Hydrodynamic Coefficient Investigation on a Partial Permeable Stepped Breakwater Under Regular Waves

YIN Zegao, ZHENG Zihan, YU Ning, and WANG Haojian

Hydrodynamic Coefficient Investigation on a Partial Permeable Stepped Breakwater Under Regular Waves

YIN Zegao1), 2), *, ZHENG Zihan1), YU Ning1), and WANG Haojian1)

1)Engineering College, Ocean University of China, Qingdao 266100, China 2) Shandong Province Key Laboratory of Ocean Engineering, Ocean University of China, Qingdao 266100, China

Traditional breakwater takes the advantage of high protection performance and has been widely used. However, it con- tributes to high wave reflection in the seaside direction and poor water exchange capacity between open seawater and an inside har- bor. Consequently, a partially permeable stepped breakwater (PPSB) is proposed to ensure safety and good water exchange capacity for an inside harbor, and a 3-D computational fluid dynamics (CFD) mathematical model was used to investigate the hydrodynamic coefficients using Reynolds-Averaged Navier-Stokes equations, Re-Normalization Group (RNG)-equations, and the VOF tech- nique. A series of experiments are conducted to measure the wave heights for validating the mathematical model, and a series of di- mensionless parameters considering wave and PPSB effects were presented to assess their relationships with hydrodynamic coeffi- cients, respectively. With the increase in thereciprocal value of PPSB slope, incident wave steepness and permeable ratio below still water level (SWL), the wave reflection coefficientdecreases. The wave transmission coefficient decreases with an increase in thereciprocal value of the PPSB slope and incident wave steepness; however, it increases with the increase in the permeable ratio below SWL. With increases in the reciprocal value of the PPSB slope, permeable ratio below SWL and incident wave steepness for rela- tively high wave period scenarios, the wave energy dissipation coefficient increases; however, it decreases slightly with increases in the incident wave steepness for the smallest wave period scenarios. Furthermore, simple prediction formulas are conducted for pre- dicting the hydrodynamic coefficients and they are well validated with the related data.

regular waves; partially permeable stepped breakwater; wave reflection coefficient; wave transmission coefficient; wave energy dissipation coefficient; mathematical model

1 Introduction

The tremendous energy of extreme waves, such as tsu- namis, can cause serious damage to coastal areas. Tradi- tional breakwaters, such as a rock mound, play a signifi- cant role in wave energy dissipation and shoreline protec- tion, and considerable attention has been directed to the interaction of a breakwater with waves (Seelig and Ahrens, 1981; Davidson, 1996; Sumer, 2005; Lee and Mizutani, 2008; Rageh and Koraim, 2010; Postacchini, 2016; Aniel-Quiroga, 2019; Santos, 2019). However, traditional breakwaters cause a high wave re- flection in the seaside direction. Problems associated with wave reflection from the breakwater are well recognized, including dangerous conditions in harbors entrances, ship navigation, and intensified sediment scour. The stepped breakwater was investigated for the first time by Saville (1955, 1956, 1957) as a special coastal protection structurebecause of the high wave energy dissipation and smallrun-up. Okayasu(2003) identified that the wave overtopping rate for stepped seawalls was smaller than that for smooth seawalls experimentally, and the front face steps were beneficial to reducing the wave reflection. Similarly, the wave transmission and energy dissipation performance of a stepped submerged breakwater were bet- ter than that of the sloping submerged breakwater (Loke- sha, 2015; Liu, 2019). Teh and Ismail (2013) studied the hydrodynamic characteristics of a stepped-slopefloating breakwater with various wave parameters, and thetriple-row stepped-slope floating breakwater was an effec- tive energy dissipater. Yin(2017) numerically and experimentally investigated the wave reflection perform- ance of a stepped breakwater, and the results showed that the slope plays a dominant role in wave reflection. In addi- tion, Kerpen and Schlurmann (2016) summarized nearly of 60 years knowledge on stepped revetments, and the wave run-up, overtopping, and loads were addressed.

