A Numerical Investigation of the Reduction of Solitary Wave Runup by A Row of Vertical Slotted Piles

To improve the current understanding of the reduction of tsunami-like solitary wave runup by the pile breakwater on a sloping beach, we developed a 3D numerical wave tank based on the CFD tool OpenFOAM® in this study. The Navier-Stokes equations were applied to solve the two-phase incompressible flow, combined with an LES model to solve the turbulence and a VOF method to capture the free surface. The adopted model was firstly validated with existing empirical formulas for solitary wave runup on the slope without the pile structure. It is then validated using our new laboratory observations of the free surface elevation, the velocity and the pressure around a row of vertical slotted piles subjected to solitary waves, as well as the wave runup on the slope behind the piles. Subsequently, a set of numerical simulations were implemented to analyze the wave reflection, the wave transmission, and the shoreline runup with various offshore wave heights, offshore water depths, adjacent pile spaces and beach slopes. Finally, an improved empirical equation accounting for the maximum wave runup on the slope was proposed by taking the presence of the pile breakwater into consideration.


Introduction
Tsunami is a giant devastating wave generated by undersea earthquake, volcanic eruptions, plate movements or landslide, and generally defined as long period waves of the order of hundreds to thousands of seconds (Jiang et al., 2019). Tsunami damage occurs mostly in the coastal areas where tsunami wave runup or rundown the beach, overtop or ruin the coastal structures, and inundate the coastal towns and villages . Breakwaters are traditionally used structures to defend the coasts against the windgenerated waves. After the 2011 East Japan tsunami, the positive role of breakwaters in mitigating coastal tsunami waves has attracted wide attentions among the scholars conducting the post-disaster surveys (Nistor and Palermo, 2015;Raby et al., 2015 and many others). Among different types of breakwaters, the rigid slotted vertical piles are sometimes built to defend the coasts against the waves due to their relatively low costs and capability of preserving the coastal water quality . A good example of pile breakwater is constructed at a coastal site of Singapore as shown in Huang and Yuan (2010).
Solitary wave has been employed in many related studies to model the leading tsunami wave because the former can represent many important properties of the latter (Lin, 2004). Existing literature on the solitary wave runup are mainly focused on the natural coasts such as sand beaches (Synolakis, 1987;Hsiao et al., 2008;Saelevik et al., 2013); mangrove forests (Irtem et al., 2009;Yao et al., 2018a) and coral reefs (Yao et al., 2018b). For the artificial pile breakwaters, their design standards or codes are generally established for wind waves. However, tsunami waves are different from wind waves in their magnitudes and dispersive properties, thus the performance of pile breakwater in absorbing the energy of tsunami waves requires further investigation.
There are numerous studies focusing on wave interac-tion with pile structures (Chaplin et al., 1992;Kamath et al., 2016;Cao and Wan, 2015;Tai et al., 2019;Fan et al., 2019). Many of these studies have focused on regular or irregular waves. Notably, Tai et al. (2019) experimentally investigated the complex interaction between a vertical pile and a plunging breaker, with the focus given to the measurement of wave forces acting on the pile. It was found that the peak force acting on the pile is strongly affected by the relative location of the pile with respect to the wave breaking location. In terms of solitary wave interaction with pile structures, earlier studies focused primarily on a single or an array of pile siting on a horizontal bottom (Mo et al., 2007;Mo and Liu, 2009). Huang and Yuan (2010) was the pioneer study to have experimentally investigated solitary wave interaction with a row of slotted piles. Wave reflection and transmission are also discussed by them to access the reduction of drag force on coastal structures protected by the piles. They found that the gap between the adjacent piles was the key factor to control solitary wave transmission. Subsequently, Liu et al. (2011) employed a shallow-waterequation-based model to reproduce the laboratory experiments of Huang and Yuan (2010), and they found that wave transmission was well captured when the ratio of wave height to water depth was less than 0.25. Recently,  applied the filtered Naiver−Stokes equations with the Large Eddy Simulation (LES) turbulence model to examine the response of a row of vertical slotted piles to the solitary wave action. The impact force on the piles was analyzed in view of the drag coefficient with the interfering effect among the piles included. Their investigation was more recently extended to solitary wave interacting with a doublerow of piles (Yao et al., 2018c).
