Reflection and transmission of regular waves from/through single and double perforated thin walls

In this paper, reflection and transmission coefficients of regular waves from/through perforated thin walls are investigated. Small scale laboratory tests have been performed in a wave flume firstly with single perforated thin Plexiglas plates of various porosities. The plate is placed perpendicular to the flume with the height from the flume bottom to the position above water surface. With this thin wall in the flume wave overtopping is prohibited and incident waves are able to transmit. The porosities of the walls are achieved by perforating the plates with circular holes. Model settings with double perforated walls parallel to each other forming so called chamber system, have been also examined. Several parameters have been used for correlating the laboratory tests’ results. Experimental data are also compared with results from the numerical model by applying the multi-domain boundary element method (MDBEM) with linear wave theory. Wave energy dissipation due to the perforations of the thin wall has been represented by a simple yet effective porosity parameter in the model. The numerical model with the MDBEM has been further validated against the previously published data.


Introduction
Seawalls are coastal structures commonly used to protect shores form actions of waves and currents. Vertical seawalls are highly preferred because of less space, materials and construction time. However, due to their vertical and impermeable front surface, strong wave reflections may occur, leading to large disturbances nearby. When the actions are severe, strong scouring may occur at the toes of the seawalls sometime. These structures can be made porous for attenuating part of the incident wave energy in order to decrease scouring. Moreover, the water level oscillations in front of perforated sea walls can be dropped with less disturbances to coastal water areas. Wave run-up also can be decreased by reducing the wave overtopping and spray.
Experimental and theoretical studies on the vertical perforated seawalls are not scarce. A good review of hydraulic performance and wave loadings on perforated/slotted single and multiple vertical seawalls is given by Huang et al. (2011). They reported that some of the earliest experimental and analytical studies were devoted to calculating reflection and transmission coefficients of vertical thin walls with different heights in infinite water depth, and in order to reduce reflection and transmission of waves from seawalls, porous breakwaters (perforated or slotted) were proposed. Additional information can be referred to Porter and Evans (1995), and Rageh and Koraim (2010).
Some other structures are also added to the sea wall to further reduce wave reflections and transmissions. For example, a perforated vertical wall is added in front of the impermeable sea wall forming a wave absorbing chamber (Jarlan type breakwater). Many experimental and theoretical studies have been performed for reducing wave reflections and transmissions from/through multiple (two or more) fully or partially perforated/slotted vertical seawalls (e.g. Das et al., 1997;Bergmann and Oumeraci, 1999;Huang, 2007;Liu and Li, 2011;Liu et al., 2014). The result of these studies indicates that the porosity of the front wall and the ratio of the wave chamber width to the wavelength are primary parameters related to wave reflection and transmission. Modifications to the original Jarlan type breakwater are also investigated in order to promote further reduction of reflection and transmission coefficients. For example, Isaacson et al. (2000) studied analytically the performance of a fully perforated Jarlan type structure with a surface piercing rock core. Yip and Chwang (2000) modified the Jarlan type breakwater with a submerged horizontal solid plate in the chamber between the fully perforated front wall and the impermeable rear wall. Liu et al. (2007) proposed a similar Jarlan-type breakwater but with an internal submerged horizontal porous plate. Koraim et al. (2014) suggested to put a submerged breakwater in front of a seawall consisting of a front steel screen, back solid wall, and filled rock-core. Recently He and Huang (2016) have investigated an oscillating water column (OWC) in front of a vertical impermeable seawall. They found that both wave energy extraction and reduction of wave reflection could be efficiently achieved with this kind of structures.
In this paper experimental results of reflection and transmission coefficients against single and double walls with various porosities in regular waves are presented. The experimental data are also compared with the model results with the MDBEM. Seawalls with the height larger than the water depth are investigated experimentally, although the numerical model with the MDBEM is developed for truncated walls.