However, the water exchange capacity dramatically weak- ens between the inside harbor basin and outside sea to de- teriorate the inside water environment because of the hin- dering effect by traditional breakwater. Transport of a pol- lution source in the harbor basin with traditional break- water protection is difficult to the outside sea because of weakening seawater circulation, and a well-mixed condi- tion of the inside harbor basin and outside sea is no longer maintained, even in the spring tide. Therefore, attention in recent decades has been drawn to a permeable breakwater because of its relatively high wave energy dissipation and good water exchange capacity. First proposed by Jarlan (1961), and hydrodynamic performance of a permeable breakwater was examined by the behavior of the incident and reflection waves (Quinn, 1972; Kondo, 1979). Fugazza and Natale (1992) proposed a design formula to optimize the Jarlan breakwater, and the related experimental data were used to validate the formula. Rageh and Koraim (2010) experimentally and theoretically explored the wave transmission, reflection, and energy dissipation perform- ance of a permeable wall with regular waves. Tanimoto (1982) derived the head loss coefficient and effective in- ertia length to analytically calculate the reflection coeffi- cient, and the optimum size was discussed with the mini- mum reflection coefficient. Teng(2004) divided the fluid domain to develop an analytic solution for the inci- dent wave interaction with perforated caissons, continuous conditions of water velocity were satisfied at the front walls of the caissons, and the reflection coefficient agreed well with the energy conservation law using infinite po- rous assumption. George and Cho (2019) theoretically studied the wave interaction problem with a vertical slot- ted breakwater, consisting of impermeable upper and lower parts, and a permeable middle part. Li(2020) ex- amined the hydroelastic interaction between water waves and a flexible submerged perforated semi-circular break- water using linear potential theory.

These investigations are essential for assessing the hy- drodynamic characteristics of stepped or permeable break- waters. For full use of their advantages and to avoid their disadvantages, their combination is proposed as a par- tially permeable stepped breakwater (PPSB) for relatively high protection performance and good exchange capabil- ity between water inside and outside the harbor. A series of horizontal permeable holes are drilled in the underpart to promote water exchange between the inside harbor basin and outside seawater. At its upperpart, impermeable steps are used to dissipate the extreme wave energy and provide safety to the inside harbor basin. In this paper, the hydro- dynamic coefficients of the PPSB are investigated numeri- cally and experimentally to improve the general understanding. The manuscript is organized as follows: the math- ematical model and experimental tests are described in Section 2, and Section 3 is devoted to the hydrodynamic coefficients relationships with the structure size and wave parameters, and the dimensional homogeneity theory and the least squares method are used to conduct several sim- ple formulas for predicting the hydrodynamic coefficients of PPSB. The conclusions are summarized in Section 4.

2 Methods

This section introduces a 3-D CFD mathematical model in FLOW-3D software (Flow Science, N.M., USA), in- cluding the governing equations and boundary conditions. In addition, the discretization errors of grid partition are depicted in the numerical results. Eventually, the physical experimental setup and procedure are described for a hy- drodynamic performance investigation of a PPSB.

2.1 Governing Equations and Boundary Conditions

FLOW-3D is a full-featured commercial software that does not require additional grid generation modules or post-processing modules. A fully integrated graphical user interface allows fast completion of the simulation project setting to result output. In this paper, a numerical wave flume is conducted in the commercial CFD software FLOW-3D 10.0, and used for examining the hydrody- namic coefficients of the PPSB. The models based on the Reynolds-Averaged Navier-Stokes (RANS) approxima- tion provide an acceptable approach to study the hydro- dynamic behavior for engineering purposes with reason- able computation efforts and simple assumptions (Garcia, 2004). The continuity and momentum equations are denoted as follows,

As a simple turbulent model without major adjustments to constants or functions, the Re-Normalization Group (RNG)-model is proven to be able to predict complex flow behavior successfully (Yakhot, 1992). The governing equations are written as follows,

whereis the turbulent kinetic energy,is the turbulent dissipation rate,vis the turbulent viscosity andv=C·2/.C,C1,C2,σ, andσare the model coefficients.