Nevertheless, the piles in the above studies were all located on flat seabeds. For practical considerations, a larger amount of pile structures is constructed in the coastal highhazard areas such as in the surf zone, where wave breaking and high-speed roller runup can produce extremely large wave forces during hurricane or tsunami events (Xiao and Huang, 2015). Therefore, a reasonable reproduction of the surf-zone process on the slope is vital to interpreting the hydrodynamics around the piles. Mo et al.'s (2013) study was one of the pioneer studies to validate a Naiver− Stokesequation-based approach based on the LES model to investigate the solitary wave breaking and interacting with a slender cylinder over a sloping beach. Subsequently, Xiao and Huang (2015) used Reynolds-averaged Navier−Stokes (RANS) equations with a model for the turbulence to examine wave-pile interaction with the pile located at seven different positions on a slope. More recently, Chella et al. (2017) also applied a similar RANS approach to investigate the effects of the breaking characteristics, the geometric properties, the relative pile positions and the incident wave heights on the breaking-wave force characteristics of a single pile on the slope. However, all the literature concen-trated on a single pile rather than a row of piles on a sloping beach. For the latter case, the effect of interference among the piles on the wave runup process on the slope may become important.
Therefore, to improve our current understanding of solitary wave runup reduction by a pile breakwater on a sloping beach, a set of numerical simulations were performed to investigate wave reflection from a row of vertical slotted piles, wave transmission through the piles as well as wave runup on the slope behind the piles. A 3D (three-dimensional) numerical wave tank based on the open-source CFD (Computational Fluid Dynamics) tool OpenFOAM® was developed. OpenFOAM® supports two-phased incompressible flow modeling, which has been proved an efficient and powerful tool to explore the complicated nearshore wave dynamics (Higuera et al., 2013b). Importantly, the turbulence model LES was applied in the present study in that LES was widely used to simulate fast and highly unsteady motions (Pope, 2000), such as the flows in the present pile slots. A recent study of Yao et al. (2018c) also showed that the LES turbulence model can provide more accurate predictions of flow and vorticity fields than the RANS model around a double-row of vertical slotted piles located on a horizontal seabed. In addition, the free surface motions were tracked by the commonly used VOF (Volume of Fluid) method. Supplementary laboratory experiments were also conducted to validate the numerical model adopted in this study.
The rest of the paper is organized as follows. Section 2 looks at the numerical approaches including the governing equations and the wave generation method. The laboratory and numerical settings are introduced in Section 3, respectively. Section 4 describes model validations based on the free surface elevation, the velocity, the dynamic pressure and the shoreline runup. Model applications in view of a detailed analysis of wave runup reduction by the pile breakwater are given in Section 5. The conclusions obtained from this study are shown in Section 6.

Governing equations
In the LES approach, it is essential to separate the velocity field that contains the large-scale components, the filtered Navier−Stokes equations is obtained by filtering the velocity field (Leonard, 1975). The filtered velocity is defined as: where is the filter kernel. The eddy sizes are described by using a characteristic length scale ( ), which is defined as: ∆x ∆y ∆z where , , and are the grid size in streamwise, span-∆ ∆ wise and vertical directions, respectively. Eddies larger than are roughly considered as large eddies, and they are directly solved. Those who are smaller than are considered as small eddies.
The filtered continuity and momentum equations are where is the filtered pressure, is the strain rate of large scales, which is defined as: and is the residual stress that uses a sub-grid scale (SGS) model to get a full solution for the Navier−Stokes equations above.
The SGS stress is usually calculated by a linear relationship with the rate of strain tensor based on the Boussinesq hypothesis. The one-equation eddy viscosity mode, which is supposed to be better than the well-known Smagorinsky model for solving the highly complex flow and shear flow (Menon et al., 1996), is employed in this study. Based on the one-equation model of Yoshizawa and Horiuti (1985), the sub-grid stress is defined as: δ ij ν s where is the Kronecker-delta, is the SGS eddy viscosity, which is given by k s and the SGS kinetic energy is solved by , and as recommended by the OpenFOAM Foundation (2016). The flow field associated with wave breaking consists of water and air phases, and they are solved via the VOF method (Hirt and Nichols, 1981). In this method, the fluid density is defined as: where is the density of water, is the density of air, and is the volume fraction of water contained in a mesh cell. The transport of is modeled by an advection equation The third term on the left-hand side of Eq. (10) is an artificial compression term to avoid the excessive numerical diffusion and the interface smearing, the new introduced velocity is suitable to compress the interface.
