Special Section on Particle Methods and their Applications in Ocean and Coastal Engineering, Part 1Free Access

SPH Modeling of Tsunami Wave Overtopping over the Reef-Flat with a Vertical Revetment

College of Ocean Engineering and Energy, Guangdong Ocean University, Zhanjiang 524088, P. R. China

The State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, P. R. China

E-mail Address: 2338709193@163.com

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Zhisheng Xia

Shandong Energy Group New Energy Co., Ltd, Shandong 250014, P. R. China

E-mail Address: zsxia@vip.qq.com

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, and

Hongjie Wen

School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510641, P. R. China

E-mail Address: wenhongjie@scut.edu.cn

Corresponding author.

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https://doi.org/10.1142/S252980702340002XCited by:0 (Source: Crossref)

Abstract

Artificial revetments are commonly constructed on reef-flats to protect the rear infrastructure. Evaluating the performance of a revetment under extreme wave conditions requires accurate prediction of wave overtopping. To simulate tsunami wave overtopping on an artificial revetment above a coral reef-flat, a numerical model based on the weakly compressible smoothed particle hydrodynamics (WCSPH) method is developed. This numerical model includes an additional dissipative term in the continuity equation and a dynamic boundary condition enhancement to correct the pressure distortion from particles at the boundary. Reef topography is simplified as a steep reef-face and a horizontal reef-flat, and tsunami waves are described as solitary waves. This research explores the characteristics of solitary wave overtopping on the revetment above the reef-flat with varying slope angles and reef-flat lengths.

Keywords:

1. Introduction

Coral reef coasts are prevalent in tropical and subtropical seas, they are significant for the development of marine biological, mineral, and space resources. Typically, they have a steep reef-face and a flat reef-flat. Coral reef coasts are situated in remote locations in the open ocean and are vulnerable to extreme ocean environments, such as tsunami waves and storm surges due to the lack of effective protection measures. Thus, studying the breaking and overtopping of extreme waves on coral reef coasts is crucial for designing and evaluating the safety of coral reef coastal structures, which is essential for the sustainable development and ecological protection of coral reefs.

Field observation is the first research method to study wave propagation and deformation over coral reef coasts. Munk and Sargent [1948] discovered that the water level on the reef-flat in the Marshall Islands was 0.45–0.6m higher than the surrounding sea area due to wave breaking at the reef edge and over the reef-flat. Lugo Fernández et al. [1994] found that waves propagated through Margarita Reef in Puerto Rico resulted in a decrease of 82% in wave height and dissipation of 96% in wave energy. Lowe et al. [2009] identified that the primary driving force of water circulation in Kaneohe Bay, Hawaii was the water level gradient generated by wave setup.

Physical model experiment is a widely used research method for investigating wave propagation and deformation on coral reef geomorphology. Previous experimental data was summarized by Nelson [1994], who found that the maximum relative wave height over the horizontal reef-flat was approximately 0.55. Gourlay [1996] conducted a series of physical model experiments to systematically study wave propagation, wave setup, and wave-induced current on coral reefs. Buckley et al. [2016] studied the effects of surface roughness on wave propagation and wave setup. Xu et al. [2020] investigated the propagation and deformation characteristics of waves on coral reefs using a physical model experiment. Liu et al. [2020] studied the process of wave propagation and overtopping on coral reefs with seawalls using a large-scale wave tank. Zhu et al. [2020] studied the effects of vertical walls on wave deformation and wave breaking on an ideal-shaped reef-flat through a two-dimensional physical model experiment.

Numerical simulation is also a well-established approach for investigating wave propagation and deformation on coral reef topography. Boussinesq-type wave models are widely used for studying wave evolution over the reef-flat due to their low computational cost and ability to reasonably predict wave propagation and deformation. Roeber et al. [2010] established a nearshore wave propagation model for coral reefs based on the shock-capturing Boussinesq model and a classical eddy viscosity model. Fang et al. [2022] developed a two-layer Boussinesq model to simulate the propagation and shoaling processes of solitary waves on the reef topography. Numerical models based on the Navier–Stokes equations have also been developed to simulate the flow field in detail and study wave propagation and deformation on coral reef topography. Yao et al. [2019] used the finite volume method to develop a numerical model to study the interaction between solitary waves and the island reef topography, and the vertical distribution of flow velocity is obtained. Wen et al. [2019, 2020a, 2020b] systematically studied the influences of reef permeability on the propagation, deformation, and breaking processes of waves on coral reef topography, as well as the spatial distribution of wave setup, infragravity waves, and wave-induced currents on the reef-flat by establishing an anisotropic porous media model based on the SPH model.