Model set-up and experimental tests
The experiments were carried out in the wave flume of the Hydraulics and Coastal Engineering Laboratory of the Civil Engineering Department at Yildiz Technical University. The test set up is shown in Fig. 1. The flume is 20 m long, 1 m wide and 1 m deep. The flume is equipped with a regular/irregular wave generator and HR (DAQ) system software for wave data acquisition and analysis. A displacement piston wave maker is used to generate sinusoidal and random waves. The wave maker measures the incoming wave, and in the mean time it corrects the paddle motion to absorb the reflected wave. The generated wave is predictable even with highly reflective models.
The vertical perforated walls were represented by Plexiglas plates with the thickness of 15 mm in the experiments. They were placed 14.5 m away from the wave generator. The porosity of the wall was represented by circular holes with porosities in order of 20%, 26% and 40%. The distance between the axes of each perforation wall was kept constant. The porosity (P) is defined as the ratio between the area of one single circular hole (A=πD 2 /4) and the area of a square (e 2 ) as shown in Fig.2a, i.e. P=A/e 2 . The diameters of the circular holes were 5.2, 5.9 and 8.5 cm for porosities of 20%, 26% and 40%, respectively, as shown in Fig. 2b. To prevent wave overtopping the crest of a wall was 70 cm above the bottom, with still water depth (d) of 33 cm for all experiments.
Water surface displacements were measured by parallel wire capacitance-type wave gauges to obtain the wave parameters. First one wave gauge (Probe 6) was placed two meters away from the wave generator to observe the incoming waves. To measure wave transmission, another wave gauge (Probe 1) was placed behind the walls. The reflected waves were measured by the other four wave gauges (Probes 2, 3, 4, 5) in front of the walls, as shown in Fig. 1. The reflected and incoming waves were separated by the tech-  Nadji CHIOUKH et al. China Ocean Eng., 2017, Vol. 31, No. 4, P. 466-475 467 nique of Wallingford. The incident wave heights varied between 3.5 cm and 17.6 cm, with wave periods in the range 0.8-2.2 s. H i , H r , and H t are used to represent the incident, the reflected, and the transmitted waves. The coefficients of reflection and transmission are calculated respectively as C r =H r /H i and C t =H t /H i .

Governing equations and boundary conditions
The idealised geometry of the two-dimensional wavestructure interaction problem is shown in Fig. 3. Regular waves with small amplitude a, period T and wavelength L, propagate from the left in water depth d. By assuming an irrotational flow and incompressible fluid motion, the problem is formulated with a velocity potential Φ(x, y, t) =Re , in which Re denotes the real part, is the time independent spatial velocity potential, , is the wave angular frequency, and t is time. The wave number is the solution of the dispersion relation , where g is the gravitational acceleration. The wave field can be solved if is known. Two porous walls of height W H separated by a distance B and extending from above the water surface to a certain distance above the channel floor are placed perpendicular to the direction of the wave propagation, as shown in Fig. 3. With this disposition the fluid domain is divided in three regions. Region (1) is in front of the left wall (seaward side), Region (3) is behind the right wall (leeward side), and Region (2) is between the two walls. With sufficient porosity of the walls the incoming waves are transmitted to the leeward side. The flow in each region could still be described by a velocity potential under the assumptions of the linear wave theory. Special matching conditions at the interfaces of the flow regions would ensure a smooth transfer of the mass flow from one region to the next. In this study the perforated walls are treated as solid homogeneous porous materials.
The spatial velocity potential in each region (j) satisfies the following boundary conditions: ∂y 2 = 0 fluid region j = 1, 2, 3; (1) = 0 j = 1, 2, 3 and y = 0 (free surface boundary Γ 4 ); (2) where n is the normal to the boundary pointing out of a flow region, and φ I is the incident velocity potential defined as: The radiation condition is treated by using far field potentials at two fictitious vertical boundaries at sufficiently large distances and representing the left boundary of Region (1) and the right boundary of Region (3), respectively. This ensures that the disturbances must be outgoing waves only (Yueh and Chuang, 2012): where and are unknown complex coefficients to be determined.