In the solver interFoam, the algorithm PIMPLE, which is a mixture of the PISO (Pressure Implicit with Splitting of Operators) and SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithms, is applied to solve the coupled velocity and pressure fields. The MULES (multi-dimensional universal limiter for explicit solution) method is used to maintain boundedness of the volume fraction independent of the underlying numerical scheme, mesh structure, etc. Euler scheme is utilized for the time derivatives, Gauss linear scheme is employed for gradient term, and Gauss linear corrected scheme is selected for the Laplacian term. Detailed implementation of the model can be found in the OpenFOAM Foundation (2016).

Wave generation and absorption
The official version of OpenFOAM® does not include the wave generation and absorption module, which is a key element for the numerical wave tank. Therefore, supplementary tools such as waves2Foam (Jacobsen et al., 2012) and IHFOAM (Higuera et al., 2013a) were developed. The IHFOAM is selected in this study by considering that it imposes an active wave absorbing boundary condition without a relaxation zone as required by the waves2Foam. Moreover, it supports various wave theories including the solitary wave theory. In the present numerical study, a combined wave surface and velocity inlet boundary condition provided by IHFOAM is used. The prescribed free surface and velocities for generation of solitary wave at the inlet boundary are (Lee et al., 1982) where is the wave height, is the free surface elevation, is the water depth, is the wave celerity, and are the velocities in the streamwise and vertical directions, respectively. Active wave absorption function is also included in the IHFOAM boundary condition used in the present numerical study. As for details of this active wave absorption, please refer to Higuera et al. (2013a). For a detailed numerical setup including the layout of boundary conditions, please refer to Section 3.2. typed wavemaker was located at one end of the flume to generate the design solitary waves. The beach with a slope of ( is the slope inclination angle) was installed with its toe located 20 m away from the wavemaker. A group of vertical slotted piles ( Fig. 1b) was designed based on the Froude law of similarity. A geometric scale factor of 1:10 was used for a prototype pile breakwater in a coastal position of Singapore (Huang and Yuan, 2010). The individual pile diameter was and the center-tocenter space between adjacent piles was , thus the ratio of space to diameter was . The piles were arranged in a row along the flume width, and the row of pile models was placed on the slope which is 3 m shoreward of the slope toe. Both the piles and the sloping beach were made of PVC materials.
The water surface was monitored at five locations along the flume, including S1 at 3 m upstream of the slope, S2 at the slope toe, S3 at 0.4 m seaside of pile breakwater, S4 at 0.4 m leeside of pile breakwater and S5 at 0.8 m leeside of pile breakwater (Fig. 1c). The non-intrusive ULS-100 ultrasonic wave gauges (Sinfotek, China) with a nominal instrument error of 0.1 mm are used, considering the potential error induced by wave breaking. The actual accuracy of these wave gauges in the present setup is determined to be smaller than 1 mm. An Acoustic Doppler Velocimetry (Nortek Vectrino, Norway, referred to as an ADV hereafter) was employed at 2 m upstream of the pile breakwater to measure the current at a depth of 0.05 m under the still water level (Fig. 1c). The accuracy of the ADV depends on the ac-curacy of the sound speed model, with tap water used in the present experiment, the relative accuracy of the ADV is smaller than 2%. Four pressure transducers (TEST, China) were used to record the dynamic pressure around the pile surface, being placed on the front (P1), rear (P3), and two side faces (P3 and P4) of the most centered pile in the row (Fig. 1d). The transducers were put 0.01 m above the pile bottom so that they were submerged throughout the experiments. All the above measurement instruments were sampled with a frequency of 50 Hz via a data acquisition system. It must be admitted that due to the relatively low sampling frequency (50 Hz), the pressure measurements may have missed the real peak and fine fluctuations of pressure time series, but these fine structures would have been of a very small temporal and spatial scale anyway and should not play a significant role in the overall forcing acting on the pile structures. In order to measure the maximum runup, which is defined as the maximum elevation of wave uprush above the still water level, marker lines with a spacing of 1 cm have been drawn on the slope. A video camera (C1 in Fig. 1a, Logitech, Switzerland) was located on the top of the flume, behind the pile breakwater to capture the water uprush process. Due to the effects of slotted piles, side walls and slight non-uniformity of the slope roughness, it was observed during the experiments that the moving shoreline on the slope was not straight. Therefore, the maximum vertical runup height was obtained by examining the upmost waterline on the slope. The error is believed to be smaller than 1 cm (within two adjacent marker lines), cor-  China Ocean Eng., 2020, Vol. 34, No. 1, P. 10-20 13 responding to a vertical runup height of smaller than 0.5 mm.