In recent years, various revetment structures of different shapes have been built on the reef-flat to protect rear facilities and ensure personnel safety, with the aim of developing coral reef resources more effectively. While scholars have conducted extensive research on wave evolution over natural coral reefs, research on wave evolution over coral reefs with artificial revetments is lacking (Wen et al. [2019]). These revetments can block wave-generated currents, increase wave setup, and even induce infragravity wave resonance over the reef-flat, which can intensify and increase the probability of wave overtopping on the revetment. Severe wave overtopping can cause submersion, leading to building collapse, damage to coastal design facilities, and employee injuries. The presence of the horizontal reef-flat means that wave overtopping on reef revetments is significantly different from that of the traditional coastal protection. Therefore, basic research is necessary to determine whether the empirical formula for wave overtopping discharge for traditional revetments is applicable to reef revetments.

Wave overtopping phenomenon is a complex hydrodynamic problem that involves the separation and aggregation of free water surfaces, posing significant challenges for numerical simulation techniques. Traditional methods based on Euler grids require complex techniques for interpolation and reconstruction to address these problems, resulting in decreased computational accuracy and efficiency. In contrast, meshless SPH methods offer unparalleled advantages over grid methods due to their Lagrangian characteristics and adaptability, particularly when dealing with large free surface deformation problems such as wave overtopping. With the continuous improvement of the accuracy and efficiency of the SPH method, it has been successfully employed to simulate wave breaking, wave impact, and other highly nonlinear problems (He et al. [2021, 2023]; Kazemi & Luo [2022]; Luo et al. [2021]; Lyu et al. [2023]; Ming et al. [2018]; Shi et al. [2018]; Sun et al. [2021]; Wang & Liu [2020]; Zheng et al. [2014]). This study systematically investigates the wave overtopping process on a reef revetment under tsunami waves using a validated SPH model. It also examines the influences of reef topography, revetment position, and water depth on wave overtopping.

2. Numerical Model

2.1. Governing equations

The present model assumes water as the weakly compressible fluid and describes fluid motion through Favre average N-S equations as follows :

dρdt=−ρ∇⋅u+ψ,<math display="block" altimg="eq-00002.gif"><mfrac><mrow><mi>d</mi><mi>ρ</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mi>ρ</mi><mo>∇</mo><mo>⋅</mo><mstyle mathvariant="bold-italic"><mi>u</mi></mstyle><mo>+</mo><mi>ψ</mi><mo>,</mo></math>(1)

dudt=−1ρ∇P+g+(υ+υt)∇2u,<math display="block" altimg="eq-00003.gif"><mfrac><mrow><mi>d</mi><mstyle mathvariant="bold-italic"><mi>u</mi></mstyle></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>ρ</mi></mrow></mfrac><mo>∇</mo><mi>P</mi><mo>+</mo><mstyle mathvariant="bold-italic"><mi>g</mi></mstyle><mo>+</mo><mo stretchy="false">(</mo><mi>υ</mi><mo>+</mo><msub><mrow><mi>υ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">)</mo><msup><mrow><mo>∇</mo></mrow><mrow><mn>2</mn></mrow></msup><mstyle mathvariant="bold-italic"><mi>u</mi></mstyle><mo>,</mo></math>(2)

where

$u$ , P and

$ρ$ denote the filtered velocity, pressure and fluid density, respectively.

$g$ is the gravitational acceleration.

$υ$ and

$υ_{t}$ represent the molecular viscosity of water (10

$^{- 6}$ m²/s) and the turbulence eddy viscosity, respectively. The turbulence eddy viscosity is determined through the Smagorinsky model

$ν_{t} = {(C_{s} d x)}^{2} | S |$ , where

$C_{s} = 0.1$ is the Smagorinsky constant and dx is the particle spacing.

$| S | = {(2 S_{c d} S_{c d})}^{1 / 2}$ and

$S_{c d} = 0.5 (\partial u_{c} / \partial x_{d} + \partial u_{d} / \partial x_{c})$ , where c and d refer to spatial coordinates.