Moreover, at the interfaces between Regions (1) and (2), and between Regions (2) and (3), special matching conditions are imposed, see for example Liu and Li (2011). Along the fictitious boundaries and , continuity requires that: For the porous walls (boundaries and respectively relating to the back and front sides of each porous wall), the boundary condition for slotted/perforated walls of Yu (1995) is imposed: where G is a dimensionless complex quantity. The superscripts 1 and 2 indicate the front and back wall, respectively. This quantity is known as the porous effect parameter and can be evaluated by different methods (Huang et al., 2011). The method of Yu (1995) due to its simplicity is adopted in this study: where δ is the thickness of the wall, f is the linearized resistance (friction) coefficient, is an inertia coefficient, and θ is the argument of the complex part of G. For thin perforated walls the argument θ, which is associated with the added mass (C m ), is usually not significant and can be negligible (θ≈0 and C m ≈0). Hence it is customary to use s=1. The linearized friction coefficient f is estimated from the empirical relation of Li et al. (2006), , which has been reported to work well in the range of . More details can be referred to Huang et al. (2011) and Liu and Li (2011).

MDBEM formulation and solution techniques
The physical problem described above is governed by the Laplace Eq. (1), and the boundary conditions given by Eqs.
(2), (3), (6), (7), and (8). This is a boundary value problem that is first transformed to integral equations using the Green's theorem, and then solved by the MDBEM. For smooth (constant) elements the general form of the integral equation is written as: This equation relates the potentials of the source points, , to and its normal derivative of the field points, , lying on a boundary of a flow region (j=1, 2, 3). It is well understood here that the index (p) is related to source points. For Region (1) the boundary , for Region (2) the boundary , and for Region (3) the boundary (see Fig. 3). Q is the free space fundamental solution of the Laplace equation. It depends only on the distance , and is given together with its normal derivative as: (∂r/∂n) The quantity defines the direction cosines of the normal to an element. (10) is applied consecutively to all source points in each fluid region (j). For the numerical implementation of the MDBEM the total boundary of each region is discretised into a number of N j elements (N 1 , N 2 , N 3 correspond to the number of elements-nodes respectively for Regions 1, 2, and 3). Variations of the variables over the elements are assumed to be constant and the unknowns are defined at the mid-elements nodes. The resulting discretised integral equations are written in the following general discrete forms: is the length of an element (m) part of the boundary , and δ pm is the Kronecker delta. The discretised form for a region (j) can be rewritten in a matrix form as: where and are complex coefficients involving integrals of the fundamental solution and its normal derivative along the elements of each flow region (j=1, 2, 3), e.g.
∆Γ m These coefficients relate a source point (p) to a field point (m) belonging to an element . The integrals in Eq. (14) are evaluated using Gaussian quadrature rules. The discretised system of Eq. (13) is further expanded to include all flow regions (j=1, 2, 3) in one system: Nadji CHIOUKH et al. China Ocean Eng., 2017, Vol. 31, No. 4, P. 466-475 469 where N is the total number of constant elements from all flow regions, e.g. .

∂φ/∂n
Finally the boundary conditions, expressed by Eqs.
(2), (3), (6), (7) and (8), are introduced into Eq. (15). The resulted algebraic system of equations is further rearranged such that all unknowns are moved to one side. It is then solved numerically using a Gaussian elimination algorithm to yield the vector of unknowns (velocity potentials φ (or ) and the coefficients and ). Reflection and transmission coefficients are determined from the following expressions: The wave energy equilibrium requires that: where is the incident wave energy, is the reflected wave energy, is the transmitted wave energy, and is the dissipated wave energy describing the portion of the incident wave dissipated by the perforated walls. H d is the loss in wave height corresponding to . By inserting the energy terms in Eq. (17) and dividing by , it leads to a similar nondimensional expression: where is the reflection coefficient, is the transmission coefficient, and is the wave height loss coefficient. The corresponding energy coefficients are, the reflected wave energy coefficient E r = , the transmitted wave energy coefficient , and the wave energy dissipation (loss) coefficient E d = . The wave energy loss coefficient E d is expressed in a non-dimensional form, (19) It is worth to mention that fully extended porous walls are treated simply by setting W H =d. For single porous wall breakwaters they are solved by setting the porosity of the front wall to be sufficiently large (G 1 =∞). By this way the front wall vanishes and the structure reduces to a single porous wall of porosity P 2 and porous parameter G 2 . In all subsequent computations a large value of G 1 =∞ has been taken as 10 8 .