Four offshore solitary wave heights ( , 0.04, 0.06, 0.08 m) were examined with a fixed offshore water depth of . Plunging breakers were observed in all these wave conditions because the slope parameter ( ) ranged from 0.12 to 0.24, and it fell into the type of plunging breaker as suggested by Grilli et al. (1997). Each tested case was run for three times to guarantee the data repeatability. Serval minutes elapsed between the tests so that the flume can become relatively calm. Fig. 2 shows the general layout of the computational domain and the details in mesh decomposition. In order to maintain the balance between the simulation accuracy and efficiency, the numerical model was set to reproduce main aspects of the laboratory experiments. The aforementioned

Numerical setup
wave generation and absorption inlet boundary was applied at the beginning of numerical domain (left end of the domain as shown in the upper panel of Fig. 2). The solitary wave length can be estimated by the horizontal distance that contains 95% of its total mass, i.e., (Dean and Dalrymple, 1991). For the tested scenarios in this study, the above formula gives the largest for and . Therefore, the numerical waves were set to be generated at 4 m seaside of the slope toe, which was supposed to be sufficiently far. For the other boundaries, the top boundary was free to the atmosphere, while both the bottom and the end of domain were no-slip wall conditions with a wall function. To reduce the numerical cost, the most three centered piles in the row of piles were chosen for simulations. We thus adopted the symmetric boundary conditions at both side faces of the domain by considering the symmetric distribution of the slotted piles along the domain centerline. For the pile surface, we also imposed the no-slip wall condition with a wall function.
We applied the structured mesh to discretize the numerical domain, and varying grid size was utilized to reduce the total number of mesh cells. Grid size was maintained a constant 4 mm in the spanwise direction. In vertical direction, the grid size increased gradually from 1.5 mm to 4 mm within a region 50 mm high near the slope bottom to capture the sheet flow associated with the wave runup, and it was then kept a constant of 4 mm all the way to the domain top (see Fig. 2). In the streamwise direction, the grid size changed slowly from 20 mm at the domain inlet to 4 mm at a location 25 mm upstream of the piles. The core region covering from 25 mm seaside to 25 mm leeside of the piles set a constant size of 4 mm. Grid refinement close to the pile surface was realized by a few layers of structured cells with a uniform size of 2.5 mm (see also Fig. 2). Further downstream, grid size increased gradually, and it went back to 10 mm at the end of domain. The total grid cells was 6.1 million.
The time step was adjusted automatically during the  A numerical convergence test was performed by varying the grid size around the pile surface using the typical wave condition of and . In addition to the default grid size of 2.5 mm, we also tested another three sizes (1.25, 5 and 10 mm). Fig. 3 shows the results for the normalized maximum free surface elevation and pressure on the seaside face of the pile, the normalized maximum streamwise velocity in the mid-depth of the slot, as well as the normalized maximum runup on the slope. The differences in the simulations of wave, flow, pressure and runup were generally smaller than 3% when the grid size around the pile surface declined from 2.5 mm to 1.25 mm, indicating that our selection of the grid size of 2.5 mm was sufficient for present simulations in view of both computational accuracy and efficiency.

