coefficients (and) are determined from the following expressions:

The coefficient of the wave energy dissipation due to the permeability of the porous breakwaters is

3 Numerical Solution by ISBM in Multi Domain

For the numerical solution, the ISBM along with the DDM is used to solve the problem stated previously. The whole computational domain is decomposed into separate domains, that is, the fluid domain (region (2)) and the do- mains of the breakwaters (regions (3) and (4)), as shown in Fig.3 for dual breakwaters. Only the boundaries of the subdomains are represented with nodes. For subdomain (2), the total boundary isΓ(2)=Γ+Γ(2)s+Γ(3)b+Γ(4)b+Γ−+Γ+. For subdomain (3), the total boundary isΓ(3)=Γ(3)s+Γ(3)b. For subdomain (4), the total boundary isΓ(4)=Γ(4)s+Γ(4)b. In Fig.3, the arrows emanating from the nodes show the direction of the outward normal. Now each region (subdomain) is subjected to a different type of boundary conditions. Subdomain (=2) is subjected to Eqs. (2), (3), (5a and 5b), and (7a and 7b), subdomain (=3) is subjected to Eqs. (3) and (7a), and subdomain (=4) is subjected to Eqs. (3) and (7b).

It is worth mentioning that at the interfaces between subdomains (boundariesΓ(3)bandΓ(4)b), the nodes on two linked subdomains are different, but they can be related by the continuity equations. For example, in Fig.4, where the connecting boundary (Γ(3)b) between subdomains (2) and (3) is shown, the nodes in region (2) are numbered 5, 6, 7, ···, 12, 13, 14, while those in region (3) are numbered 5, 6, 7, ···, 12, 13, 14. Each node () and its corresponding node () are related by the continuity relations of Eq. (4a).

Fig.3 Domain decomposition and discretization for dual breakwaters.

Fig.4 Numbering of nodes on an interface between two linked subdomains.

Numerical discretization is carried out by means of the ISBM for each subdomain. In the ISBM, the nodal values of the potentials and their fluxes are expressed as linear combinations of the fundamental solutions and their derivatives (Gu, 2012b; Gu and Chen, 2013):

The boundary conditions for each subdomainare satisfied by a linear combination of Eqs. (12) and (13). The discretization process leads to the following equation system:

1) For subdomain= 2:

Eq. (2) for nodesxÎΓ(number of nodesx=),

Eq. (3) for nodesxÎΓ(2)s(number of nodesx=(2)s),

Eq. (5a) for nodesxÎΓ−(number of nodesx=−),

Eq. (5b) for nodesxÎΓ+(number of nodesx=+),

Eq. (7a) for nodesxÎΓ(3)b(number of nodesx=(3)b),

Eq. (7b) for nodesxÎΓ(4)b(number of nodesx=(4)b),

where2 is the total number of nodes in subdomain (2) (2=N+(2)s+−+++(3)b+(4)b). The resulting discretized Eqs. (16)–(21) are written in a more compact matrix as follows:

which can be further written as follows:

whereandare defined as follows:

2) For subdomain=3:

Eq. (3) for nodesxÎΓ(3)s(number of nodesx=(3)s),

Eq. (7a) for nodesxÎΓ(3)b(number of nodesx=(3)b),

where3 is the total number of nodes in subdomain (3) (3=(3)s+(3)b). Eqs. (24)–(25) are written in a more compact matrix as follows:

which can be further written as follows:

whereis defined as follows:

3) For subdomain=4:

Eq. (3) for nodesxÎΓ(4)s(number of nodesx=(4)s),

Eq. (7b) for nodesxÎΓ(4)b(number of nodesx=(4)b),

where4 is the total number of nodes in subdomain (4) (4=(4)s+(4)b). Eqs. (28)–(29) are written in a more compact matrix as follows:

which can be further written as follows:

whereis defined as follows:

Finally, the subsystems of Eqs. (23), (27), and (31), re- spectively, of each subdomain, are written in a single matrix system as follows:

4 Validation of the Numerical Method

In the previous formulation, the mathematical model was developed for a dual breakwater system. A single breakwater is easily constructed by the method which is described as follows. The single breakwater system is re- presented by the back breakwater with appropriate values of porosity, friction coefficient, and inertial mass (respectively,4,4, and4). The front breakwater is assigned a porosity of unity (3=1) so that the porous medium becomes water, and the friction coefficient is set to3=0. Moreover, the auxiliary right vertical boundaryΓ+is fixed at an appropriate distance,, +x=w+xand settingX=w. This way, the Cartesian coordinate system () is placed in the middle of the back breakwater (see Fig.1a), and the fictitious vertical boundariesΓ−and Γ+are located at equal distances from the originof the Cartesian coordinate system.

Additionally, the inertial mass coefficient for both breakwaters is taken equal to unity,3=4=1,, by default no added mass henceC≈0, and the linearized friction coefficientis assumed to be a constant. Details about these assumptions could be found in Sollitt and Cross (1972), Sulisz (1985), Liu and Li (2013, 2014), and Koley (2019). Henceforth, the subscripts are dropped in the variables of the porosity, the friction coefficient, and the inertial mass, which will be simply written as,, and, respectively. Thus, for the single breakwater system, it will be understood that the values of the porosity, the friction coefficient,and the inertial mass correspond to those of the back break- water (4,4, and4), and they are assumed to be similar for the dual breakwaters. Hence,,, andcorrespond to those for both breakwaters.

In the numerical computations, the whole boundary of the computational domain is discretized with 920 source nodes, and the vertical boundaries are selected, such that [(x+x)−(X−w)]/2w≥3. The computational errors were found to remain small and not to exceed 10−2. Further details are provided in Li(2016), Chioukh(2019), and Senouci(2019).

The numerical results of the ISBM for a number of limiting cases are compared against those of other investigators to demonstrate the validity of the present method. Both impermeable and permeable single and dual breakwaters are tested. The first case examined is a bottom-standing single impermeable rectangular breakwater, such thatw/=w/=1,/=0.75,=0, and=0. This case was previously studied by Abul-Azm (1994) using the eigenfunction expansion method (EFEM) and Chen(2011) using both the RMM and the BEM. Fig.5 presents the variations of the coefficients of reflection and transmission (and) with respect to, including those of the present investigation. The agreement among the results of all methods is shown to be fairly high. At large values of, the results of the ISBM still follow those of the EFEM and BEM, but those of the RMM show some fluctuations.

Fig.5 Variations of Cr and Ctversuskd for a single impermeable breakwater.

Comparison has also been made for a single porous rectangular breakwater with/=0.8,w/=w/=1,=0.3, and=1. Liu and Li (2013) and Twu(2001) previously studied this case using separate semi-analytical methods based on the EFEM. Fig.6 presents the variations ofandwith respect to, including those of the present investigation. A strong agreement is observed among the results of all methods.

Fig.6 Variations of Cr and Ctversuskd for a single permeable breakwater.

Another case inspected is a structure of bottom-standing dual impermeable trapezoidal breakwaters, such that/=0.5,w/=w/w=0.5,X/=3,=0, and=0. Cho(2004) experimentally studied this case and provided analytical solutions using the EFEM. Similarly, Ouyang(2016) carried out numerical tests of the same case using the RMM. Fig.7 shows the results of, including those of the present study. In general, the results of all methods are in close agreement.

Fig.7 Variations of Crversuskd for dual impermeable breakwaters.

The last case examined is a structure of bottom-standingdual trapezoidal breakwaters. The comparison is made both for impermeable breakwaters (=0 and=0) and porous breakwaters (=0.5 and=1.5). Cho(2004) experimentallystudied these two examples, for which/=0.5,w/=2,w/=0.5, andX/=4.5. Fig.8 displays the results of, including those of the present study. In general, the results of the present method follow closely the experimental values.

Fig.8 Variations of Crversuskd for dual impermeable and permeable breakwaters.

5 Results and Discussion

The hydrodynamic characteristics of submerged single porous breakwaters are known to depend on a number of factors, including the material porosity, friction coefficient, relative water depth, relative depth of submergence/, and relative breakwater bottom widthw/and top widthw/w. For dual breakwaters, the relative interspacingX/(orX/) is also an important correlating factor. In this work, only a subset of the data gathered from this investigation is shown.