Therefore, the RANS equations and RNGequations are used to model wave motions and wave turbulence. The volume of fluid (VOF) method presented by Hirt and Nichols (1981) is used for tracking the wave surface and the governing equation can be expressed as follows,

whereis the volume fraction of water in a cell,=1 means the cell is full of water,=0 means the cell is full of air, and 0<<1 denotes the cell contains a free boundary.

Fig.1(a) shows the side view of the boundary conditions for the mathematical model, where the width, height, and length of the wave flume are 1.0m, 1.2m, and 30.0m, respectively, consistent with the following experimental wave flume. The horizontal, vertical, and transverse coordinates are,, anddirections, respectively, and the origin pointis at the intersection location between the balance location of the wave generator and horizontal straight line through the flume bottom center. The PPSB height is 0.8m, consisting of eight steps. For a single step, its heightand widthare both 0.1m. At the upper side of the wave flume, AGis specified as atmospheric pres- sure. At AB, the wave boundary is utilized to generate the desired regular waves. A no slip wall boundary is ap- plied on the bottom wall of BC, the two-side vertical walls of the wave flume and the walls of PPSB. At the rear area of flume CEFG, the porous material with a porosity of 0.8 is set to absorb the transmitted waves through PPSB, minimizing unwanted wave reflection, and it is proven to be able to perform satisfactorily in wave absorbing. The still water depthis 0.4m for all the following experimental and numerical scenarios. Fig.1(b) shows the 3D model of the PPSB with eight horizontal steps. The PPSB size in the transverse directionis equal to the flume width andis the diameter of circular holes.

Fig.1 Experimental sketch and boundary conditions for the mathematical model: (a) side view, (b) 3-D shape of PPSB in the mathematical model, and (c) photo of the experimental PPSB.

2.2 Discretization Error Analysis

To examine the grid independency on the numerical results, the extrapolated relative error (ERE) and grid con- vergence index (GCI) are calculated to estimate the dis- cretization error (Celik, 2008).

For a three-dimensional mathematical model, a typical grid with the size ofis

where ?Vis the volume of theth grid andis the total number of grids.

Grid numbers with 7200000, 900000, and 120000 and corresponding gird heights of1=0.005m,2=0.01m, and3=0.02m are used for the relatively far field, respec- tively, and the local grids are refined near the wave sur- face and PPSB to ensure high accuracy of the numerical results, such as good tracking performance on the wave surface elevation.

GCI is defined as follows,

where

where32=3?2,21=2?1, andrepresents the nu-merical solution on a grid. Sign is the sign function. The result will be monotone convergence or divergence for=1, and oscillation convergence for=?1,21=2/1, and32=3/2.

The extrapolated value is

ERE is

Table 1 shows the discretization errors for the velocities and pressure values at=10m,=0.2m, and=0m for=19.8s. The regular incident wave heightHis 0.1m and the wave periodis 2.1s. With an increase in the grid number from 900000 to 7200000, the GCI errors are as small as 0.8% for the velocities and 0.2% for the pressure values, respectively.

Fig.2 shows that slight variation occurs in the wave surface elevationbetween the Grid 1 and Grid 2 sce- narios, and a relatively largedifference appears between the Grid 2 and Grid 3 scenarios. The grid size of2=0.01m is relatively independent of the computational results, so Grid 2 is chosen for the following computation after considering the balance between numerical accuracy and computational speed.

Table 1 Analysis of discretization error

Note: ‘–’ means no values.

Fig.2 Wave surface elevation histories at x=10m location using three grids for a=b=0.1m, D=0.05m, d=0.4m, Hi=0.1m and T=2.1s.