Solitary wave runup on a sloping beach in the absence of slotted piles
For the traditional plane beaches, Synolakis (1987) is the pioneer to propose a power-law empirical formula to characterize the maximum wave runup on a slope under the action of breaking solitary waves Subsequently, Hsiao et al. (2008) suggested the following empirical expression by including the slope effect To simulate this scenario by the present model, the numerical setup was the same as those described in Section 3.2 except for the removal of the pile. Taking the aforementioned wave condition ( and ) as an example, we firstly compared the numerical generated solitary wave profile at S1 with the theoretic profile calculated by Eq. (11). The adopted model accurately reproduced the leading solitary waves, except for some small tail wave oscillations (Fig. 4a). The tail wave oscillation that deviates from the theoretical profile is likely to be induced by the high-order non-linear effects that the prescribed wave generating boundary condition did not include. We remark that the present study focuses on the primary soliton, and tail waves are not important. We then ran the four tested wave conditions in the laboratory experiments and compared the simulations with the predictions from both Eq. (14) and Eq. (15) in Fig. 4b. It appears that Hsiao et al.'s (2008) formula predicted slightly less wave runup than those predicted by Synolakis (1987) for the current laboratory slope of 1:20. The simulated wave runups in general agreed fairly well with the predictions from Hsiao et al. (2008) formula, indicating that the present model is robust to model the breaking solitary wave process on a plane slope without the slotted piles.

Solitary wave runup on a sloping beach in the presence of slotted piles
In the section, model validation in terms of the present experiments was performed. The above wave condition ( and ) was again chosen to show the model robustness. Fig. 5a compares the time series of normalized free surface elevations among the wave sampling locations (S1−S5). The time is normalized by and indicates the onset of numerical generation of offshore solitary wave. Good agreements between measurements and simulations of the leading solitary waves can be observed at all locations. Those reflected waves (second peaks in the time series) and tails wave oscillations (at S4 Fig. 3. Variations of the normalized maximum free surface elevation on the seaside face of the pile, maximum pressure along the depth in front of the pile, maximum mid-depth velocity in the slot and maximum wave runup on the slope with different grid sizes around the slotted piles.  and S5) caused by transmitted wave breaking and runup on the slope were reasonably captured as well. The adopted model also satisfactorily reproduced the measurements of normalized streamwise velocity and vertical velocity associated with the leading solitary wave at the ADV sampling location (Fig. 5b). Fig. 5c presents the time series of normalized dynamic pressure at three pressure sampling locations (P1−P3). Again, the adopted model reasonably simulated the peak pressure at P1 (the seaside face of pile) associated with the leading wave impact as well as the pressure drop at both P2 (the side face of pile) and P3 (the leeside face of pile) due to velocity acceleration both in the pile slot and behind the pile. Some fluctuations in the laboratory data largely result from the entrapped air bubbles around the slot, and they can be visible during the laboratory observations. There is a slightly larger deviation between the experimental and numerical results during the pressure drop phase after the peak value for P1. This is likely due to larger measurement error of the pressure sensor P1 induced by the relatively shallow submergence due to the thin turbulent flow after the peak during the experiment. Note that since P2 and P3 are mounted much closer to the sloping beach surface (please refer to Fig. 1 for pressure sensor locations), the submergence of P2 and P3 is sufficiently deep so that the measurement accuracy is not affected like P1. It is interesting to note a pressure drop at P2 and P3 prior to the arrival of the peak dynamic pressure, which is not captured by P1. The cause of this phenomenon is likely associated with flow separation at the side and leeside of the pile. As the wave front approaches, the wave induced flow increases in intensity and induces a flow separation, which in turn induces a pressure drop before the turbulent bore of the broken wave washes in and creates a dynamic pressure peak. For P1, the pressure gauge is located at the front stagnation point. No flow separation occurs here, hence there is no pressure fluctuation.