5.1 Single Breakwaters

Fig.9 shows the variations of,, andfor different values of/. The calculating structural conditions arew/=1,w/w=0.5,=0.2, and=1. The results in Fig.9(a) show the occurrence of multiple peaks in. The larger primary peaks occur at lower values of. On the other hand,values demonstrate only one peak around=1. It is also shown that increasing the breakwater height increases the reflection and lowers the transmission, indicating that elevated breakwaters provide better shelter. This is because structures with a large depth of submergence will reflect more waves; hence the transmission of the waves will be lower. As demonstrated in Fig.9(b), porous structures with a large depth of submergence will dissipate more wave energy. For low depth of submergence, there is hardly any wave energy dissipation leading to very small reflections and nearly full transmissions. It is also worth mentioning that the oscil-latory behavior of the reflections,and not the transmissions and the dissipations, indicated that the variations ofare associated to the changes in, which is not the case for.

Fig.10 shows the variations of,, andwithfor various values of w/. The calculating structural conditions arew/w=0.5,/=0.8,=0.2, and=1. The results in Fig.10(a) show that increasing the breakwater width leads to the appearance of more peaks inbut not in. For a particular width of the breakwater, the magnitudes of the peaks indecrease with the increase in. Increasing the width increases the reflection but decreases the transmission. However, when comparing Fig.9(a) and Fig.10(a), it appears thatandare more influenced by the changes in the breakwater’s height than the breakwater’s width especially in the lower range of. For example, the peak value ofhas decreased from 0.38 to 0.11 when/has decreased from 0.8 to 0.2, whereashas decreased from 0.42 to 0.31 whenw/has decreased from 2 to 0.5. As shown in Fig.10(b), the energy dissipation increases if the breakwater width increases. It is known that for a submerged breakwater, the incident waves penetrate the front and the upper horizontal faces of the submerged porous structure. By increasing the width of the breakwater and hence its upper horizontal face, the porosity effect will be more prominent, leading to an increase in the wave energy dissipation.

Fig.9 Variations of Cr, Ct, and Ed with kd for various values of h/d.

Fig.10 Variations of Cr, Ct, and Ed with kd for various values of wb/d.

Fig.11 shows the variations of,, andwithfor different values ofand forw/=1,w/w=0.5,/=0.8, and=1. As shown in Fig.11(a), the effects of increasing the porosity decrease bothand, except in the range≈1.7–2.4, where effects of the porosity are reversed for. For nonporous structures (=0 and=0), a zero minimum inis marked at≈2.15, but for higher porosities,does not show a zero minimum. This is reasonable since for porous structures,increases with the increase of. Whenattains a zero minimum for the nonporous structure (=0),will not fall to zero for the porous structures since the value ofhas increased to 2. The reason behind this is that when the porosity of the breakwater is nil, and for certain wavelengths (≈2.15 for this case), the reflected wave components from the different sides of the breakwater cancel out due to total destructive interference (out of phase), leading to a zero minimum in the reflection coefficient. However, a porous breakwater allows for some incident wave energy to dissipate within the porous medium. There is still destructive interference between the reflected wave components, but they no longer perfectly nullify because they do not have the same amplitudes. A very small value of thereflection coefficient will still manage to subsist. As shown in Fig.11(b), increasing the porosity produces more wave energy dissipation. This is because a submerged breakwater with larger porosity allows more waves to penetrate into the structure and dissipate before they finally pass through the structure. For nonporous breakwaters, there is no wave energy dissipation, and the variations in the reflections and transmissions are simply due to the presence of the structure in front of the incident waves.

Fig.12 shows the variations of,, andfor various values of. The calculating conditions are asw/=1,w/w=0.5,/=0.8, and=0.5. As shown in Fig.12(a), increasing the friction coefficient results in high- er reflection and lower transmission. The wave energy dissipation also increases with increasing, as shown in Fig.12(b).

Fig.11 Variations of Cr, Ct, and Ed with kd for various values of ε.

