water

Article Shock-Capturing Boussinesq Modelling of Broken Characteristics Near a Vertical

Weijie Liu 1,2 , Yue Ning 1, Yao Zhang 3,* and Jiandong Zhang 3

1 College, Zhejiang University, Zhoushan 316021, China; [email protected] (W.L.); [email protected] (Y.N.) 2 Key Laboratory of Coastal Disasters and Defense of Ministry of Education, Nanjing 210098, China 3 National Marine Hazard Mitigation Service, Ministry of Natural Resources of the People’s Republic of China, Beijing 100000, China; [email protected] * Correspondence: [email protected]; Tel.: +86-010-62492529

 Received: 1 November 2018; Accepted: 12 December 2018; Published: 19 December 2018 

Abstract: Broken wave characteristics in front of a vertical seawall were modeled and studied using a shock-capturing Boussinesq wave model FUNWAVE-TVD. Validation with the experimental data confirmed the capability of FUNWAVE-TVD in predicting the wave characteristics via the shock-capturing method. Compared to the results obtained from the Boussinesq model coupled with an empirical breaking model, the advantage of the present shock-capturing model for the broken near a vertical seawall was clearly revealed. A preliminary investigation of the effects of the key parameters, such as the incident , water level at the seawall, and slope, on the wave kinematics (i.e., the root mean square of the surface fluctuations and depth-averaged horizontal velocity) near the seawall was then conducted through a series of numerical experiments. The numerical results indicate the incident wave height and the water depth at the seawall are the important parameters in determining the magnitude of the wave kinematics, while the effect of the seabed slope seems to be insignificant. The role of the breaking point locations is also highlighted in this study, in which case further breaking can reduce the wave kinematics significantly for the coastal structures predominately subjected to broken waves.

Keywords: broken waves; vertical ; Boussinesq wave model; shock-capturing methods

1. Introduction The vertical seawall is one of the most widely-used coastal structures for shore and harbor protections from water wave actions. In general, there are three kinds of wave types forming in front of a vertical seawall, which are non-breaking standing waves, breaking waves and broken waves [1]. The non-breaking standing waves are the waves which do not break in front of a vertical seawall (Figure1a); the breaking waves are the waves which break above the foundation bed close to the seawall (Figure1b); the broken waves are the waves which break around antinodes at least a half wavelength from a vertical seawall on the sloping beach and usually form when the foundation bed is low or underground (Figure1c). Table1 shows the primary conditions to form these three wave types [1].

Water 2018, 10, 1876; doi:10.3390/w10121876 www.mdpi.com/journal/water Water 2018, 10, 1876 2 of 15 Water 2018, 10, x FOR PEER REVIEW 2 of 16

FigureFigure 1. 1. ThreeThree wave wave types types in in front front of ofvertical vertical seawalls seawalls [1]. [ 1(].a) non (a) non-breaking‐breaking standing standing waves; waves; (b) breaking(b) breaking waves; waves; (c) broken (c) broken waves. waves. Table 1. The conditions to form three wave types in front of a vertical seawall. Table 1. The conditions to form three wave types in front of a vertical seawall. The Size of Foundation-Bed Forming Conditions Wave Type The Size of Foundation‐Bed Forming Conditions Wave Type h1/hs ≥ 2/3 hs ≥ 2H Non-breaking standing waves h1/hs ≥ 2/3 hs ≥ 2H Non‐breaking standing waves hs < 2H, i ≤ 1/10 Broken waves 2/3 ≥ h 1/hs > 1/3 hs < 2Hh1, i≥ ≤ 1.81/10H Non-breakingBroken waves standing waves 2/3 ≥ h1/hs > 1/3 h1 ≥ h11.8

Water 2018, 10, 1876 3 of 15 storm surges, there is a dawning realization that broken wave loads may endanger the endurance of structures significantly due to their long durations [11]. Moreover, except for the wave loading, broken waves may extend over a broad region and the morphological change, especially the scouring in front of a vertical seawall which can affect the stability of the structure, caused by broken waves is reported to be more serious than that by the other two wave types [1]. Therefore, reliable predictive models and the study of broken wave characteristics near vertical seawalls still have important engineering significance for the design of the seawall subjected to the broken waves. There are vast ranges of existing models for predictions of nearshore wave hydrodynamics and these models may be mainly classified into three categories. The first category is based on energy balance equations (e.g., [12,13]), which demonstrate reasonable predictive skills of phase-averaged nearshore current field. However, these models are based on the assumptions of linear progressive waves and may not be applied to the wave field where waves have skewed and asymmetric profiles. Moreover, these kinds of models completely fail in modeling waves in front of structures due to the lack of wave reflection mechanism. The second category refers to a number of Computational (CFD) models (e.g., [14–16]), which directly solves the Navier–Stokes equations with certain turbulence models. Velocity filed can be directly obtained from these models and wave impact associated with wave-structure interactions can be estimated conveniently. However, these models are computationally expensive so far and may be still limited to small-scale phenomena, such as the breaking waves in front of seawalls. The third category mainly refers to the depth-integrated Boussinesq-type wave model, which employs a polynomial approximation to the vertical profile of velocity field, reducing the dimensions of a three-dimensional problem by one. This category may be regarded as somewhat between the above-mentioned two categories due to the efficiency and applicability of Boussinesq equations. Over the last decades, considerable efforts (e.g., [17–20]) have been made to improve the dispersive and nonlinear property of the classical Boussinesq equations [21]. Meanwhile, several empirical sub-models have also been proposed to accommodate wave breaking in surf zones. The most common approach is to employ an ad-hoc dissipation term to the momentum equation. There are two primary types of breaking models: (1) -viscosity type (e.g., [22,23]), (2) surface-roller type (e.g., [24–26]). Even though the two types stem from different ideas, their overall effect in momentum equation is similar. Incorporating energy dissipation into the time-domain Boussinesq-type model, the breaking-induced nearshore circulation can be also predicted by averaging the modeled wave velocity field [27]. Coupled with these breaking models, Boussinesq-type wave models have been widely applied to investigate various practical problems so far (e.g., [28–31]), however, the energy dissipation and breaking trigger mechanism of these widely-used breaking models are still limited to the assumptions of progressive waves, in which case the strong reflected wave components from structures may make the empirical breaking model invalid. In order to check the applicability of surface-roller and eddy-viscosity breaking models for broken waves near a vertical seawall, Liu and Tajima [32] applied an original image-based measuring system to wave flume experiments and experimental data of broken wave characteristics in front of a vertical seawall with high resolution both in time and space domain was collected for model validation. They found that both models tended to overestimate the energy dissipation due to the excess energy dissipation in reflected waves. Table2 summarizes the main properties of the three categories of existing models for predictions of nearshore wave hydrodynamics. Water 2018, 10, 1876 4 of 15

Table 2. Three categories of existing models for predictions of nearshore wave hydrodynamics.