Validation of the MDBEM wave model
To validate the present method, results of the boundary element formulation are compared against those of a number of cases.
The first case examined is the single impermeable partially immersed barrier of height W H . It has been studied by Porter and Evans (1995) by using a multi-term Galerkin method. Liu and Li (2011) also developed analytical solu-tions by means of the eigenfunction expansion method and a least square approach for such case. To simulate a single impermeable barrier with the MDBEM model in the present study the porosity parameters of front and back walls are set to (|G 1 |=∞) and (|G 2 |=0), respectively. Fig. 4 shows the comparison of reflection and transmission coefficients for W H /d=0.5. From Fig. 4 a good agreement between the three methods can be observed in the entire range of all wave numbers.
The second case examined is a structure of double partially immersed impermeable barriers (|G 1 |=|G 2 |=0) separated by a distance B. Das et al. (1997) obtained the reflection coefficients (C r ) by using a multi-term Galerkin method. Similarly Liu and Li (2011) presented analytical results of the same test case using the eigenfunction expansion method. As shown in Fig. 5 the results of the presented MD-BEM agree well with the results of two previous methods.
Model results of the MDBEM method are compared to representative data pertaining to wave period T=1 s of the present experiment. For a wave chamber with front wall porosity P 1 =40% and back wall porosity P 2 =26%, and with different values of the relative chamber gaps (B/L). By using an estimated value of the linearized friction coefficient f=5.5, the numerical results of C r and C t agree well with the experimental data. Fig. 4. Comparison of C r and C t with different methods for a single impermeable barrier (W H /d=0.5, |G 1 |=∞ and |G 2 |=0).

Results of single perforated walls
Relationships between coefficients C r and C t and wave steepness H i /L are shown in Fig. 7 for the three different single perforated walls used in the present experiments. Linear regression lines the data for each wall porosity are also shown in this figure. The variations of C r and C t appear to be almost linear in the range of the wave steepness investigated. It is obvious that C r increases with the wave steepness whereas C t decreases. This is attributed to larger incident waves inducing stronger velocity fields. Hence the flow resistance of the structure increases resulting in the reduction of the transmission coefficients and an increased reflection.
It can be found from Fig. 7 that for P=40% the variations of C r and C t with H i /L are almost constant. It is also observed that as the porosity increases, C r decreases and C t increases. As expected, the reflection in front of the wall is found to be the largest for the 20% porosity and the smallest for the 40% porosity. Transmission on the other hand shows the reverse trend. In addition, the discrepancies between the three curves appear to become larger when the wave steepness increases. It can be clearly seen that the differences of C r and C t in the range of porosities investigated (P=20% to P=40%), are around 10% at the lower range of H i /L (kd=0.5) and 20% at the higher range of H i /L (kd=2). This is in accordance with experimental and analytical results of Li et al. (2006) from single thin slotted walls.