Fig.12 Variations of Cr, Ct, and Ed with kd for various values of f.

5.2 Double Breakwaters

In this study, the double breakwaters are considered to be of the same height and width. Fig.13 shows the variations in,,andwithfor different values of. The calculating conditions are suchw/=0.5,w/w=0.5,X/=3,=0.2, and=1. Fig.13(a) shows resonating curves especially for, with several peaks occurring near the integer values of. The larger primary resonance occur ataround 1. Increasing the breakwater height increases the reflection and lowers the transmission accordingly. The overall width of both breakwaters is 2w/=1, which is similar to the single breakwater presented in Fig.9. However, a much better performance is observed by using two breakwaters separated by a distance than when using a single breakwater. By using a system of dual breakwaters, the reflections increase, transmissions decrease, and extent of the spectrums ofandbecomes much larger and wider, suggesting better protection and hence, better performance. In Fig.13(b), the variations ofgive an indication of the amount of the wave energy dissipated when using two breakwaters separated by a distance. The energy dissipated becomes largest when the breakwater height is increased. This is evident since for an increased structural height the amount of pores of the whole medium also increases, leading to enhanced dissipation. This is also noticed to be true forwhen using two breakwaters instead of a single breakwater, as shown in Fig.9(b). Here, it is also indicated that the variations ofsynchronize better withrather than with.

Fig.14 shows the variations in,,andfor different values ofw/. In this case, w/w=0.5,/=0.8,X/=3,=0.2, and=1. As indicated in Fig.14(a), the differences between the resonating curves ofandare fairly small, suggesting that the changes made inw/affectandonly slightly compared with the changes induced by/. These small changes are attributed to enlarged structural size leading to increased porosity and inevitably to an augmented dissipation. This can be clearly seen in Fig.14(b) for. The overall width of both breakwaters is analogous to the single breakwater of Fig.10. Comparing Fig.14 and Fig.10 confirms that using two porous breakwaters separated by a distance performs much better than using a single breakwater. This is due to increased wave dissipations leading to increased reflections and decreased transmissions. Again, the changes inseem to be in accordance more withrather than with.

Fig.13 Variations of Cr, Ct, and Ed with kd for various values of h/d.

Fig.14 Variations of Cr, Ct, and Ed with kd for various values of wb/d.

Fig.15 shows the variations of,, andwith respect toand different values of, and forw/=0.5,w/w=0.5,/=0.8,X/=3, and=1. As shown in Fig.15(a), the effects of increasing the porosity are shown to decrease, butvalues increase in the lower range ofand decrease otherwise. As shown in Fig.15(b), for nonporous breakwaters (=0), there is no wave energy dissipation, and the larger reflections and small transmissions are simply due to the presence of the structure in front of the incident waves. When the porosity is increased, the wave energy dissipation is also increased. Comparing theresultsofFig.11forasinglebreakwaterofsimilarwidthto the overall width of the dual breakwaters of Fig.15, using two breakwaters still performs better. This is because for the dual breakwater system, wave dissipation is increased leading to increased reflections and lower transmissions.

Fig.16 shows the variations of,, andwith respect tofor different values ofand forw/=0.5,w/w=0.5,/=0.8,X/=3, and=0.5. The results in Fig.16(a) clearly indicate that an increase in the friction coefficient increases the reflections and decreases the trans- missions. The wave energy dissipation in Fig.16(b) is also seen to increase with increasing. Comparison of Fig.16 and Fig.12 once again confirms that using two porous breakwaters separated by a distance perform much better than using a single breakwater.

Figs.17–20 show the effect of varying the distancebetween the breakwaters on,, andfor different values of/,w/,, and, respectively. The distance between the breakwaters is expressed in terms of twice the relative spacing 2X/. All features examined previously in Figs.13–16 can be confirmed here as well. Allandcurves are found to oscillate periodically with the increasing values of 2X/. The formation of the resonating peaks and troughs inandis due to a phase change in the various wave components (incident, reflected, and transmitted) interacting outside/inside the confined region between the dual porous breakwaters. The maxi- mums are due to the constructive interferences (in phase), whereas the minimums are due to the destructive interferences (out of phase).