Category No. Governing Equations Scope of Application Deficiency Phase-averaged nearshore Limited to the phenomena suitable 1 Energy-balanced equations current field for linear wave assumptions Velocity filed and wave impact Limited to the small-scale 2 Navier–Stokes equations associated with phenomena due to the high cost wave-structure interactions of computation Near shore wave processes, Limited to the progressive waves Boussinesq equations with 3 including shoaling, dissipation, due to the applicability of empirical an ad-hoc dissipation term diffraction, refraction and reflection breaking models

Recently, research efforts have been devoted to extending Boussinesq-type models to include shock-capturing capabilities for a more realistic account of wave breaking processes, creating a series of shock-capturing Boussinesq models (e.g., [33–36]). The main feature of a shock-capturing Boussinesq model is its ability of capturing breaking waves as shock waves by switching Boussinesq equations to nonlinear (NSWE) via disregarding dispersive terms when necessary. Wave breaking is thus treated in a more natural approach, since the energy dissipation due to wave breaking is predicted implicitly by the shock theory and does not require further parameterization based on empirical assumptions [36]. The shock-capturing Boussinesq models have been shown to be robust in predicting progressive wave processes in the nearshore, including shoaling, breaking, refraction, diffraction as well as wave run-up on the plane and natural beaches [35,37], while their capability in prediction of broken wave characteristics near vertical seawalls is still unclear so far. Thus, in light of the aforementioned works, this study utilized a well-known shock-capturing Boussinesq wave model, named FUNWAVE-TVD [35], to evaluate the ability of the shock-capturing method to predict the broken wave characteristics in front of a vertical seawall. Numerical results of the surface water profiles obtained from FUNWAVE-TVD, version 3.0 released in December 2016, will be validated with the laboratory measurements conducted by Liu and Tajima [32]. The validated numerical model was also applied to a preliminary investigation of the effects of the key parameters, such as the incident wave height, water level at the seawall, and seabed slope, on the wave kinematics under broken waves near the vertical seawall to give some insights into the design of the vertical seawall predominately subjected to broken waves. The remaining of this paper is organized as follows. In Section2, the governing equations and numerical methods of FUNWAVE-TVD are briefly introduced. In Section3, laboratory experiments conducted by Liu and Tajima [32] are reviewed and experimental results of one typical case are presented. In Section4, numerical simulations are validated with the experimental data to demonstrate the applicability of the present model. In Section5, numerical experiments are implemented to investigate the effects of key parameters on the wave kinematics near the seawall. Finally, in Section6, discussions about the numerical results and conclusions are summarized.

2. Numerical Model The shock-capturing Boussinesq wave model FUNWAVE-TVD [35] adopts the fully nonlinear Boussinesq equations of Chen [38], including the depth-integrated conservation equation and momentum equation: ηt + ∇ · M = 0 (1)

uα,t + (uα · ∇)uα + g∇η + V1 + V2 + V3 + R = 0 (2) where η is the surface elevation, the subscript t indicates partial derivatives with regard to time, ∇ is the horizontal gradient operator. The vector M is the horizontal volume flux expressed as

n ¯ o M = D uα + u2 (3) Water 2018, 10, 1876 5 of 15

where D = h + η denotes the total local water depth, h is the still water depth. uα represents the horizontal velocity at a reference elevation, where zα = ζh + βη with ζ = −0.53 and β = 0.47 [39]. u2 is the depth-dependent correction of velocity at o(µ2)(µ is the ratio of water depth to wave length) and written as 1   u (z) = (z − z)∇A + z2 − z2 ∇B (4) 2 α 2 α ¯ where A = ∇ · (huα) and B = ∇ · uα. u2 is the depth-averaged contribution to the horizontal velocity field given by ¯ 1 R η u2 = D −h u2(z)dz h 2 i h i (5) zα 1 2 2 1 = 2 − 6 (h − hη + η ) ∇B + zα + 2 (h − η) ∇A

V1 and V2 in Equation (2) are dispersive Boussinesq terms which are expressed as

 2   2  zα η V1 = ∇B + zα∇A − ∇ BtηAt (6) 2 t 2

 1   1  V = ∇ (z − η)(u · ∇)A + z2 − η2 (u · ∇)B + [A + ηB]2 (7) 2 α α 2 α α 2 2 The term V3 represents the vertical vorticity at o(µ ) and is written as

z ¯ z V3 = ω0i × u2 + ω2i × uα (8) where z ω0 = (∇ × uα) · i = vα,x − uα,y (9)

 ¯  z  ω2 = ∇ × u2 · i = zα,x Ay + zαBy − zα,y(Ax + zαBx) (10) in which iz is the unit vector in the z direction. The term R stands for diffusive and dissipate terms which include subgrid lateral turbulent mixing and bottom friction. The bottom friction in this study is calculated by a quadratic friction law incorporating a Manning coefficient:

gn2 R f = uα|uα| (11) (h + η)4/3 where n is the Manning coefficient. The value of the Manning coefficient can be found in standard textbooks of hydraulics or fluid mechanics for the commonly used surface materials. The main feature of a shock-capturing Boussinesq model is its numerical treatment of wave breaking as shock waves by disregarding dispersive terms. In FUNWAVE-TVD, the fully nonlinear Boussinesq equations of Chen [38] are well organized and reformulated in a well-balanced conservative form. Starting from the conservative form of the governing equations, a combined finite-volume and finite-difference shock-capturing scheme is implemented. Wave breaking treatment then follows the approach of Tonelli and Petti [40], who successfully used the shock-capturing ability of NSWE with a TVD solver to model moving hydraulic jumps. Comparing to the empirical breaking models, this treatment of wave breaking does not require several empirical parameters as breaking criteria based on progressive assumptions to tune the additional breaking model. Only the ratio of wave height to water depth ε is used to trigger wave breaking and all the dispersive terms are set to be zero when ε > 0.8. Notably, the present model with shock-capturing capability has been proved to be robust with no need of tuning ε over different bathymetries [35], including the sharply varying such as fringing reefs [37] where empirical parameters of traditional breaking models have to be tuned [41]. More details about the numerical scheme can be referred to Shi et al. [35]. Water 2018, 10, 1876 6 of 15

3.Water Laboratory 2018, 10, x FOR Experiments PEER REVIEW 6 of 16 Laboratory experiments were performed in the two-dimensional wave flume at the University of Laboratory experiments were performed in the two‐dimensional wave flume at the University Tokyo. The flume is 30 m long, 0.6 m wide, 1 m high and equipped with a piston-type wave maker as of Tokyo. The flume is 30 m long, 0.6 m wide, 1 m high and equipped with a piston‐type wave maker shown in Figure2. A solid vertical wall as a vertical seawall was installed on a 1:30 sloping bed, whose as shown in Figure 2. A solid vertical wall as a vertical seawall was installed on a 1:30 sloping bed, surface material was smooth steel. The toe of the sloping bed and the vertical seawall was located at whose surface material was smooth steel. The toe of the sloping bed and the vertical seawall was 12 m and 19.795 m from the wave maker. An original image-based measuring system was applied to located at 12 m and 19.795 m from the wave maker. An original image‐based measuring system was capture the water surface fluctuations in the side glass wall plane as high-resolution data sets both in applied to capture the water surface fluctuations in the side glass wall plane as high‐resolution data time and space domains. Two video cameras with high resolution capture of 1920 × 1080 pixels and sets both in time and space domains. Two video cameras with high resolution capture of 1920 × 1080 a frame rate of 30 fps were used to record successive still images of the instantaneous water surface pixels and a frame rate of 30 fps were used to record successive still images of the instantaneous water boundary along the cross section of about 2 m (two steel frames of the wave flume) near the seawall surface boundary along the cross section of about 2 m (two steel frames of the wave flume) near the where the water and background were colored in blue and yellow, respectively. Obtained images were seawall where the water and background were colored in blue and yellow, respectively. Obtained firstly rectified based on the eight reference points with actual XY-square-coordinates on the glass images were firstly rectified based on the eight reference points with actual XY‐square‐coordinates wall. Based on the RGB-values, which are the integer numbers stored in computers for indicating how on the glass wall. Based on the RGB‐values, which are the integer numbers stored in computers for much red, green, and blue is included in any human perceptional color, in each pixel, the surface water indicating how much red, green, and blue is included in any human perceptional color, in each pixel, boundary was detected through the following parameter A: the surface water boundary was detected through the following parameter A:

AA==RGBR+ G − B (12)(12)

For the present laboratory experiments, the yellow background should have large values of both R andand GGand and small smallB Bsince since yellow yellow is is created created by by adding adding red red to green to green and and the bluethe blue water water must must have largehave Blargeand B small and smallR and RG and. Therefore, G. Therefore, the parameter the parameterA computed A computed by Equation by Equation (12) should (12) should have larger have valueslarger onvalues the yellowon the backgroundyellow background and smaller and valuessmaller on values the blue on the water blue and water decrease and decrease abruptly atabruptly the air-water at the boundary.air‐water boundary. In the laboratory In the laboratory experiments, experiments,A was always A was greater always than greater 200 on than the yellow 200 on background the yellow whilebackground less than while 100 less on thethan blue 100 wateron the evenblue water on the even breaking on the and breaking broken and wave broken water wave where water the where water colorthe water tended color to tended be brighter to be than brighter the deeper than the water. deeper Therefore, water. Therefore, a single critical a single value criticalA = value 150 was A = used 150 towas determine used to determine the surface the water surface boundary. water boundary. Detected pixel Detected coordinates pixel coordinates of the water of surfacethe water boundary surface wereboundary then transferredwere then transferred to the actual to XY-coordinates the actual XY‐coordinates on the glass on wall the of glass the flume.wall of Validated the flume. by Validated the data ofby wavethe data gauges of wave located gauges at certainlocated positions,at certain positions, it was shown it was that shown this that image-based this image measuring‐based measuring system wassystem able was to captureable to surfacecapture fluctuationssurface fluctuations within acceptable within acceptable errors both errors in non-breaking both in non‐breaking and breaking and areasbreaking [32]. areas [32].

Figure 2. Experimental setup. Figure 2. Experimental setup.

Five tests were run for regular waves with incident wave heights Hi varying from 4.2 cm to 5.5 cm, Five tests were run for regular waves with incident wave heights Hi varying from 4.2 cm to 5.5 wave periods Ti from 1.2 s to 1.6 s and water depths at the seawall hs from 3 cm to 4.5 cm (water depth overcm, wave flat bottom periodsh Tfromi from 30 1.2 cm s to to 31.5 1.6 s cm). and Incident water depths wave at conditions the seawall and hs waterfrom 3 levels cm to were 4.5 cm designed (water todepth form over broken flat bottom waves withh from the 30 presence cm to 31.5 of thecm). seawall Incident in wave which conditions case waves and start water to break levels about were designed to form broken waves with the presence of the seawall in which case waves start to break 3/2 or one wave length from the seawall. For one wave case with Hi = 4.7 cm, Ti = 1.2 s, hs = 3 cm, progressiveabout 3/2 or waves one wave with length the same from incident the seawall. wave conditionsFor one wave were case also with performed Hi = 4.7 without cm, Ti = the 1.2 presence s, hs = 3 ofcm, the progressive seawall for waves comparison. with the Figure same3 showsincident the wave time conditions and spatial were variation also performed of root mean without square the of presence of the seawall for comparison. Figure 3 shows the time and spatial variation of root mean the surface water fluctuations, ηrms, near the seawall for the above-mentioned case and progressive square of the surface water fluctuations, ηrms, near the seawall for the above‐mentioned case and progressive wave case. ηrms was computed based on the extracted surface water level data of each single wave period so that time‐variation of its values can be observed. The origin of the horizontal axis was set at the location of the seawall and the vertical axis is time. The gray color in Figure 3

Water 2018, 10, 1876 7 of 15

wave case. ηrms was computed based on the extracted surface water level data of each single wave Waterperiod 2018, so 10, that x FOR time-variation PEER REVIEW of its values can be observed. The origin of the horizontal axis was7 of set 16 at the location of the seawall and the vertical axis is time. The gray color in Figure3 represents the steel representsframe of wavethe steel flume, frame behind of wave which flume, no data behind was captured which no by data the camera. was captured As indicated by the by camera. the red colorAs indicatedin Figure by3a, the antinodes red color of standingin Figure wave 3a, antinodes features canof standing be successfully wave features captured can by thebe successfully image-based capturedmeasuring by the system. image The‐based positions measuring of antinodes system. remainThe positions nearly of the antinodes same, which remain implies nearly steady the same, partial whichstanding implies waves steady form partial in front standing of the verticalwaves form seawall. in front It was of the also vertical observed seawall. during It thewas experiments also observed that duringwaves the started experiments to break that around waves the antinodestarted to which break is around 3/2 wave the length antinode (i.e., which around is x3/2= −wave140 cm) length from (i.e.,the around vertical x wall. = −140 Compared cm) from tothe the vertical progressive wall. Compared waves which to the start progressive to break around waves whichx = −100 start cm to as breakindicated around in Figurex = −1003b, cm it can as beindicated seen that in theFigure formation 3b, it can of antinodes be seen that made the the formation breaking of points antinodes further madefrom the the breaking vertical wallpoints than further the progressive from the vertical wave case.wall than More the details progressive about the wave image-based case. More measuring details aboutsystem the and image data‐based reports measuring can be referred system toand Liu data and reports Tajima can [32 ].be referred to Liu and Tajima [32].

FigureFigure 3. 3.TimeTime and and spatial spatial variation ofof the the root root mean mean square square of theof the surface surface fluctuations fluctuations near thenear seawall the seawallfor (a) for broken (a) broken waves waves and (b and) progressive (b) progressive waves. waves.

4. Model-Data Comparison 4. Model‐Data Comparison In model validation, model setup was the same as Liu and Tajima’s experiments. Shi et al. [35] have In model validation, model setup was the same as Liu and Tajima’s experiments. Shi et al. [35] validated the present model with the laboratory experiments conducted by Hansen and Svendsen [42] have validated the present model with the laboratory experiments conducted by Hansen and for progressive and breaking with different breaking types over a uniform slope. It was Svendsen [42] for progressive wave shoaling and breaking with different breaking types over a found that the grid spacing is important for FUNWAVE-TVD, which uses the TVD technique in uniform slope. It was found that the grid spacing is important for FUNWAVE‐TVD, which uses the TVDconjunction technique with in conjunction a nonlinear with shallow a nonlinear water equation shallow to water deal withequation wave to breaking deal with implicitly. wave breaking The grid implicitly.resolution The is especiallygrid resolution important is especially to introduce important the rightto introduce amount the of theright energy amount dissipation of the energy at the dissipationbreaking point.at the In breaking Hansen andpoint. Svendsen’s In Hansen experiments, and Svendsen’s regular experiments, waves with incidentregular waves wave heightswith incidentfrom 3.6 wave cm heights to 6.7 cm from and 3.6 wave cm to periods 6.7 cm fromand wave 1.0 s periods to 3.33 s from were 1.0 generated s to 3.33 s on were a flat generated bottom withon a 0.36flat mbottom water depthwith 0.36 and propagatedm water depth over aand 1:34.26 propagated slope. Shi over et al. [a35 1:34.26] demonstrated slope. Shi the modelet al. [35] could x demonstratedunderestimate the the model peak could wave underestimate height with a the coarse peak grid wave size height (i.e., with d = a 0.05 coarse m) grid compared size (i.e., to d ax fine = x 0.05grid m) size compared (i.e., d to= a 0.02 fine m)grid due size to (i.e., numerical dx = 0.02 dissipation m) due to numerical resulting from dissipation the coarse resulting grid resolution.from the coarseSince grid the experimentalresolution. Since conditions, the experimental especially theconditions, wave lengths especially of incident the wave waves lengths of Liu of and incident Tajima’s wavesexperiments, of Liu areand similar Tajima’s to Hansen experiments, and Svendsen’s are similar experiments, to Hansen numerical and Svendsen’s results of experiments, these two grid x x numericalsizes (d results= 0.05 m of and these d two= 0.02 grid m) sizes are also(dx = presented 0.05 m and in thisdx = study 0.02 m) to are demonstrate also presented the sensitivity in this study of the n togrid demonstrate spacing. Thethe valuesensitivity of Manning of the grid coefficient spacing.employed The value in of the Manning bottom frictioncoefficient term n wasemployed set to 0.012 in thefor bottom the smooth friction steel term surface. was set Internal to 0.012 wavemaker for the smooth [43] was steel implemented surface. Internal for incident wavemaker regular waves[43] was and implementedsponge layer for was incident placed regular at the offshore waves and side sponge to absorb layer the was reflected placed waves at the from offshore the vertical side to seawall.absorb the reflectedValidation waves results from the of two vertical wave seawall. cases with Hi = 4.7 cm, Ti = 1.2 s, hs = 3 cm and Hi = 5.5 cm, Ti = 1.2 s, hs = 4.5 cm are presented here. Figures4 and5 demonstrate the predicted and measured Validation results of two wave cases with Hi = 4.7 cm, Ti = 1.2 s, hs = 3 cm and Hi = 5.5 cm, Ti = 1.2 spatial distribution of mean water level and ηrms near the seawall for these two wave cases. Mean water s, hs = 4.5 cm are presented here. Figures 4 and 5 demonstrate the predicted and measured spatial level and ηrms were computed from the measured and predicted time series of surface fluctuations of distribution of mean water level and ηrms near the seawall for these two wave cases. Mean water level ten wave cycles when waves reached a periodic equilibrium state. The origin of the horizontal axis and ηrms were computed from the measured and predicted time series of surface fluctuations of ten wave cycles when waves reached a periodic equilibrium state. The origin of the horizontal axis was also set at the location of seawall. To quantify the performance of the model, the model skill is calculated as [44]:

Water 2018, 10, 1876 8 of 15

Water 2018, 10, x FOR PEER REVIEW 8 of 16 was also set at the location of seawall. To quantify the performance of the model, the model skill is calculated as [44]: N 2 N  ii2 XXi  i ∑Xpredpred − measXmeas skill 1 =i1 N i 1 2 (13) skill = 1 − ii (13) N  XXXX  2 i pred meas i i1 − + − ∑ Xpred X Xmeas X i=1 i i where Xi pred donates the predicted value, i Xmeas indicates the measured value, X represents the where Xpred donates the predicted value, Xmeas indicates the measured value, X represents the mean meanmeasured measured value andvalueN andis the N numberis the number of data of points. data points. As shown in Figures 44 andand5 ,5, the the model model with with fine fine grid grid size size (i.e., (i.e., d dx = 0.02 m) captures the characteristics of broken waves with reasonable accuracy. The formations of antinodes and their positions are well reproduced by thethe presentpresent modelmodel bothboth inin meanmean waterwater levellevel and andη ηrmsrms. The coarse grid size tendstends toto underpredict underpredict the the peak peak of of antinodes antinodes since since wave wave breaking breaking mainly mainly happens happens around around the theantinodes antinodes under under broken broken waves waves near anear vertical a vertical seawall seawall and coarse and grid coarse size grid leads size to more leads numerical to more numericaldissipation dissipation similar to thesimilar underprediction to the underprediction of peak wave of peak height wave of progressive height of progressive wave. In addition, wave. In it addition,is noted in it Figureis noted4 that in Figure the antinode 4 that the of antinode mean water of mean level water which level is located which at is around located xat= around−50 cm x is= −obviously50 cm is obviously underestimated underestimated by the present by the model present for model the case for withthe caseHi = with 5.5 cm,Hi =T 5.5i = cm, 1.2 s,Ti h=s 1.2= 4.5 s, h cm.s = 4.5Figure cm.6 Figureshows the6 shows typical the snapshots typical snapshots of the collision of the of collision incident of broken incident wave broken and reflected wave and wave reflected at this wavelocation at this (around locationx = −(around50 cm) x for = − the50 wavecm) for case the with waveH icase= 4.7 with cm, HTii == 4.7 1.2 cm, s, h sT=i = 3 1.2 cm s, and hs = wave 3 cm caseand withwaveH casei = 5.5with cm, HiT =i 5.5= 1.2 cm, s, This == 1.2 4.5 s, cm. hs = As4.5 showncm. As inshown Figure in6 Figure, the collision 6, the collision of broken of broken incident incident waves wavesand reflected and reflected waves inwaves the case in the with caseH iwith= 5.5 H cm,i = 5.5Ti cm,= 1.2 T s,i =h 1.2s = s, 4.5 hs cm,= 4.5 which cm, which has larger has larger incident incident wave waveheight height and increasing and increasing nonlinearity, nonlinearity, becomes becomes more violent more atviolent this location at this location so that more so that water more splashes water splashesout. The out. splashing The splashing water could water be could captured be captured by the cameraby the camera during during the experiments, the experiments, while itwhile could it couldnot be not simulated be simulated by the by present the present model, model, resulting resulting in an in underestimation an underestimation of mean of mean water water level level by the by themodel. model. Moreover, Moreover, the the splashing splashing water water released released from from the the breaking breaking antinodes antinodes also also make make thethe surface water fluctuationsfluctuations more more complicated complicated so so that that the the present present model model still still cannot cannot reproduce reproduce the surface the surface water waterprofiles profiles very accurately. very accurately. Figure Figure7 shows 7 shows the predicted the predicted spatial spatial distribution distribution of root of root mean mean square square of ofdepth-averaged depth‐averaged horizontal horizontal velocity, velocity,U rmsUrms(U (U = = P P//DD,, thethe vectorvector M inin Equation (1) is defined defined as ( (PP,, QQ)) in the numerical scheme scheme of of the the present present model, model, in in which which case case PP andand QQ isis the the component component of of MM alongalong x directionx direction and and y ydirection),direction), near near the the seawall. seawall. As As seen seen in in this this figure, figure, Urms isis relatively relatively smaller smaller at the positions of antinodes, which confirms confirms that the present model surely captures the key characteristics of partial near the seawall.

Figure 4. ComparisonsComparisons of of predicted predicted and and measured measured spatial spatial distribution distribution of of(a) ( meana) mean water water level level and and (b)

η(brms) η forrms thefor wave the wave case casewith with Hi = H4.7i = cm, 4.7 T cm,i = 1.2Ti =s, 1.2hs = s, 3 hcm.s = 3 cm.

Water 2018, 10, x FOR PEER REVIEW 9 of 16 Water 2018, 10, 1876 9 of 15 Water 2018, 10, x FOR PEER REVIEW 9 of 16 Water 2018, 10, x FOR PEER REVIEW 9 of 16

Figure 5. Comparisons of predicted and measured spatial distribution of (a) mean water level and (b) ηrms for the wave case with Hi = 5.5 cm, Ti = 1.2 s, hs = 4.5 cm. Figure 5. ComparisonsComparisons of of predicted predicted and and measured measured spatial spatial distribution distribution of of(a) ( meana) mean water water level level and and (b) ηrms for the wave case with Hi = 5.5 cm, Ti = 1.2 s, hs = 4.5 cm. η(brms) η forrms thefor wave the wave case casewith with Hi = H5.5i = cm, 5.5 T cm,i = 1.2Ti =s, 1.2hs = s, 4.5hs cm.= 4.5 cm.