Influence of the structure porosity on C r and C t could be represented by the analytical relations of Yu (1995) for thin porous walls extending from the free surface to the bed: where G is the porosity parameter given by Eq. (9), which is function of the porosity P, the wave number k, the friction coefficient f, the plate thickness δ and the inertia coefficient s. In Fig. 8, C r and C t calculated by the analytical expressions of Eq. (20) are shown against the present experimental data and the MDBEM model results. The linear regression lines of the data are also shown in the same figure. The general trends of C r and C t appear to be well represented by the MDBEM method and the analytical expressions for the three porosities investigated. All MDBEM computations and analytical calculations were performed by using the friction coefficient f=5.5 with the wall thickness of 15 mm and the water depth of 33cm. It can be observed from Fig. 8 that the MDBEM model slightly over predicts C r and underpredicts C t . This means that the flow resistance due to the presence of the structure has been slightly overestimated. The analytical results agree well with the experimental data and the MDBEM results, indicating that C r and C t are correctly influenced by the porosity. Bergmann and Oumeraci (1999) have expressed the performance of the single perforated walls in terms of a socalled Reflection Parameter (RP) given by the following relation: The results of their experiments on four single perforated walls with different porosities (11%, 20%, 26% and 40.5%) confirm the existence of a high correlation between RP and reflected, transmitted and dissipated wave energies. The maximum wave energy dissipation is observed at values of RP between 3 and 4 and reaches about 50% of the incident wave energy.
In the present work a slightly modified reflection parameter is recommended: Fig. 6. Comparison of experimental and numerical data of C r and C t versus B/L. Wave chamber with front wall porosity P 1 =40% and back wall porosity P 2 =26%, and wave period T=1 s. Fig. 7. Variations of the experimental values of C r and C t versus the wave steepness for single perforated walls with different wall porosities.
Nadji CHIOUKH et al. China Ocean Eng., 2017, Vol. 31, No. 4, P. 466-475 471 The correlation between RP of Eq. (22) and reflected, transmitted and dissipated wave energies for the present three perforated walls are shown in Fig. 9. It can be clearly seen from Fig. 9 that reflection and transmission energies correlate well with the recommended RP. The wave energy dissipation rises with the increase of RP, and it reaches the maximum around RP=3, i.e. .
The correlation has also been made between the dissipated wave energy, derived from Eq. (18) and expressed here as (H d ) 2 =(H i ) 2 -(H r ) 2 -(H t ) 2 , and the Reynolds number of the flow through the perforated wall pores, . U is the maximum horizontal component of the orbital velocity, D is the diameter of the pores, and ν is the kinematic viscosity. Under shallow water condition, as it is for the present case, the maximum horizontal orbital velocity can be approximated according to the small amplitude wave theory, e.g. U= . The relationships between the energy dissipation and Reynolds number based on the experimental and the MDBEM results are shown in Fig. 10 for three porosities. It can be observed from Fig. 10 that the dissipated wave energy rises as Re increases. Effects of the porosity P are noticeable. For a particular Re number, the dissipated wave energy is larger for the smaller porosity. The Reynolds parameter, Re/P, also named as "Porosity Scaled Reynolds Parameter" is proposed in this study to remove the porosity effects. The dissipated energy is plotted against the Reynolds parameter for experimental and model results in Fig. 11. Besides the big accordance between the experiments and the predictions, apparently much of the scattering in the data disappears.

Results of double walls (single wave chambers)
In this study, variations of C r and C t of single wave chambers with one front perforated wall and one back perforated or impermeable wall are also presented. Effects of the spacing B between the walls (relative gap ratio B/L) are  also examined. To optimize B, both the reflection and transmission coefficients, C r and C t , were obtained for different relative gap ratios (B/L). For a constant incident wave length (L), the gaps between the porous walls were changed systematically, such that B was taken as 0.1L, 0.2L, …, and L. Variations of C r and C t are shown in Fig. 12, for porosities P 1 =40% and P 2 =26% of the front and back walls, respectively. Effects of the wave period and the predictions from the MDBEM model are also shown in the figure. From Fig. 12 it can be clearly observed that C r oscillates with the minimum values occurring at B/L≈0.25 and 0.75, and the maximum values at B/L≈0.0, 0.5, and 1.0. This agrees with results of earlier studies such as wave reflection on breakwaters of double slotted barriers (Huang, 2007). The experimental values are within the envelope of the MDBEM model predictions. Effects of the wave period are not clear from the experimental results, but the numerical results of the MD-BEM model indicate that C r decreases with the increasing wave period. Variations of C t are seen to be similar to those of C r , but the amplitudes of oscillations are smaller. The numerical results of the MDBEM model predictions agree well with the experimental data. The numerical results reveal that effects of the increasing wave periods tend to increase C t values. Due to construction limitations of the chamber width practically, the best distance between the perforated walls should be taken as the smallest distance between the walls leading to minimum values of C r and C t , i.e. B/L≈0.25 (see Fig. 12). To determine the optimum chamber width B, the value of L should be taken as the predominant incident wave length. At the minimum distance B/L≈0.25, the overall wave energy loss becomes the largest leading to the minimum values of the coefficients C r and C t . This can be clearly interpreted by the simple relation given in Eq. (19).