$ψ = χ h c_{0} \nabla^{2} ρ$ is the adding diffusive term that suppresses the non-physical pressure oscillations (Antuono et al. [2012]). Here, h and

$c_{0}$ represent the smoothing length and the reference speed of sound, respectively. Following Marrone et al. [2011], the value of

$χ$ is set to 0.1 in our model. The pressure is directly calculated by the equation of state as shown below :

P=B[(ρρ0)7−1],<math display="block" altimg="eq-00017.gif"><mi>P</mi><mo>=</mo><mi>B</mi><mfenced separators="" open="[" close="]"><mrow><msup><mrow><mfenced separators="" open="(" close=")"><mrow><mfrac><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>7</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow></mfenced><mo>,</mo></math>(3)

where

$B = c_{0}^{2} ρ_{0} / 7$ is the parameter to limit the fluid compressibility,

$ρ_{0} = 1000 kg / m^{3}$ denotes the reference density and

$c_{0}$ must be at least ten times greater than the maximum fluid velocity to keep density variation less than 1% (Monaghan [1994]).

By applying the standard SPH discretizations, governing equations (1) and (2) can be discretized as follows :

dρidt=ρiN∑j=1mjρj(ui−uj)⋅∇i˜Wij+γhc0N∑j=1mjρj(ρj,f−ρi)rij⋅∇i˜Wij[r2ij+(0.1h)2],<math display="block" altimg="eq-00021.gif"><mfrac><mrow><mstyle><mtext mathvariant="normal">d</mtext></mstyle><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mstyle><mtext mathvariant="normal">d</mtext></mstyle><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi></mrow></msub><munderover accentunder="true" accent="true"><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi></mrow></munderover><mfrac><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><msub><mrow><mstyle mathvariant="bold-italic"><mi>u</mi></mstyle></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow><mstyle mathvariant="bold-italic"><mi>u</mi></mstyle></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>⋅</mo><msub><mrow><mo>∇</mo></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mover accent="true"><mrow><mi>W</mi></mrow><mo>̃</mo></mover></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><mi>γ</mi><mi>h</mi><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub><munderover accentunder="true" accent="true"><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi></mrow></munderover><mfrac><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfrac><mfrac><mrow><mo stretchy="false">(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>j</mi><mo>,</mo><mi>f</mi></mrow></msub><mo>−</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mstyle mathvariant="bold-italic"><mi>r</mi></mstyle></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>⋅</mo><msub><mrow><mo>∇</mo></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mover accent="true"><mrow><mi>W</mi></mrow><mo>̃</mo></mover></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mo stretchy="false">[</mo><msubsup><mrow><mi>r</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>.</mo><mn>1</mn><mi>h</mi><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy="false">]</mo></mrow></mfrac><mo>,</mo></math>(4)

duidt=−N∑j=1mj[piρ2i+pjρ2j]∇i˜Wij+g+N∑j=14(υ+υt)mjrij⋅∇i˜Wij⋅(ui−uj)(ρi+ρj)[r2ij+(0.1h)2],<math display="block" altimg="eq-00022.gif"><mfrac><mrow><mstyle><mtext mathvariant="normal">d</mtext></mstyle><msub><mrow><mstyle mathvariant="bold-italic"><mi>u</mi></mstyle></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mstyle><mtext mathvariant="normal">d</mtext></mstyle><mi>t</mi></mrow></mfrac><mo>=</mo><mo>−</mo><munderover accentunder="true" accent="true"><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi></mrow></munderover><msub><mrow><mi>m</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="" open="[" close="]"><mrow><mfrac><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>j</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></mfenced><msub><mrow><mo>∇</mo></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mover accent="true"><mrow><mi>W</mi></mrow><mo>̃</mo></mover></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><mstyle mathvariant="bold-italic"><mi>g</mi></mstyle><mo>+</mo><munderover accentunder="true" accent="true"><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi></mrow></munderover><mfrac><mrow><mn>4</mn><mo stretchy="false">(</mo><mi>υ</mi><mo>+</mo><msub><mrow><mi>υ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>j</mi></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>⋅</mo><msub><mrow><mo>∇</mo></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mover accent="true"><mrow><mi>W</mi></mrow><mo>̃</mo></mover></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>⋅</mo><mo stretchy="false">(</mo><msub><mrow><mstyle mathvariant="bold-italic"><mi>u</mi></mstyle></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow><mstyle mathvariant="bold-italic"><mi>u</mi></mstyle></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msubsup><mrow><mi>r</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>.</mo><mn>1</mn><mi>h</mi><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy="false">]</mo></mrow></mfrac><mo>,</mo></math>(5)

where N represents the total number of particles within the support domain of particle i.