Figure 6. Typical snapshots of the collision of incident broken wave and reflected wave for (a) the

waveFigure case 6. Typical with H snapshotsi = 4.7 cm, ofTi the= 1.2 collision s, hs = 3 of cm incident and (b broken) wave wave case with and reflected Hi = 5.5 cm, wave Ti for = 1.2 (a) s, the hs wave= 4.5 Figure 6. Typical snapshots of the collision of incident broken wave and reflected wave for (a) the cm. casewave with caseH withi = 4.7 Hi cm, = 4.7T icm,= 1.2 Ti s,= h1.2s = s, 3 h cms = and3 cm ( band) wave (b) wave case with caseH withi = 5.5 Hi cm,= 5.5T icm,= 1.2 Ti s,= 1.2hs = s, 4.5 hs = cm. 4.5 wave case with Hi = 4.7 cm, Ti = 1.2 s, hs = 3 cm and (b) wave case with Hi = 5.5 cm, Ti = 1.2 s, hs = 4.5 cm.

FigureFigure 7. Predicted spatialspatial distribution distribution of of root root mean mean square square of depth-averagedof depth‐averaged horizontal horizontal velocity velocity near nearthe seawall. the seawall. Figure 7. Predicted spatial distribution of root mean square of depth‐averaged horizontal velocity

near the seawall.

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In order to further demonstrate the advantages of the shock‐capturing method in predicting waveIn characteristics order to further under demonstrate broken thewaves advantages near a ofseawall, the shock-capturing comparisons method of the innumerical predicting results wave obtainedcharacteristics from underthe Boussinesq broken waves wave near model a seawall, with an comparisons empirical surface of the‐ numericalroller breaking results model obtained [25] from and the Boussinesqpresent model wave with model dx = with 0.02 an m empirical are presented surface-roller in Figure breaking 8 for the model case with [25] andHi = the 4.7 presentcm, Ti = model 1.2 s, withhs = 3 d cm.x = The 0.02 notable m are presented model of in Madsen Figure8 et for al. the [25] case has with beenH successfullyi = 4.7 cm, Ti applied= 1.2 s, h tos = model 3 cm. Thecross notable‐shore motionsmodel of of Madsen regular et waves al. [25] including has been successfully various types applied of breaking to model on cross-shore plane beaches motions and over of regular submerged waves bars.including It is variousalso reported types ofthe breaking surface onroller plane type beaches breaking and model over submerged demonstrates bars. the It is most also reportedappropriate the predictivesurface roller skills type of breaking time‐varying model profiles demonstrates of surface the mostwater appropriate compared to predictive the other skills empirical of time-varying breaking modelsprofiles of[28]. surface As seen water in comparedFigure 8, for to thethe other present empirical study, breakingmean water models level [28 predicted]. As seen by in the Figure surface8, for‐ rollerthe present breaking study, model mean has water an obvious level predicted setup after by wave the surface-rollerbreaking while breaking ηrms has modelan obvious has andecreasing obvious trend.setup afterIt is also wave noted breaking that the while profileηrms ofhas ηrms an predicted obvious decreasing by the surface trend.‐roller It is model also noted becomes that the smoother profile thanof ηrms thepredicted ones of FUNWAVE by the surface-roller‐TVD and model experimental becomes data, smoother especially than theat the ones left of part FUNWAVE-TVD of the figure (i.e., and −experimental180 < x < −100 data, cm). especially As revealed at the in Liu left and part Tajima of the figure [32], empirical (i.e., −180 breaking < x < −100 models cm). including As revealed surface in Liu‐ rollerand Tajima model [32 and], empirical eddy‐viscosity breaking model models developed including based surface-roller on the progressive model and wave eddy-viscosity assumptions model take thedeveloped whole partial based on standing the progressive wave field wave as assumptionspure progressive take thewaves whole with partial their standing breaking wave criteria field and as simulatedpure progressive energy waves dissipation with their happens breaking both criteria in andincident simulated waves energy and dissipationreflected happenswaves while both experimentalin incident waves data and indicates reflected that waves dissipation while experimentalof reflected waves data indicates is insignificant that dissipation actually. ofTherefore, reflected wavesexcess isdissipation insignificant computed actually. by Therefore, these breaking excess dissipationmodels leads computed to an obvious by these setup breaking and decreasing models leads of ηtorms an and obvious smaller setup reflected and decreasingwaves lead ofto aη rmssmootherand smaller profile reflected of ηrms. To waves the authors’ lead to delight, a smoother the present profile modelof ηrms .using To the shock authors’‐capturing delight, method the present improves model these using features. shock-capturing The results methodpresented improves here indicate these thefeatures. shock‐ Thecapturing results method presented using here the indicate ratio of wave the shock-capturing height to the local method depth using as the the breaking ratio of criteria wave isheight more to appropriate the local depth for modeling as the breaking the broken criteria waves is more near appropriate a vertical seawall for modeling by eliminating the broken the wavesexcess dissipationnear a vertical in reflected seawall by waves. eliminating the excess dissipation in reflected waves.

Figure 8. Comparisons of predicted spatial distribution ofof ((aa)) meanmean waterwater levellevel andand ( (bb))η ηrmsrms between Madsen et al. [25] [25] and FUNWAVE-TVDFUNWAVE‐TVD forfor thethe wavewave case case with withH Hii == 4.8 cm, Ti = 1.2 1.2 s, s, hhss == 3 3 cm. cm. 5. Numerical Experiment Results 5. Numerical Experiment Results As seen in Table1, the wave types in front of vertical seawalls are mainly determined by the As seen in Table 1, the wave types in front of vertical seawalls are mainly determined by the wave height, water depth at the seawall, and the seabed slope. Thus, after validation of the model, a wave height, water depth at the seawall, and the seabed slope. Thus, after validation of the model, a series of numerical experiments were implemented to investigate the effects of these key parameters, series of numerical experiments were implemented to investigate the effects of these key parameters, such as the incident wave height Hi, wave depth at the vertical seawall hs, and bed slope tanθ, on the such as the incident wave height Hi, wave depth at the vertical seawall hs, and bed slope tanθ, on the wave kinematics (i.e., the root mean square of the surface fluctuations and depth-averaged horizontal wave kinematics (i.e., the root mean square of the surface fluctuations and depth‐averaged horizontal velocity) under broken waves near vertical seawalls. Numerical experiment setup is shown in Figure9. velocity) under broken waves near vertical seawalls. Numerical experiment setup is shown in Figure The maker is 20 m from the seawall and the computational domain with a total length 9. The internal wave maker is 20 m from the seawall and the computational domain with a total of 26 m is applied with a 3 m sponge layer at the offshore side. The numerical experimental wave length of 26 m is applied with a 3 m sponge layer at the offshore side. The numerical experimental conditions for all model runs are listed in Table3. The tested range of parameters in this numerical wave conditions for all model runs are listed in Table 3. The tested range of parameters in this study is designed based on the Froude similarity with a geometric scale factor of 1:100. Therefore, at numerical study is designed based on the Froude similarity with a geometric scale factor of 1:100. the prototype scale, incident wave heights are from 4 m to 10 m, representing the typical wave heights Therefore, at the prototype scale, incident wave heights are from 4 m to 10 m, representing the typical during typhoon and super typhoon and wave depths at the vertical seawall are from 2 m to 6 m, wave heights during typhoon and super typhoon and wave depths at the vertical seawall are from 2

Water 2018, 10, x FOR PEER REVIEW 11 of 16 Water 2018, 10, 1876 11 of 15 m to 6 m, representing the typical water depths near the vertical seawall subjected to broken waves. Forrepresenting each group the in typicalTable 3, water only one depths parameter near the was vertical changed seawall while subjected keeping other to broken parameters waves. the For same. each Gridgroup size in Tabledx =3 ,0.02 only m, one Manning parameter coefficient was changed n = 0.012 while keepingwere used other in parametersall numerical the experiments same. Grid size as validateddx = 0.02 above. m, Manning coefficient n = 0.012 were used in all numerical experiments as validated above.