Variation of C r with B/L was also investigated for single wave chambers with a perforated front wall and an impermeable back wall (P 2 =0%). In this case C t =0. Three different porosities (P 1 =20%, 26% and 40%) of the perforated front wall were investigated. In this paper only results per-   Fig. 12. Variations of C r and C t versus B/L. Wave chamber with front wall porosity P 1 =40% (a) and back wall porosity P 2 =26% (b).
Nadji CHIOUKH et al. China Ocean Eng., 2017, Vol. 31, No. 4, P. 466-475 473 taining to P 1 =20% and 40% are shown. Since in practice B/L is usually smaller than 0.5, the results only for B/L smaller than 0.5 are shown in Fig. 13 together with the predictions of the MDBEM model. Apparently C r reaches the minimum value at B/L=0.2-0.25. This minimum value increases with the increase of the porosity. Furthermore the general trends of the experimental values are adequately re-produced by the MDBEM predictions. It is clear that C r rises as the wave period increases. The predictions of C r are larger than the experimental values. It seems that wave energy dissipation is overestimated by the present method, leading to an over prediction of the reflection coefficients.
Apparently this discrepancy has decreased by using a slightly lower value of the linearised friction coefficient (f).
Effect of wave height variation on the wave reflections from the single wave chamber is also investigated. The results of C r against H i /d are shown in Fig. 14 for B/L=0.1, 0.2, 0.3, and 0.4, for front wall porosity P 1 =20% and P 1 =40%. From Fig. 14 it can be noticed that the variation of C r with H i /d is small. It is suggested that the wave height does not have a predominant influence. The scatter is due mainly to the data with different wave lengths, owing to the limitation of experimental mechanism (not being able to fix the wave height and vary the wave length, or vice versa).

Conclusions
Based on small scale experimental model tests and a MDBEM numerical model, wave reflections and transmissions from/through single and double perforated walls in regular waves are investigated in this study. The MDBEM model is first validated against data of well known test cases. Numerical model results are also compared with the experimental data , and a good agreement is found. It is concluded that with suitable friction and added mass coefficients, the present numerical model could be used with confidence.
In general, the performance of a single wall is affected by both the porosity and the wave steepness. The reflection coefficient increases with the increase of the wave steepness and decreases with the increase of the porosity. The transmission coefficient shows a reverse trend. The relationship between reflected, transmitted, and dissipated wave en-ergies is well described by the reflection parameter RP obtained from the experiment. The wave energy dissipation increases with the rise of the reflection parameter, and reaches the maximum (50% of the incident wave energy) at RP=3. Similarly wave energy dissipation increases with the "Porosity Scaled Reynolds" parameter, Re/P.
For double perforated walls the performance is primarily determined by the relative chamber width besides wall porosities and wave periods. The wave height seems to have little influence. The numerical results indicate that for a particular constant relative chamber width, effects of increasing the wave periods are to decrease reflection and increase transmission. When the back wall of the chamber is impermeable, the reflection coefficients increase with the increase of the wave period. The double perforated walls seem to be a better option than single perforated walls. The optimum chamber width could be easily calculated based on  B/L=0.2-0.25, if the predominant incident wave length L is known. Under such condition, wave energy dissipation is the largest leading to the minimum values of reflection and transmission coefficients. With appropriate wall porosities for the chamber system reflection and transmission coefficients could be smaller than those from single porous wall.