$m_{j}$ and

$ρ_{j}$ denote the mass and density associated with neighboring particle j.

$ρ_{j, f}$ is the modified density of the fluid particle j, while

$\nabla_{i} {\tilde{W}}_{i j}$ denotes the normalized derivative of the kernel function, which can be written as

ρj,f=ρ0(pj+ρjg⋅rijB+1)1/7,<math display="block" altimg="eq-00027.gif"><msub><mrow><mi>ρ</mi></mrow><mrow><mi>j</mi><mo>,</mo><mi>f</mi></mrow></msub><mo>=</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mfenced separators="" open="(" close=")"><mrow><mfrac><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>+</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>j</mi></mrow></msub><mstyle mathvariant="bold-italic"><mi>g</mi></mstyle><mo>⋅</mo><msub><mrow><mstyle mathvariant="bold-italic"><mi>r</mi></mstyle></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mi>B</mi></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>1</mn><mo stretchy="false">/</mo><mn>7</mn></mrow></msup><mo>,</mo></math>(6)

∇i˜Wij=∇iWij∑Nj=1mjρj(rj−ri)⊗∇iWij,<math display="block" altimg="eq-00028.gif"><msub><mrow><mo>∇</mo></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mover accent="true"><mrow><mi>W</mi></mrow><mo>̃</mo></mover></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mo>∇</mo></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mstyle mathvariant="bold-italic"><mi>W</mi></mstyle></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi></mrow></msubsup><mfrac><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><msub><mrow><mstyle mathvariant="bold-italic"><mi>r</mi></mstyle></mrow><mrow><mi>j</mi></mrow></msub><mo>−</mo><msub><mrow><mstyle mathvariant="bold-italic"><mi>r</mi></mstyle></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>⊗</mo><msub><mrow><mo>∇</mo></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>W</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mfrac><mo>,</mo></math>(7)

where ⊗ is the tensor product and

$r_{k}$ is the position corresponding to particle k.

The present model utilized the quintic kernel proposed by Wendland [1995] as the kernel function, which can be expressed as

W(r,h)=74πh2(1−q2)4(2q+1)0≤q≤2,<math display="block" altimg="eq-00030.gif"><mi>W</mi><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>7</mn></mrow><mrow><mn>4</mn><mi>π</mi><msup><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><msup><mrow><mfenced separators="" open="(" close=")"><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>4</mn></mrow></msup><mo stretchy="false">(</mo><mn>2</mn><mi>q</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="1em" class="quad"></mspace><mn>0</mn><mo>≤</mo><mi>q</mi><mo>≤</mo><mn>2</mn><mo>,</mo></math>(8)

where

$q = r / h$ , r represents the distance between the neighboring particles. To integrate time, the explicit second-order symplectic scheme that preserves high resolution for long-time simulation is utilized in this study (Leimkuhler & Patrick [1996]). The variable time step used in the model is dependent on the Courant–Friedrich–Levy condition, the force terms, and the viscous diffusion term (Monaghan & Kos [1999]).

2.2. Boundary conditions

The dynamic boundary condition proposed by Crespo [2007] is usually implemented to simulated the solid boundary condition. The boundary particles, arranged in two layers similar to the fluid particles, are governed by the same equations as the fluid particles. The densities and pressures of the boundary particles undergo updates at every time step, while the positions and velocities remain unchanged or are externally imposed. To suppress the fluctuating pressure field in the vicinity of the solid boundaries and to prevent fluid particles from crossing the solid boundaries, the pressure of boundary particle $p_{k}$ is corrected using the subsequent approach (Ren et al. [2014]) :