Figure 9. Numerical experiment setup. Figure 9. Numerical experiment setup. Table 3. Numerical experimental wave conditions. Table 3. Numerical experimental wave conditions. Hi (m) hs (cm) tanθ Ti (s) Group 1 Case 1Hi 0.04(m) hs (cm) 6.0 tanθ 1:30Ti (s) 1.2 Group 1 Case 2 1 0.04 0.07 6.0 6.0 1:30 1:30 1.2 1.2 Case 3 2 0.07 0.1 6.0 6.0 1:30 1:30 1.2 1.2 Group 2 Case 4 3 0.1 0.07 6.0 6.0 1:30 1:30 1.2 1.2 Case 5 0.07 4.0 1:30 1.2 Group 2 Case 4 0.07 6.0 1:30 1.2 Case 6 0.07 2.0 1:30 1.2 Group 3 Case 7 5 0.07 0.07 4.0 6.0 1:30 1:15 1.2 1.2 Case 8 6 0.07 0.07 2.0 6.0 1:30 1:20 1.2 1.2 Group 3 Case 9 7 0.07 0.07 6.0 6.0 1:15 1:25 1.2 1.2 CaseCase 10 8 0.07 0.07 6.0 6.0 1:20 1:30 1.2 1.2 Case 9 0.07 6.0 1:25 1.2 Figure 10 shows the computed Case spatial 10 distributions0.07 6.0 of ηrms1:30and Urms1.2 near the seawall for Cases 1–3 of Group 1 whose incident wave heights are 0.04 m, 0.07 m and 0.1 m respectively. Urms can be somehowFigure regarded 10 shows as the a physicallycomputed relevantspatial distributions indicator of of the ηrms potential and Urms surrounding near the seawall scour andfor Cases wave 1–3impact of Group on the 1 seawall. whose incident As seen wave in Figure heights 10a, are the positions0.04 m, 0.07 of antinodesm and 0.1 are m respectively. nearly the same Urms for can these be somehowwave cases regarded since they as havea physically the same relevant indicator relationship. of the potential It is noted surrounding both Case 1 scour and Case and 2wave start impactto break on at the the seawall. antinode As aroundseen in Figurex = −0.5 10a, m, the which positions is 1/2 of wave antinodes length are from nearly the the seawall same for since these the waveelevations cases ofsince antinodes they have forming the same near dispersion the seawall relationship. continues It to is increase. noted both For Case these 1 and two Case wave 2 cases,start tolarger break incident at the antinode wave height around induces x = higher−0.5 m,η whichrms near is the1/2 seawallwave length and higher from the runup seawall on the since seawall the elevationsconsequently. of antinodes On the other forming hand, near waves the of seawall Case 3 withcontinues the largest to increase. incident For wave these height, two 0.1wave m, cases, in this largergroup incident break further wave fromheight the induces seawall higher at the antinodeηrms near aroundthe seawallx = − and1.75 higher m, which runup is 3/2 on wavethe seawall length consequently.from the seawall. On the However, other hand,ηrms wavesof Case of 2 Case and Case 3 with 3 becomethe largest nearly incident the same wave between height, 0.1 the m, antinode in this groupat around breakx =further−0.5 m from and the the seawall seawall at and the the antinode consequent around runups x = −1.75 on the m, seawallwhich is of 3/2 these wave two length cases fromare also the close.seawall. As However, seen in Figure ηrms of 10 Caseb, similar 2 and to Caseηrms 3, forbecome Case nearly 1 and Casethe same 2, which between break the at antinode the same atlocation, around incident x = −0.5 m wave and with the seawall larger incident and the waveconsequent height runups induces on higher the seawallUrms near of these the seawall two cases and are for alsoCase close. 2 and As Case seen 3 U inrms Figureare nearly 10b, thesimilar same to near ηrms, the for seawall. Case 1 and Case 2, which break at the same location, incident wave with larger incident wave height induces higher Urms near the seawall and for Case 2 and Case 3 Urms are nearly the same near the seawall.

Water 2018, 10, 1876 12 of 15 Water 2018, 10, x FOR PEER REVIEW 12 of 16

Figure 10. Computed spatial distributions of (a) ηrms and (b) Urms near the seawall for Cases 1–3. FigureFigure 10. 10.Computed Computed spatial spatial distributions distributions of of ( a(a))η ηrmsrms and ( b) Urms nearnear the the seawall seawall for for Cases Cases 1–3. 1–3.

rms rms Figure 11 showsshows thethe computedcomputed spatialspatial distributionsdistributions of ofη ηrmsrms and UUrmsrms nearnear the the seawall for Cases s s 4–6 inin GroupGroup 2 2 whose whose water water levels levels at the at seawall,the seawall,hs, are hs, 6 are cm, 6 4 cm, cm and 4 cm 2 cmand respectively. 2 cm respectively.hs is adjusted hs is adjustedby changing by changing the deep-water the deep depth‐water over depth flat bottomover flat in bottom this study. in this As study. seen in As Figure seen in11 a,Figure the positions 11a, the positionsof antinodes of antinodes are different are for different these wave for these cases wave due to cases different due dispersionto different relationships. dispersion relationships. Locations of s Locationsbreaking points of breaking move points further move quickly further from quickly the seawall from the as hseawalls decreases, as hs decreases, in which case in which the η rmscasenear the rms ηtherms seawallnear the and seawall runups and on runups the seawall on the also seawall decrease also significantly. decrease significantly. As seen in Figure As seen 11 b,in shallower Figure 11b,hs s rms shallowerinduces smaller hs inducesUrms smallernear the U seawallrms near asthe well. seawall as well.

FigureFigure 11. 11.Computed Computed spatial spatial distributions distributions of of ( (aa))η ηrmsrms and ( b) Urms nearnear the the seawall seawall for for Cases Cases 4–6. 4–6. Figure 11. Computed spatial distributions of (a) ηrms and (b) Urms near the seawall for Cases 4–6.

Figure 12 shows the computed spatial distributions ofrmsηrms andrms Urms near the seawall for Figure 12 shows the computed spatial distributions of ηrms and Urms near the seawall for Cases Cases 7–10 in Group 3 whose bed slope angles, tanθ, are 1:15, 1:20, 1:25 and 1:30 respectively. 7–10 in Group 3 whose bed slope angles, tanθ, are 1:15, 1:20, 1:25 and 1:30 respectively. As seen in As seen in Figure 12a, the positions of antinodes are slightly different for these wave cases and Figure 12a, the positions of antinodes are slightly different for these wave cases and the locations of the locations of breaking points are all at the antinodes which are 1/2 wave length from the seawall. breaking points are all at the antinodes which are 1/2 wave length from the seawall. Spatial Spatial distributions ofrms the ηrms do not vary much as the seabed slope changes and the runups on the distributions of the ηrms do not vary much as the seabed slope changes and the runups on the seawall seawall get a little smaller when the bed slope becomes milder. As seen in Figure 12b, Urmsrms also seem get a little smaller when the bed slope becomes milder. As seen in Figure 12b, Urms also seem insensitive to the seabed slope. insensitive to the seabed slope.

Water 2018, 10, 1876 13 of 15 Water 2018, 10, x FOR PEER REVIEW 13 of 16

FigureFigure 12. Computed12. Computed spatial spatial distributions distributions of of ( a(a))η ηrmsrms andand ( (bb) )UUrmsrms nearnear the the seawall seawall for Cases for Cases 7–10. 7–10.