2.3 Physical Experiments and Mathematical Model Validation

2.3.1 Physical experiments

To validate the aforementioned mathematical model, a series of physical experiments was conducted in a wave flume in the hydraulic lab of Ocean University of China. The length, width, and height of the wave flume are 30m, 1.0m, and 1.2m, respectively. At the left side of the flume, a piston type wave generator is used to generate the regu- lar waves by adjusting its operating frequency and stoke length. At the right side of the flume, porous material is used to absorb the wave, minimizing the unwanted wave reflection as shown in Fig.1(a). Fig.1(c) shows that the experimental model of PPSB is made by wood with eight steps, and its height is 0.8m, where=1.0m and=0.1m remained constant in the following experiments and nu- merical simulations. On each step at the four underpart steps, 10 circular holes with a diameter of=0.05m are created for permeability, and the distance of adjacent hole centers is 0.1m, and the vertical distance between the holes’ centers and their step top is 0.05m. Four wave gauges (with a sampling frequency of 50Hz and an accuracy of 0.3%), numbered with 1#, 2#, 3#, and 4#, are used to measure the wave height histories in the front and back of the PPSB. 1#, 2#, and 3# wave gauges are fixed in front of the PPSB to separate the incident and reflected wave heights from their records using the three-point method proposed by Zelt and Skjelbreia (1992). The distances be-tween 1# and 2#, 2# and 3# are determined based on the method of Zelt and Skjelbreia (1992). 4# is placed behind the device to measure the transmitted wave, and the dis- tance between the PPSB end and the 4# wave gauge is a constant of 1.0m. All of the wave gauges are calibrated before the experiments to ensure measurement accuracy. The side view of experiment is shown in Fig.1(a).

The breakwater location was determined to ensure ac- tion with a fully developed wave and attenuate the secon- dary wave reflection effect on the device because of the wave paddle. Before each experiment, special care is taken to fix the end of PPSB at the location of=12.5m, and two iron bricks are placed on top of the PPSB to ensure stability under a wave effect. The wave flume is filled with fresh tap water until the still water depthreached 0.4m and the water temperature is 17℃. The wave generator starts to generate the regular waves. The incident wave periodvaries from 1.5s to 2.7s at a step of 0.3s, and the incident wave heightivaries from 0.08m to 0.16m at a step of 0.02m. Note that all of the wave parameters are consistent with the model limits reported by Hughes (1993) and Frostick(2011). The scenario summaries of nu- merical computations and physical experiments are shown in Table 2, where=0m denotes the vertical breakwater scenario without steps.

Table 2 Summary of experimental and numerical scenarios

Notes:=0.1m and=1.0m.

2.3.2 Mathematical model validation

Fig.3 shows the theoretical and numerical wave surface elevations without PPSB at=5m,=9m, and=15mlocations (i=0.1m,=1.5s). The numerical wave surface elevation profiles fit the theoretical data very well, show- ing that the aforementioned mathematical model is able to investigate the wave behavior in the flume. Additionally, a slight wave height attenuation occurs in the numerical val- ues because of the water viscosity effect.

Fig.4 shows the numerical wave surface elevations and experimental data with PPSB at=9.5m and=13.5m locations (==0.1m,i=0.1m,=1.8s and 2.4s). The numerical wave profiles fit the experimental data well. Their discrepancies are smaller than 15%, showing that the mathematical model is able to investigate the wave interactions with PPSB in the wave flume. Additionally, the comparatively gradual front slope and steeper back slope occurred in front of the PPSB, as shown in Figs.4(a) and (c), and the significant nonlinear characteristics ap- peared because of the complex effect of incident and re- flected waves, particularly at wave crests and troughs. In contrast, the wave surface elevation variation is obviously small behind the PPSB in Figs.4(b) and (d), and the geo- metrical nonlinear characteristics were reduced to a great extent.

It is expected that the wave breaking phenomenon of- ten occurs at the wave surface on PPSB. The wave break- ing is an important issue in coastal hydrodynamics, which has drawn a considerable attention (Jachowski, 1964). Ker- pen and Schlurmann (2016) demonstrated that when the step height ratio to wave height is greater than unity, the SWL position with respect to the step height plays a sig- nificant role in the wave breaking. For the given wave pa- rameters, the incident wave is prone to breaking with small, and the water level increases in front of the PPSB as a result. Interestingly, in the experimental scenarios with relatively largei, a number of air bubbles generated at the holes’ exit near the wave surface behind PPSB, and their effects on the hydrodynamic coefficients deserve further investigation as much as the mass transfer with water.