p' j = p j + ρ j g \cdot r k j, <math display="block" altimg="eq-00033.gif"><msubsup><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow><mrow><mi>'</mi></mrow></msubsup><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>+</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>j</mi></mrow></msub><mstyle mathvariant="bold-italic"><mi>g</mi></mstyle><mo>\cdot</mo><msub><mrow><mstyle mathvariant="bold-italic"><mi>r</mi></mstyle></mrow><mrow><mi>k</mi><mi>j</mi></mrow></msub><mo>,</mo></math> (9)

pk,f=∑Nj=1p′jW(rkj,h)Δvj∑Nj=1W(rkj,h)Δvj,<math display="block" altimg="eq-00034.gif"><msub><mrow><mi>p</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>f</mi></mrow></msub><mo>=</mo><mfrac><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi></mrow></msubsup><msubsup><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow><mrow><mi>′</mi></mrow></msubsup><mi>W</mi><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>k</mi><mi>j</mi></mrow></msub><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi></mrow></msubsup><mi>W</mi><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>k</mi><mi>j</mi></mrow></msub><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfrac><mo>,</mo></math>(10)

p k = β p k, f + (1 - β) p' k . <math display="block" altimg="eq-00035.gif"><msub><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mi>β</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>f</mi></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>-</mo><mi>β</mi><mo stretchy="false">)</mo><msubsup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>'</mi></mrow></msubsup><mo>.</mo></math> (11)

Here, j represents the fluid particle within the support area of boundary particle k;

$p_{k, f}$ is the interpolated pressure of boundary particle k using its support area’s fluid particles;

$p_{k}^{'}$ is the boundary pressure calculated according to Eq. (3); N is the number of fluid particles supporting boundary particle k; In our simulations, we use

$β = 0.7$ , which effectively reduces the pressure difference between boundary and fluid particles, leading to a smooth pressure field near the boundary.

The horizontal displacement of the wave maker plate for solitary wave is written as

ξ(t)=Hkdtanh{k[c(t−T2)−(ξ(t)−S2)]},<math display="block" altimg="eq-00039.gif"><mi>ξ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>H</mi></mrow><mrow><mi>k</mi><mi>d</mi></mrow></mfrac><mo>tanh</mo><mfenced separators="" open="{" close="}"><mrow><mi>k</mi><mfenced separators="" open="[" close="]"><mrow><mi>c</mi><mfenced separators="" open="(" close=")"><mrow><mi>t</mi><mo>−</mo><mfrac><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mo>−</mo><mfenced separators="" open="(" close=")"><mrow><mi>ξ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><mfrac><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow></mfenced><mo>,</mo></math>(12)

where H and T are the solitary wave height and wave period, respectively; c is the solitary wave celerity and S is the piston stroke.

3. Model Verification

This section evaluates the performance of the developed model by comparing its predicted results for solitary wave overtopping on a seawall with Kobel et al.’s [2017] experimental data. The computational domain setup adheres to Kobel et al.’s physical model test and is depicted in Fig. 1. A piston type wavemaker is situated at the left end of the numerical flume to generate waves, with a distance of 5.0m from the toe of the seawall. The water depth in front of the breakwater is 0.25m, the height of the breakwater is 0.3m, and the top width is 0.02m. Under various wave conditions, wave overtopping discharge and maximum water body thickness at the top of the seawall are tested. The primary computational conditions are presented in Table 1. The numerical model utilizes a particle spacing of $d x = 0.5$ cm.

Fig. 1. Computational setup for wave overtopping on a seawall (unit: m).

**Table 1. Computational conditions for wave overtopping on a seawall.**
Water depth D	Wave height H	Slope angles $β$	Seawall height w	Seawall width $b_{K}$
0.25m	0.075m	18.4°	0.3m	0.02m
	0.125m	45°
	0.14m	90°
	0.175m

Figures 2 and 3 qualitatively describe the overtopping process when the wave crest reaches the top of seawall under varying conditions of wave height and seawall front face angle, respectively. The high-speed images captured by Kobel et al. [2017] are used as a reference. The predicted overtopping process compares favorably with the observed experimental images in terms of wave steepening and free surface curvature at the seawall crest. Figure 2 specifically focuses on the effect of relative wave amplitude ( $ε = H / D$ ) when $β$ is equal to 90°. At $ε = 0.3$ (Fig. 2(a)), the overflowing surface profile is similar to that of a standard weir overflow, except for the sloping free surface caused by the approaching solitary wave. At higher values of $ε$ , such as 0.5 in Fig. 2(b), a more peaked free surface profile is evident. At $ε = 0.7$ (Fig. 2(c)), these effects are further intensified, as an almost sharp peak wave crest and a steep upstream free surface slope become apparent.