6. Discussions6. Discussions and and Conclusions Conclusions InIn this this paper, paper, the the fully fully nonlinear nonlinear Boussinesq Boussinesq wave wave model model FUNWAVE-TVD FUNWAVE‐TVD was was applied applied to model to wavemodel characteristics wave characteristics under broken under waves broken in frontwaves of in a verticalfront of seawall. a vertical Validation seawall. Validation with the experimental with the dataexperimental confirmed data the ability confirmed of FUNWAVE-TVD the ability of FUNWAVE in capturing‐TVD the in key capturing characteristics the key characteristics in such conditions in usingsuch a conditions shock-capturing using a shock method.‐capturing The grid method. size isThe important grid size is to important predict the to predict peak height the peak around height the antinodesaround the accurately. antinodes The accurately. shock-capturing The shock method‐capturing is more method appropriate is more forappropriate modeling for the modeling broken waves the nearbroken a seawall waves than near the a empiricalseawall than breaking the empirical model based breaking on progressive model based wave on assumptions.progressive wave As the Boussinesq-typeassumptions. As models the Boussinesq only yield‐type the models velocity only based yield on the potential velocity flow based theory, on potential additional flow flow theory, model canadditional be further flow coupled model for can predictions be further of coupled the vertical for predictions profile of the of the horizontal vertical shearprofile velocity of the horizontal field, which is necessaryshear velocity for estimations field, which of surroundingis necessary scourfor estimations and cross-shore of surrounding sediment transport.scour and In cross this‐shore study, a preliminarysediment transport. investigation In this of the study, effects a preliminary of the key parameters,investigation such of the as theeffects incident of the wave key parameters, height, water levelsuch at as the the seawall, incident and wave seabed height, slope, water on level the wave at the kinematicsseawall, and (i.e., seabed the rootslope, mean on the square wave ofkinematics the surface fluctuations(i.e., the root and mean depth-averaged square of the horizontal surface fluctuations velocity) near and verticaldepth‐averaged seawalls horizontal was conducted velocity) through near a vertical seawalls was conducted through a series of numerical experiments. series of numerical experiments. The numerical experiment results from this study suggest the incident wave height and the The numerical experiment results from this study suggest the incident wave height and the water depth at the seawall are the important parameters in determining the magnitude of the wave water depth at the seawall are the important parameters in determining the magnitude of the wave kinematics near the seawall. The role of the breaking point locations is also highlighted in this study. kinematics near the seawall. The role of the breaking point locations is also highlighted in this study. When incident waves have the same breaking locations, larger incident wave height can induce larger Whenwave incident kinematics waves near have the theseawall, same in breaking which case locations, the potential larger incidentsurrounding wave scour height and can wave induce impact larger waveon the kinematics seawall are near larger. the seawall, However, in whichthe breaking case the point potential can move surrounding further from scour the and seawall wave as impact the onincident the seawall wave are height larger. increases However, and the the wave breaking kinematics point cannear move the seawall further can from be mitigated the seawall by this as the incidentfurther wave breaking. height increases and the wave kinematics near the seawall can be mitigated by this furtherOver breaking. a uniform plane beach, locations of breaking points seem to be more sensitive to the water depthOver at athe uniform seawall. plane The breaking beach, locations point moves of breaking further pointsfrom the seem seawall to be quickly more sensitive as the water to the depth water depthat the at seawall the seawall. gets shallow, The breaking and as point a result, moves the magnitude further from of the seawallwave kinematics quickly asnear the the water seawall depth atdecrease the seawall significantly. gets shallow, This and implies as a result,seawalls the subjected magnitude to the of thebroken wave waves kinematics are under near higher the seawall risk decreaseduring significantly. the high water This‐level implies events, seawalls such as subjected the high to the and broken storm wavessurge. areThe under rising highersea‐level risk due during to thethe high climate water-level change events,which results such asin thea net high increase tide andin water storm depth surge. at the The seawall rising sea-levelwill endanger due tothe the climatecoastal change structures which as resultswell. Moreover, in a net increase the water in depth water at depth the seawall at the seawallcan be adjusted will endanger as the locations the coastal structuresof seawalls as well. change. Moreover, Therefore, the waterseawalls depth which at the are seawall planned can to be build adjusted at the as back the locations of the beach of seawalls and change.predominantly Therefore, subjected seawalls to which broken are waves planned should to buildbe well at theconsidered back of for the their beach construction and predominantly sites if subjectedpossible. to broken waves should be well considered for their construction sites if possible. The seabed slope seems to be insignificant, affecting the wave kinematics near the seawall. The The seabed slope seems to be insignificant, affecting the wave kinematics near the seawall. authors must state that natural beaches are more complex than the idealized uniform beaches used The authors must state that natural beaches are more complex than the idealized uniform beaches in this study. A realistic nearshore bathymetry which may significantly affect the breaking points used in this study. A realistic nearshore bathymetry which may significantly affect the breaking should modify the results in this study. Furthermore, artificial obstacles, such as submerged points should modify the results in this study. Furthermore, artificial obstacles, such as submerged breakwaters, which are placed in front of seawalls to make the wave breaking earlier, may be an breakwaters,efficient way which to reduce are the placed structural in front risk of induced seawalls by to broken make waves. the wave breaking earlier, may be an efficient way to reduce the structural risk induced by broken waves.

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Author Contributions: Numerical simulations and data analysis were completed by W.L., Y.N., Y.Z. and J.Z. Draft was prepared by W.L. Funding: This work was funded by the National Natural Science Foundation of China (grant numbers 51609043, 51809234), the Key Laboratory of Water-Sediment Sciences and Water Disaster Prevention of Hunan Province (under the awards 2018SS03), the Key Laboratory of Coastal Disaster and Defense, Ministry of Education, China (under the awards 201708) and the Bureau of Science and Technology of Zhoushan (under the awards 2018C81040). Conflicts of Interest: The authors declare no conflict of interest.