Fig.3 Numerical wave surface elevations with theoretical values for Hi =0.10m and T=1.5s: (a) x=5m, (b) x=9m, and (c) x=15m.

Fig.4 Numerical wave surface elevations with experimental values: (a) T=1.8s (at x=9.5m in front of PPSB), (b) T=1.8s (at x=13.5m behind PPSB), (c) T=2.4s (at x=9.5m in front of PPSB), and (d) T=2.4s (at x=13.5m behind PPSB).

3 Results and Discussion

It is well accepted that the wave reflection, transmission and energy dissipation behaviors play an important role in coastal structure design and engineering applications. The- oretically, the incident wave energy is equal to the sum of the reflected wave energy, the transmitted wave energy, and the dissipated wave energy because of the wall friction and turbulence caused by PPSB. In this section, the wave reflection coefficientr, the wave transmission co- efficientt, and the wave energy dissipation coefficientNare used to investigate their relationships with wave parameters and PPSB geometrical parameters. The vari- ables are denoted as

whererandtare the heights of wave reflection and transmission, respectively.

The three-point method is used to determine the inci- dent wave heightiand reflected wave heightrfrom composite waves records in front of the PPSB (Goda and Suzuki, 1976; Mansard and Funke, 1980; Zelt and Skjel- breia, 1992), and the wave height behind the PPSB is roughly regarded as the transmitted wave heightt. To investigate the hydrodynamic characteristics of PPSB, the aforementioned mathematical model is used to simulate its coupling behaviors with waves, and 375 groups of nu- merical scenarios are shown in Table 2. In addition, a se- ries of dimensionless parameters are produced as/,i/, andπ2/4to explore their effect on hydrody- namic coefficients, whereis the reciprocal value of a stepped breakwater slope representing the step effect of PPSB,i/is the incident wave steepness (is the inci- dent wave length), andπ2/4illustrates the ratio of permeable area to the cross section area below SWL, andis the number of holes below SWL.

3.1 The Hydrodynamic Coefficients Relationships with Slope Effect of PPSB

Fig.5 shows the hydrodynamic coefficients relationship with/for0.05mandi0.12m. With increasing/, bothrandtdecrease as shown in Figs.5(a) and (b), illustrating that the wave reflection and transmission ca- pabilities reduce. However,Nincreases as a result, as shown in Fig.5(c), showing that a higher wave energy dispassion occurs because of PPSB. A possible explana- tion is that with increasing/, the wall friction effect on waves, including the steps’ wall and the inner wall of holes, increases. The wave turbulence strengthens coupling with wave run-up and the breaking processes, and the wave energy dissipation increases consequently. In particular,the decreasing amplitude ofrandtand the increasing amplitude ofNare relatively high for/<1, illustrating that the step effect plays a significant role in hydrody- namic behaviors for a relatively steep PPSB. With increas- ing/for/≥1 scenarios, the variation amplitudes of hydrodynamic coefficients change slowly and tend to be gradually stable, especially forNvalues at the/=2.0 location. In addition,rvalues are often greater than thetvalues, showing that the transmitted wave energy through the PPSB is smaller than its reflected wave energy.

Fig.5 Relationships between the hydrodynamic coefficients of (a) Kr, (b) Kt, (c) KN, and b/a for D = 0.05m and Hi= 0.12m.