Fig. 2. Comparisons of wave overtopping process between the experiment and numerical models under various wave height conditions (a) $ε = 0.3$ (b) $ε = 0.5$ and (c) $ε = 0.7$ .

Figure 3 illustrates the effect of the seawall front face angle on the overtopping process. All images in Fig. 3 represent the conditions prior to the jet impact on the tailwater. As shown in Fig. 3(a), the upstream free surface profile nearly follows the channel bottom, with slight wave crest curvature observed at the seawall crest when $β = 18 . 4^{\circ}$ . The compactness of the overflow process is increased as the angle increases to $β = 4 5^{\circ}$ (Fig. 3(b)), with a strong surface curvature noticeable at the seawall crest. At $β = 9 0^{\circ}$ (Fig. 3(c)), these effects are intensified even further, resulting in an almost sharp peaked wave crest and a steep upstream free surface slope.

Based on the experimental results, Kobel et al. [2017] gave the following dimensionless empirical formula for estimating the maximum water body thickness ( $d_{0} / w$ ) and wave overtopping discharge ( $V / D^{2}$ ) over the seawall crest :

d0w=1.32[ε(Dw)4[(β/90∘)−0.21−ε](β90∘)0.16]=1.32E2,<math display="block" altimg="eq-00073.gif"><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow><mrow><mi>w</mi></mrow></mfrac><mo>=</mo><mn>1</mn><mo>.</mo><mn>3</mn><mn>2</mn><mfenced separators="" open="[" close="]"><mrow><mi>ε</mi><msup><mrow><mfenced separators="" open="(" close=")"><mrow><mfrac><mrow><mi>D</mi></mrow><mrow><mi>w</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>4</mn><mo stretchy="false">[</mo><msup><mrow><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">/</mo><mn>9</mn><msup><mrow><mn>0</mn></mrow><mrow><mo>∘</mo></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mo>−</mo><mn>0</mn><mo>.</mo><mn>2</mn><mn>1</mn></mrow></msup><mo>−</mo><mi>ε</mi><mo stretchy="false">]</mo></mrow></msup><msup><mrow><mfenced separators="" open="(" close=")"><mrow><mfrac><mrow><mi>β</mi></mrow><mrow><mn>9</mn><msup><mrow><mn>0</mn></mrow><mrow><mo>∘</mo></mrow></msup></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>0</mn><mo>.</mo><mn>1</mn><mn>6</mn></mrow></msup></mrow></mfenced><mo>=</mo><mn>1</mn><mo>.</mo><mn>3</mn><mn>2</mn><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo></math>(13)

VD2=1.35[ε(Dw)(2/ε)(β/90∘)0.25(D+H/2−wbK)0.12]0.7=1.35W2,<math display="block" altimg="eq-00074.gif"><mfrac><mrow><mi>V</mi></mrow><mrow><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mn>1</mn><mo>.</mo><mn>3</mn><mn>5</mn><msup><mrow><mfenced separators="" open="[" close="]"><mrow><mi>ε</mi><msup><mrow><mfenced separators="" open="(" close=")"><mrow><mfrac><mrow><mi>D</mi></mrow><mrow><mi>w</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">/</mo><mi>ε</mi><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">/</mo><mn>9</mn><msup><mrow><mn>0</mn></mrow><mrow><mo>∘</mo></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mn>0</mn><mo>.</mo><mn>2</mn><mn>5</mn></mrow></msup></mrow></msup><msup><mrow><mfenced separators="" open="(" close=")"><mrow><mfrac><mrow><mi>D</mi><mo>+</mo><mi>H</mi><mo stretchy="false">/</mo><mn>2</mn><mo>−</mo><mi>w</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>K</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>0</mn><mo>.</mo><mn>1</mn><mn>2</mn></mrow></msup></mrow></mfenced></mrow><mrow><mn>0</mn><mo>.</mo><mn>7</mn></mrow></msup><mo>=</mo><mn>1</mn><mo>.</mo><mn>3</mn><mn>5</mn><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo></math>(14)

where

$E_{2}$ and

$W_{2}$ are the water body thickness parameter and the wave overtopping volume parameter, respectively.