References

1. Gao, X.; Inouchi, K. The characteristics of scouring and depositing in front of vertical breakwaters by broken clapotis. Coast. Eng. J. 1998, 40, 99–113. [CrossRef] 2. Oumeraci, H.; Klammer, P.; Partenscky, H.W. Classification of breaking wave loads on vertical structures. J. Waterw. Port Coast. Ocean Eng. 1993, 119, 381–397. [CrossRef] 3. Goda, Y. Dynamic response of upright breakwaters to impulsive breaking wave forces. Coast. Eng. 1994, 22, 135–158. [CrossRef] 4. Oumeraci, H.; Kortenhaus, A. Analysis of the dynamic response of caisson breakwaters. Coast. Eng. 1994, 22, 159–183. [CrossRef] 5. Oumeraci, H.; Kortenhaus, A. Wave impact loading: Tentative formulae and suggestions for the development of final formulae. In Proceedings of the 2nd Task 1 MAST III Workshop (PROVERBS), Edingburgh, UK, 2–4 July 1997. 6. Karim, M.F.; Tingsanchali, T. A coupled numerical model for simulation of wave breaking and hydraulic performances of a composite seawall. Ocean Eng. 2006, 33, 773–787. [CrossRef] 7. Bullock, G.N.; Obhrai, C.; Peregrine, D.H.; Bredmose, H. Violent breaking wave impacts. Part 1: Results from large-scale regular wave tests on vertical and sloping walls. Coast. Eng. 2007, 54, 602–617. [CrossRef] 8. Cuomo, G.; Allsop, W.; Bruce, T.; Pearson, J. Breaking wave loads at vertical seawalls and breakwaters. Coast. Eng. 2010, 57, 424–439. [CrossRef] 9. Mokrani, C.; Abadie, S.; Grilli, S.T.; Zibouche, K. Numerical Simulation of the Impact of a Plunging Breaker On a Vertical Structure and Subsequent Overtopping Event Using a Navier-Stokes VOF Model. In Proceedings of the Twentieth International Offshore and Polar Engineering Conference, Beijing, China, 20–25 June 2010; p. 8. 10. Kisacik, D.; Troch, P.; Van Bogaert, P. Experimental study of violent wave impact on a vertical structure with an overhanging horizontal cantilever slab. Ocean Eng. 2012, 49, 1–15. [CrossRef] 11. Ramachandran, K. Broken wave loads on a vertical wall: Large scale experimental investigations. In Proceedings of the 6th International Conference on Structural Engineering and Construction Management, Kandy, Sri Lanka, 11–13 December 2015. 12. Van Dongeren, A.R.; Sancho, F.E.; Svendsen, I.A.; Putrevu, U. SHORECIRC: A Quasi 3-D Nearshore Model. Coast. Eng. Proc. 1994.[CrossRef] 13. Tajima, Y.; Madsen, O.S. Modeling Near-Shore Waves, Surface Rollers, and Velocity Profiles. J. Waterw. Port Coast. Ocean Eng. 2006, 132, 429–438. [CrossRef] 14. Bradford, S.F. Numerical Simulation of Dynamics. J. Waterw. Port Coast. Ocean Eng. 2000, 126, 1–13. [CrossRef] 15. Christensen, E.D.; Walstra, D.-J.; Emerat, N. Vertical variation of the flow across the surf zone. Coast. Eng. 2002, 45, 169–198. [CrossRef] 16. Zhao, X.; Ye, Z.; Fu, Y.; Cao, F. A CIP-based numerical simulation of freak wave impact on a floating body. Ocean Eng. 2014, 87, 50–63. [CrossRef] 17. Madsen, P.A.; Sørensen, O.R. A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry. Coast. Eng. 1992, 18, 183–204. [CrossRef] 18. Nwogu, O. Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterw. Port Coast. Ocean Eng. 1993, 119, 618–638. [CrossRef] 19. Wei, G.; Kirby, J.T.; Grilli, S.T.; Subramanya, R. A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves. J. Fluid Mech. 1995, 294, 71–92. [CrossRef] Water 2018, 10, 1876 15 of 15

20. Zhang, Y.; Kennedy, A.B.; Panda, N.; Dawson, C.; Westerink, J.J. Boussinesq–Green–Naghdi rotational water wave theory. Coast. Eng. 2013, 73, 13–27. [CrossRef] 21. Peregrine, D.H. Long waves on a beach. J. Fluid Mech. 1967, 27, 815–827. [CrossRef] 22. Nwogu, O. Numerical prediction of breaking waves and currents with a Boussinesq model. In Proceedings of the 25th Conference on Coastal Engineering, Orlando, FL, USA, 2–6 September 1996. 23. Kennedy, A.B.; Chen, Q.; Kirby, J.T.; Dalrymple, R.A. Boussinesq modeling of wave transformation, breaking, and runup. I: 1D. J. Waterw. Port Coast. Ocean Eng. 2000, 126, 39–47. [CrossRef] 24. Schäffer, H.A.; Madsen, P.A.; Deigaard, R. A Boussinesq model for waves breaking in shallow water. Coast. Eng. 1993, 20, 185–202. [CrossRef] 25. Madsen, P.A.; Sørensen, O.R.; Schäffer, H.A. Surf zone dynamics simulated by a Boussinesq type model. Part I. Model description and cross-shore motion of regular waves. Coast. Eng. 1997, 32, 255–287. [CrossRef] 26. Bredmose, H.; Schäffer, H.A.; Madsen, P.A. Boussinesq evolution equations: Numerical efficiency, breaking and amplitude dispersion. Coast. Eng. 2004, 51, 1117–1142. [CrossRef] 27. Chen, Q.; Kirby, J.T.; Dalrymple, R.A.; Shi, F.; Thornton, E.B. Boussinesq modeling of longshore currents. J. Geophys. Res. 2003, 108.[CrossRef] 28. Mohsin, S.; Tajima, Y. Modeling of time-varying shear current field under breaking and broken waves with surface rollers. Coast. Eng. J. 2014, 56, 1450013. [CrossRef] 29. Yao, Y.; Becker, J.M.; Ford, M.R.; Merrifield, M.A. Modeling wave processes over fringing reefs with an excavation pit. Coast. Eng. 2016, 109, 9–19. [CrossRef] 30. Zhang, Y.; Kennedy, A.B.; Tomiczek, T.; Donahue, A.; Westerink, J.J. Validation of Boussinesq–Green–Naghdi modeling for surf zone hydrodynamics. Ocean Eng. 2016, 111, 299–309. [CrossRef] 31. Zhang, S.; Zhu, L.; Li, J. Numerical Simulation of Wave Propagation, Breaking, and Setup on Steep Fringing Reefs. Water 2018, 10, 1147. [CrossRef] 32. Liu, W.; Tajima, Y. Image-based study of breaking and broken characteristics in front of a vertical seawall. In Proceedings of the 34th Conference on Coastal Engineering, Seoul, Korea, 15–20 June 2014. 33. Tonelli, M.; Petti, M. Simulation of wave breaking over complex bathymetries by a Boussinesq model. J. Hydraul. Res. 2011, 49, 473–486. [CrossRef] 34. Roeber, V.; Cheung, K. Boussinesq-type model for energetic breaking waves in fringing reef environments. Coast. Eng. 2012, 70, 1–20. [CrossRef] 35. Shi, F.; Kirby, J.T.; Harris, J.C.; Geiman, J.D.; Grilli, S.T. A high-order adaptive time-stepping TVD solver for Boussinesq modeling of breaking waves and coastal inundation. Ocean Model. 2012, 43–44, 36–51. [CrossRef] 36. Fang, K.; Liu, Z.; Zou, Z. Fully nonlinear modeling wave transformation over fringing reefs using shock-capturing Boussinesq model. J. Coast. Res. 2016, 32, 164–171. [CrossRef] 37. Su, S.-F.; Ma, G.; Hsu, T.-W. Boussinesq modeling of spatial variability of infragravity waves on fringing reefs. Ocean Eng. 2015, 101, 78–92. [CrossRef] 38. Chen, Q. Fully nonlinear Boussinesq-type equations for waves and currents over porous beds. J. Eng. Mech. 2006, 132, 220–230. [CrossRef] 39. Kennedy, A.B.; Kirby, J.T.; Chen, Q.; Dalrymple, R.A. Boussinesq-type equations with improved nonlinear performance. Wave Motion 2001, 33, 225–243. [CrossRef] 40. Tonelli, M.; Petti, M. Hybrid finite volume—Finite difference scheme for 2DH improved Boussinesq equations. Coast. Eng. 2009, 56, 609–620. [CrossRef] 41. Yao, Y.; Huang, Z.; Monismith, S.G.; Lo, E.Y.M. 1DH Boussinesq modeling of wave transformation over fringing reefs. Ocean Eng. 2012, 47, 30–42. [CrossRef] 42. Hansen, J.B.; Svendsen, I.A. Regular Waves in Shoaling Water: Experimental Data; Technical University of Denmark: Lyngby, Denmark, 1979. 43. Wei, G.; Kirby, J.T.; Sinha, A. Generation of waves in Boussinesq models using a source function method. Coast. Eng. 1999, 36, 271–299. [CrossRef] 44. Willmott, C.J. On the validation of models. Phys. Geogr. 1981, 2, 184–194. [CrossRef]

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