Fig.15 Variations of Cr, Ct, and Ed with kd for various values of ε.

Fig.16 Variations of Cr, Ct, and Ed with kd for various values of f.

The resonant amplitudes ofare larger than those of. Resonance increases with the increase of/,w/, and, as shown in Figs.17, 18, and 20, respectively. On the contrary, resonance decreases with increasing values of, as shown in Fig.19.The resonant peaks ofare seen to occur around the integer values of 2X/,, 2X/≈(=1, 2, 3, ···), corresponding to whenis minimum, and the resonant troughs ofoccur at 2X/≈0.5+(=0, 1, 2, 3, ···), corresponding to whenis maximum. On the other hand, the maxima and minima inare noticed to shift a little further than their counterparts in. For practical applications when designing breakwaters, only thefirst resonant mode is of interest. Accordingly, the relative spacing and porosity must be selected carefully to achieve suitable coefficients of reflection and transmission and to accomplish better sheltering.

Fig.17 Variations of Cr, Ct, and Ed with 2XS/L for different values of h/d.

Fig.18 Variations of Cr, Ct, and Ed with 2XS/L for different values of wb/d.

Fig.19 Variations of Cr, Ct, and Ed with 2XS/L for different values of ε.

Fig.20 Variations of Cr, Ct, and Ed with 2XS/L for different values of f.

6 Conclusions

In this paper, to model the interaction of normal regular waves with submerged breakwaters, a potential flow theory was employed, in conjunction with the widely known model of Sollitt and Cross (1972) representing fluid flows in porous media. Both single and dual trapezoidal breakwaters were investigated. A domain decomposition technique discretization was carried out along the boundaries of the domains, and the solution was approximated with the ISBM meshless numerical method. The resulting over-determined coupled system of algebraic equations was solved by a least-square solution procedure. The results were presented in terms of hydrodynamic quantities of re- flection, transmission, and wave energy dissipation. The correctness of the present method was confirmed by comparing the present results with previously published data of other methods, including the EFEM, BEM, RMM, and experimental investigations. The effects of major design parameters, including the breakwater’s height and width,porosity, friction coefficient, and spacing, were investigated for several wave conditions.

The results demonstrate that in places where only partial protection from waves is required (allowing for some transmitted waves), submerged breakwaters can be used successfully, since they can substantially attenuate the incident waves. Dual breakwaters are found to perform better than single breakwaters. The reflections and wave energy dissipation increase, and the transmissions decrease with the increase in submergence depth, breakwater width, and friction coefficient. Nevertheless, the effects of the width were found to be limited. On the other hand, the effects of increasing the porosity are shown to decrease both reflections and transmissions with an increase in the energy dissipation. The reflection, transmission, and energy dissipation coefficients vary periodically with the spacing relative to the wavelength. The highest values of the reflection coefficients occur at integer values twice the relative spacing to the wavelength (2X/≈with=1, 2, 3, ···), corresponding to when energy dissipation is minimum. On the other hand, the lowest values of the reflection coefficients take place when 2X/≈0.5+with=0, 1, 2, 3, ···, corresponding to when energy dissipation is maximum. The maxima and minima inare seen to shift slightly to the right with respect to their counterparts in.

Acknowledgement

This paper is supported by the Ministry of Higher Education and Scientific Research of Algeria (grant PRFU number A01L06UN310220200002).

Appendix

The equation of the flow motion in a porous medium may be written such as in Sollitt and Cross (1972), and Sulisz (1985):

whereis the angular frequency of the regular waves, andis the linearized friction coefficient. The Lorentz condition of equivalent work is used to evaluate. This principle states that the average rate of energy dissipation during one wave periodand over the entire volumeof the porous medium should be identical, whether evaluated using the true nonlinear resistance law or its linearized equivalent. Considering thatremains constant within the volumeand over the period,

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January 5, 2021

© Ocean University of China, Science Press and Springer-Verlag GmbH Germany 2021

. E-mail: chioukh.nadji@hotmail.com

(Edited by Xie Jun)