Subsequently, a comparison of the simulated wave heights at selected sampling locations as well as the maximum wave runup on the slope with those measured is given by Fig. 6 for the four tested wave conditions. At both S1 (offshore) and S3 (on the slope seaside of the pile breakwater), the model captured the offshore wave propagation on the horizonal seabed and wave shoaling on the slope very well. However, after the wave transmitted through the piles (S4), model performance was less satisfactory. Slight underprediction of the wave height could be found for small waves. As for the maximum wave runup, about 10% consistent under-prediction could also be observed. We attribute it partly to the grid size (1.5 mm) used near the bottom boundary on the slope, which may still not be sufficiently fine to capture the upmost waterline associated with the very shallow sheet flow on the smooth PVC surface in the experiments. Partly to the air-bubble entrainment in the experiments associated with accelerated flow in the slots, which may cause inaccurate free surface estimation from the adopted VOF technique when modeling wave uprush motion on the slope.

Model applications
In Section 4, the present model has been shown capable of simulating breaking solitary wave process on a sloping beach with a row of vertical slotted piles. In this section, a couple of scenarios to evaluate the impacts of four parameters (offshore wave height, offshore water depth, adjacent and . In each run, only one parameter was changed while other three parameters were kept unaltered, thus we examined a total of 16 scenarios.
With the availability of detailed wave surface distribution data, it is possible to figure out the breaking point location, which for a plunging breaker is defined as the point where a large portion of the front face of a wave becomes nearly vertical. For a spilling breaker, the breaking point is defined as the point where the whitewater begins to roll down the front face of a wave.

Wave reflection and transmission
By taking the reflected wave height estimated from the simulated timeseries at S1 ( ) and transmitted wave height estimated at S4 ( ) for examples, both wave reflection from the slotted piles and transmission through the piles can be analyzed. Dimensionless reflected wave height ( ) decreased slowly with the increase of offshore wave height (Fig. 7a). However, it slightly increased with increasing offshore wave depth (Fig. 7b) because a larger portion of the vertical piles could interact with the incident wave at a declined as the pile space increased (Fig. 7c) due to less flow blockage exerted by the piles. Similar decreasing of wave reflection with increasing beach slope was also observed (Fig. 7d), which is contrary to the case for wave interaction with a slope without the pile structure. This can be explained by Fig. 8, which shows the locations of wave breaking points. In Fig. 8, it is clearly shown that milder slope caused the incipient wave breaking further shoreward, resulting in larger wave impact on the vertical piles (thus more wave reflection). The values of generally lied between 0.25 and 0.55 within the tested ranges of the parameters, indicating that a significant portion of the incident wave energy could be reflected back due to a combined effect of the pile breakwater and the sloping beach.
Transmitted wave height shoreward of the piles is supposed to be directly related to the shoreline wave runup height. The dimensionless transmitted wave height at S4 ( ) decreased rapidly with increasing offshore wave height (Fig. 7a). Whereas, it enhanced with the increase of offshore water depth (Fig. 7b) in that larger water depth, meaning more offshore wave energy according to Eq. (11), thus more transmission. More wave energy could be transmitted through the slotted piles as the space between adjacent piles became larger (Fig. 7c) and the transmitted wave was also higher on the steeper slope due to the increased shoaling effect (Fig. 7d). Those scenarios with indicate that the transmitted wave could be still larger than Fig. 6. Comparison between the simulated and measured wave heights at S1, S3 and S4 ( , and ) as well as the maximum wave runup on the slope . Fig. 7. Variations of simulated reflected wave height at S1 ( ) and transmitted wave height at S4 ( ) with diffrent: (a) offshore wave heights ; (b) offshore water depths ; (c) adjacent pile spaces , and (d) beach slopes . the offshore wave due to wave shoaling on the slope although a substantial portion of incident wave energy may be removed by both wave reflection and turbulences generated around the piles.

Wave runup
Maximum shoreline wave runup increased with increasing offshore wave height (Fig. 9a). This is expected because the incident wave energy rose with the increase of wave height. It also increased with increasing water depth (Fig. 9b) again due to the fact that more incident wave energy is associated with larger water depth. Moreover, maximum wave-induced runup increased with the increase of adjacent pile space (Fig. 9c) because it reduced the flow blockage among the piles. Similar growth of wave runup with increasing beach slope is also observed, which was consistent with Eq. (15) for the plane slope without the pile breakwater.