3.2 The Hydrodynamic Coefficients Relationships with Incident Wave Steepness

Fig.6 illustrates the hydrodynamic coefficients rela- tionships withi/for0.05mand0.10m. With increasingi/for a givenscenario,rdecreases; in contrast,tandNvalues increase, except for theNvalues of the smallest=1.5s scenarios. With decreasingfrom 2.7s to 1.5s, ther,t, andNsensitivities reduce approximately withi/variation. One possible reason is that with increasingifor a given, the wave nonlinear effect is enhanced and the contacting area increases with steps and holes, resulting in lower wave reflection, higher wave energy dissipation and wave transmission. In addi- tion, with increasingfor a giveni, the wave nonlinear action weakens, contributing to higher wave reflection, smaller wave energy dissipation, and wave transmission. Regarding the smallest=1.5s scenarios,the strongest nonlinear characteristics appear for the wave action with PPSB. Decreases ofrandNand an increase oftare relatively slight with the increase ofi/for givensce- narios. The incident wave steepness contributes little to the wave energy dissipation, and the PPSB geometry and size play a dominant role in the hydrodynamic coefficients in contrast. Furthermore, Fig.6(b) shows that a slight de- creasing tendency oftoccurs with increasingi/for all scattered data points, which is opposite thetrelationship withi/for a givenscenario.

Battjes (1974) derived the following expression for wave reflection coefficient:

where1=0.12.

Seelig and Ahrens (1981) indicated that the Battjes’s equation overestimated therover a wider range, and it can be rewritten as

where2=0.8 and1=10 for impermeable rock structures.

Postma (1989) suggested an alternative formula as fol- lows for rock impermeable slopes:

where3=0.17 and2=0.7.

Zanuttigh and Van der Meer (2006) investigated various wave reflection coefficient equations and proposed an em- pirical equation using the hyperbolic tangent function:

where4=0.12 and3=0.87 for impermeable rock struc- tures,4=0.14 and3=0.90 for permeable rock structures, and4=0.16 and3=1.43 for smooth structures.

Fig.7 compares thervalues of 50 randomly selected computational scenarios and the aforementioned formulas proposedbyBattjes(1974),Seelig and Ahrens(1981),Post- ma (1989) and Zanuttigh and Van der Meer (2006)The results indicate that with increasing,rincreases, con- sistent with the tendency of Eqs. (17)–(20)Thervalues in our study roughly agree with the Eqs. (18)–(20) for im-permeablerock structures. However, they are slightly small-er than Eq. (20) for smooth structures, and they are greater than Eq. (20) for permeable rock structures, illustrating that the combination of step and local permeability con- tributes to the wave reflection reduction for a slope break- water. In addition, thervalues in our study are smaller than Eq. (17) proposed by Battjes (1974). A possible ex- planation is that the effect of slope geometry, material and permeability was entirely ignored in Eq. (17); however, the complex step geometry and underpart permeable holes play a vital role in the wave reflection behavior.

Fig.6 Relationships between hydrodynamic coefficients of (a) Kr, (b) Kt, (c) KN, and Hi/L for D=0.05m and b=0.10m.

Fig.7 Kr relationship with ξ.

3.3 The Hydrodynamic Coefficients Relationships with Permeable Ratio Below SWL

Fig.8 illustrates the hydrodynamic coefficient relation- ship with the permeable ratio below SWLπ2/4fori0.12mand0.10m. With increasingπ2/4for a givenscenario,rdecreases as shown in Fig.8(a); how- ever, an approximate linear increase ofNandtappears as shown in Figs.8(b) and (c), illustrating a simultaneous decrease in the reflected wave energy and an increase in the transmitted wave energy and dissipated wave energy. With the increase in the permeable ratio, the water veloc- ity and flow rate increase through the holes of PPSB, the wall friction and water disturbance effects synchronously strengthen. As a result, the wave energy dissipation and transmitted water volume increase,Nandtincrease con- sequently, and thervalues decrease.

3.4 Prediction Formulas for Hydrodynamic Coefficients

It is advantageous to develop a series of simple predic- tion formulas forr,t, andNusingi,,,,, andparameters because of the significant contributions of the aforementioned wave and PPSB parameters tohydrody- namic coefficients. The formulas of the hydrodynamic co- efficients can be expressed as follows:

The application of dimensional homogeneity to Eqs. (21), (22), and (23) yields

Note that the least squares method has been extensively utilized to minimize the sum of the error squares (also called residual error) between the true value and predicted, and it was used for deducing the following prediction formulas based on 325 groups of numerical data randomly selected by the random function in MATLAB software (MATLAB 2019b, Math Works, M.A., USA).