Figure 4 presents the comparisons between the predictions from the numerical method and the empirical fitting formula in terms of the maximum water body thickness and the wave overtopping discharge over the seawall crest. The difference between the estimated values from numerical calculations and empirical formula is within 10%, demonstrating a satisfactory simulation effect. These findings serve as a robust confirmation of the SPH numerical model’s competence in capturing the non-linear deformation of a free surface.

Fig. 4. Comparisons between the predicted results and the experimental data in terms of (a) maximum water body thickness and (b) wave overtopping discharge.

4. Wave Overtopping Over the Revetment on Reef-Flat

4.1. Computational setup

This section utilizes the validated SPH model to replicate the phenomenon of solitary wave overtopping on a reef-flat with a vertical revetment. The purpose is to investigate the changes in wave overtopping under various conditions, such as different reef face slopes, distance between the vertical revetment and the reef edge, and the revetment elevation. In this study, we assume the reef to be impermeable, and we do not account for seepage within the reef at present. The layout of the numerical model is illustrated in Fig. 5. The coral reef is approximated as a slope with a 1:m inclination and a horizontal reef-flat section with a height of 1.0m. The revetment is a simple vertical one with a height of $h_{wall}$ . The water depth in front of the reef-flat is denoted as D and the height of the solitary wave is H. The simulation of the waves is conducted using the traditional pusher plate wave generator. The values of calculation parameters are presented in Table 2.

**Table 2. Computational conditions for wave overtopping over the reef-flat with a vertical revetment.**
Slope (m)	Revetment height ( $h_{wall}$ )	Reef-flat length (L)	Water depth (D)	Wave height (H)
1, 2, 3, 4, 5	0.20, 0.25, 0.30	0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0	1.10, 1.15, 1.20, 1.25, 1.30	0.20, 0.25, 0.30

Fig. 5. Layout of the numerical model for solitary wave overtopping over the reef-flat with a vertical revetment.

4.2. Numerical results and analysis

4.2.1. Influence of the slope of the reef-face

Figure 6 presents the relationship between the slope of the reef-face and wave overtopping discharge under different wave height conditions ( $L = 5.0$ m, $h_{wall} = 0.3$ m, $D = 1.1$ m). As illustrated in the figure, the wave overtopping discharge shows a rising-falling trend with a peak value near $m = 3$ , regardless of the wave height. Moreover, the wave overtopping discharge increases with increasing wave height, irrespective of the slope of the reef-face.

Fig. 6. Relationship between the slope of the reef-face and wave overtopping discharge under different wave height conditions ( $L = 5.0$ m, $h_{wall} = 0.3$ m, $D = 1.1$ m).

Figure 7 displays how the slope of the reef-face influences wave overtopping discharge at different water depths ( $L = 5.0$ m, $h_{wall} = 0.3$ m, $H = 0.2$ m). As indicated, for water depths that are relatively shallow ( $D = 1.10$ m, 1.15m), a slight initial increase in wave overtopping discharge is observed, which is followed by a decrease as the slope of the reef-face lessens. In contrast, at progressively greater water depths, such as at a depth of $D = 1.2$ m, wave overtopping discharge is monotonically reduced with a decreased slope. As a general trend, wave overtopping discharge increases with deeper water, holding the slope of the reef-face constant.

Fig. 7. Relationship between the slope of the reef-face and wave overtopping discharge under different water depth conditions ( $L = 5.0$ m, $h_{wall} = 0.3$ m, $H = 0.2$ m).

Figure 8 illustrates how the slope of the reef-face impacts wave overtopping discharge at varied revetment elevations ( $L = 5.0$ m, $h_{wall} = 0.3$ m, $D = 1.1$ m). It is evident that the wave overtopping discharge follows a slight increase and decrease pattern, with a maximum value at $m = 3$ , as the slope decreases. Additionally, the revetment elevation plays a crucial role in the wave overtopping discharge, wherein it rapidly decreases with increasing revetment elevation.

Fig. 8. Relationship between the slope of the reef-face and wave overtopping discharge under different revetment height conditions ( $L = 5.0$ m, $D = 1.1$ m).