Subsequently, a quantitative description of the dependence of the maximum shoreline wave runup on the aforementioned four parameters (offshore wave height, offshore water depth, adjacent pile space as well as beach slope) was conducted. Motivated by the empirical formula proposed by Hsiao et al. (2008), i.e., Eq. (15), we propose a dimensionless empirical formula for predicting the wave runup. A regression analysis of the tested scenarios in Fig. 9 gives which has a Coefficient of Determination ( ) of 0.95. Note that the equation above is valid only for our tested ranges , and . Fig. 10 compares the predicted dimentionless wave runup from Eq. (16) with the simulated dimentionless wave runup from Fig. 9, very good agreement demonstrating the robustness of Eq. (16) to predict the shoreline wave runup associated with a solitary wave acting on a row of vertical slotted piles locating on a slope. We finally remark that when Eq. (16) is applied to a prototype pile breakwater, in addition to those factors explicitly considered in Eq. (16), the slope surface roughness, and the angle of wave incidence, etc., may also affect the values of empirical parameters in Eq. (16).

More on the wave runup
Finally, the tsunami-like solitary wave runup reduction by a row of vertical slotted piles was analyzed in Fig. 11 by   Comparing with the traditional plane slope, the function of a row of piles in mitigating the tsunami waves is obvious. Wave runup in Fig. 11 was reduced by 9%−62% when there was pile breakwater on the slope, and both the slope inclination and the ratio of adjacent pile space to pile diameter also counted. The maximum reduction of wave runup occurred for the mildest slope ( ) coinciding with the smallest . However, smaller means that the impacting forcing acting on the individual pile of the breakwater would be increased while the flows in the slots would be reduced due to the increased interfering effect between adjacent piles (Yao et al., 2018c). Hence in practical engineering design of the pile arrangement, a compromise should be made among such factors as the performance of the breakwater, the cost to maintain the strength of pile and the conservation of the surrounding water ecology. In defence against the tsunami waves, the resistance of pile to the collision of tsunami-induced debris may need to be considered as well.
We finally remark that the values of runup height in the present study may also be dependent on the arrangement of the slotted piles and the roughness of the slope surface, thus the conclusions for other beaches with varying pile configurations may be more or less different. Moreover, for a nature beach with the sandy or muddy bed, the pore water motion may also affect the measured runup. Nevertheless, although pile breakwaters are primarily designed to protect the coasts against the wind waves, our study clearly demonstrates a very active role that a pile breakwater can play in mitigating the tsunami wave runup on the beach, as well as a requirement in considering the effects of both beach slope and pile arrangement on the effectiveness of mitigation.

Conclusions
To evaluate the performance of a pile breakwater to mit-igate the nearshore tsunami waves during a tsunami, we developed a 3D numerical wave tank to study the tsunami-like solitary wave runup on a sloping beach with a row of piles based on the CFD tool OpenFOAM®. The Navier−Stokes equations were solved with the LES approach to model the turbulence and the VOF method to track the free surface. The numerical simulations were validated by existing empirical formulas for solitary wave runup on the slope without the pile structure as well as our new laboratory dataset for a row of piles. Simulations show that the adopted model was able to reasonably reproduce the free surface elevation, the velocity and the dynamic pressure in the vicinity of the piles and the maximum wave runup on the slopping beach. The model was subsequently employed to examine the role of different parameters (offshore wave height, offshore water depth, adjacent pile space and beach slope) in affecting the wave reflection and transmission around the pile breakwater as well as the wave runup on the slope behind the piles. It is found that wave reflection increased with the increase of offshore water depth, but decreased with increasing offshore wave height, adjacent pile space and beach slope. Following the variation of transmitted waves, shoreline wave runup not only increased with increasing offshore wave height and water depth, but also increased with the increase of pile space and beach slope. To account for the effects of investigated parameters, we finally proposed an empirical equation to predict the wave runup. The formula was then compared with an existing formula for solitary wave runup on a slope without the piles. It shows that wave runup could be reduced by 9%−62% by the slotted piles within the tested ranges. The maximum reduction of runup occurred for the piles with smallest pile space siting on the mildest slope. This study suggests that the construction of the pile breakwaters is an efficient way to de- fend the coasts against the inundation in those areas where the occurrence of tsunami is not rare.