The prediction formulas are

where=40 in our research. The main effects of inde- pendent variablesandare represented by/andπ2/4, respectively, rather than the effects of,, andbecause of the constants of,, andin our study, and their effect was neglected to simplify the research. The correlation coefficients of Eqs. (27), (28), and (29) are 0.89, 0.89, and 0.83, respectively. Subsequently, the ex- perimental data and the other 50 groups of numerical re- sults, Eqs. (14), (15), and (16) are utilized to determine ther,t, andNvalues for assessing Eqs. (27), (28), and (29) in Fig.9. All relative deviations ofrare less than 20%, and a large proportion of the relative deviations ofNare within 20%, except for three data points. The rela- tive deviations oftare relatively high because of their small values; however, a nearly 80% of the data points distribute in the ±20% error band. Consequently, Eqs. (27), (28), and (29) are acceptable for predicting the wave reflection, transmission and energy dissipation perform- ance because of PPSB under regular waves. In addition, the relative deviations of the numericalrandNvalues are generally smaller than those of the experimentalrdata; however, the relative deviations of numericaltval- ues are slightly greater than those of experimentaltdata.

Fig.8 Relationships between hydrodynamic coefficients of (a) Kr, (b) Kt, (c) KN, and βπD2/4cd for Hi=0.12m and b=0.10m.

Fig.9 Eqs. (27), (28) and (29) validation with experimental and numerical data using Eqs. (14), (15) and (16).

4 Conclusions

Based on a combination of the RANS equations, RNG-equations, and VOF technique, a 3-D CFD mathemati- cal model was conducted to investigate the hydrodynamic coefficients of PPSB for regular waves. A series of physical experiments were conducted to validate the mathematical model, and they exhibited good agreement. In addition, the PPSB size and wave parameters were considered to propose a series of dimensionless parameters for assess- ing their relationships with hydrodynamic coefficients. The results show that with increases in/, incident wave steepnessi/and permeable ratio below SWL, the wave reflection coefficientrdecreases, illustrating that a smaller wave reflection occurs for the higher wave steep- ness and permeable ratio scenarios with a gentler PPSB. The wave transmission coefficienttdecreases with in- creasing/andi/; however,tincreases with in- creasing incident wave steepness for a givenscenario and permeable ratio below SWL. The wave energy dissi- pation coefficientNincreases with increases of/, the permeable ratio below SWL, and incident wave steepness for relatively highvalues; however, it decreases slightly with increases of incident wave steepness for the smallest=1.5s scenarios in our study. In addition, the Iribarren number was used for estimating the wave reflection coef- ficientrof PPSB, and they agree roughly with the lit- erature reported by Postma (1989), Seelig and Ahrens (1981) and Zanuttigh and Van der Meer (2006) for impermeable rock structures. Furthermore, simple prediction formulas forr,t, andNwere conducted using the numerical data, the dimensional analysis and the least squares method, and they validated well with the related data.

The limitations of this study should be noted. First, re- gular waves were used to simplify our research, and the PPSB interaction with complex wave climates, such as ir- regular waves and tsunamis, should be investigated in fu- ture work. In addition, the fitting formulas were conducted based only on the given values for still water depth, single step height, PPSB width and holes’ number below SWL, and their direct effects on the wave characteristics deserve further examination.

Acknowledgements

The study was financed by the National Natural Science Foundation of China (Nos. 51879251 and 51579229), the Shandong Province Science and Technology DevelopmentPlan (No. 2017GHY15103), and the State Key Laboratory of Ocean Engineering, China (No. 1602).

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July 30, 2020;

December 8, 2020;

December 29, 2020

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

. E-mail: yinzegao@ouc.edu.cn

(Edited by Xie Jun)