4.2.2. Influence of the distance between the revetment and the reef edge

Figures 9 and 10 depict the correlation between the wave overtopping discharge and the distance from the revetment to the reef edge under different wave heights and revetment heights. The graphs reveal that, for each wave height and revetment height, wave overtopping discharge initially increases as the distance between the revetment and reef edge increases, before reaching a peak when the distance is approximately 1.0m, and then decreasing. The possible reasons are explained as follows: When the seawall is positioned at the reef edge ( $L = 0.0$ m), a significant portion of wave energy is directly reflected back to the sea, resulting in a relatively small wave overtopping discharge. When $L = 1.0$ m, a substantial amount of water accumulates on the reef-flat in front of the seawall, leading to a rapid rise in the water level accompanied by a significant wave overtopping phenomenon. Increasing the distance between the seawall and the reef edge leads to an increase in dissipated wave energy resulting from wave breaking on the reef-flat, leading to a decrease in the wave overtopping discharge. These results demonstrate that the distance between the revetment and the reef edge significantly impacts wave overtopping discharge, and thus emphasizes the need for careful consideration of this factor in the design of coastal protection measures.

Fig. 9. Relationship between the revetment position and wave overtopping discharge under different wave height conditions ( $m = 5$ , $h_{wall} = 0.3$ m, $D = 1.1$ m).

Fig. 10. Relationship between the revetment position and wave overtopping discharge under different revetment height conditions ( $m = 5$ , $H = 0.2$ m, $D = 1.1$ m).

Figure 11 displays the correlation between the wave overtopping discharge and the distance from the revetment to the reef edge at varying water depths. It is evident that, irrespective of water levels, the wave overtopping discharge increases in the initial stage and subsequently decreases with greater distance. Nonetheless, as the water level increases, the location of the revetment with respect to the maximum wave overtopping discharge gradually shifts in a backwards direction. For instance, at a water depth of 1.1m, the revetment’s location corresponding to the maximum wave overtopping discharge is roughly 1.0m, whereas at a water depth of 1.2m, the corresponding position of revetment is around 2.5m. It is suspected that the variation in the position of the revetment with respect to the maximum wave overtopping discharge is a result of the diverse locations at which waves break for various water depths.

Fig. 11. Relationship between the revetment position and wave overtopping discharge under different water depth conditions ( $m = 5$ , $h_{wall} = 0.3$ m, $H = 0.2$ m).

5. Conclusions

This study employs the WCSPH method to establish a two-dimensional numerical wave flume, aiming to investigate the process of solitary wave overtopping. The model’s performance is analyzed and compared to experimental data. The effects of the reef-face slope and the distance from the revetment to the reef edge on solitary wave overtopping over the artificial revetment placed on the reef-flat are analyzed. The following conclusions are mainly drawn:

(1)	The study finds that the developed SPH model is capable of accurately simulating the process of solitary wave overtopping over a seawall. The model is validated by comparing the predicted results with the experimental data, which showed that it can effectively predict the maximum water body thickness and wave overtopping discharge over the seawall crest. The findings demonstrate that the SPH model effectively simulates wave overtopping and large free surface deformation.
(2)	The study notes that a decrease in the slope of the reef-face leads to an initial increase in wave overtopping discharge, followed by a decrease. This is due to the reflection of the front reef face and the resulting shift in the location of wave breaking. When the revetment is situated close to the reef edge, the reef-flat can cause a blocking effect, leading to a thicker water body over the revetment crest. When the revetment is moved further away from the reef edge, the distance between the revetment and the reef edge increases. As a result, wave breaking and turbulent dissipation lead to greater energy loss, ultimately resulting in a decrease in wave overtopping discharge.
(3)	According to the study, the slope of the reef face and the distance between the revetment and the reef edge are significant factors in the process of solitary wave overtopping. Therefore, it’s crucial for coastal engineers to consider these factors carefully when designing solutions to reduce potential damage. Additionally, further research on the impact of reef surface roughness and porosity on wave overtopping is necessary to improve the applicability of SPH model.

Acknowledgments

The SPH code used in this paper is based on SPHysic open source code and the authors are grateful to all the developers for their valuable contributions. This work was supported by the National Natural Science Foundation of China (No. 52101312), Guangdong Basic and Applied Basic Research Foundation (Nos. 2023A1515011000, 2022A1515240014, 2023A1515012183) and Zhanjiang Ocean Youth Talent Project (No. 2021E05006).

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