A Comparative Numerical Study of Solitary Wave Interaction with Concave, Convex, and Sloped Seawalls: Hydrodynamics, Wave Loads, and Turbulent Flow Analysis

A Comparative Numerical Study of Solitary Wave Interaction with Concave, Convex, and Sloped Seawalls: Hydrodynamics, Wave Loads, and Turbulent Flow Analysis | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A Comparative Numerical Study of Solitary Wave Interaction with Concave, Convex, and Sloped Seawalls: Hydrodynamics, Wave Loads, and Turbulent Flow Analysis SeyedMahmood Ghassemizadeh, Farzad ShojaeeBaghdar, Amin Mazaherizaveh, and 3 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7445438/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 03 Feb, 2026 Read the published version in Water Waves → Version 1 posted 9 You are reading this latest preprint version Abstract Seawalls are critical for coastal protection, yet a comprehensive understanding of how their geometry affects hydrodynamic loads, performance, and local turbulence remains incomplete. While previous studies have investigated solitary wave forces on conventional vertical or sloped structures, a systematic comparative analysis of concave, convex, and sloped seawalls one that holistically links hydrodynamic performance (run-up, reflection), wave-induced loads, and the resulting turbulent flow structures has been a notable gap in the literature. This study addresses this gap by conducting a detailed numerical investigation using a validated RANS model coupled with a k-ε RNG turbulence scheme and the Volume of Fluid (VOF) method. We analyze the interaction of highly nonlinear solitary waves with these distinct geometries. The results demonstrate that seawall curvature is a critical design parameter. Concave seawalls significantly increase wave reflection and generate concentrated, high-energy vortices at the structure's toe, leading to amplified wave loads and a higher potential for local scour. In contrast, convex profiles promote smoother flow separation, resulting in reduced wave forces and diminished turbulence near the bed. Sloped seawalls are effective at dissipating energy along the structure's face, thereby minimizing loads but increasing the likelihood of significant overtopping. By providing an integrated analysis of pressure distribution, impact forces, and Turbulent Kinetic Energy (TKE), this research offers crucial insights for the optimal design of coastal defenses, enabling engineers to balance structural stability, hydrodynamic efficiency, and scour mitigation. solitary wave seawall geometry wave-structure interaction wave loads turbulent flow numerical modeling Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Figure 21 Highlights 1. A comparative analysis links seawall geometry to wave loads, hydrodynamics, and turbulence. 2. Concave seawalls amplify wave reflection and impact loads, creating intense toe vortices. 3. Convex profiles reduce wave loads and near-bed turbulence by promoting smoother flow separation. 4. Sloped geometry is most effective at dissipating energy but shows the highest overtopping risk. 5. Seawall curvature is identified as a critical parameter for optimizing structural stability and hydrodynamic performance. 1. Introduction Coastal structures, such as seawalls and dikes, serve as essential defenses against wave attacks, erosion, and flooding. Seawalls, often built as vertical or sloped barriers along shorelines, primarily reflect wave energy, while dikes, typically earthen embankments, aim to contain water and prevent inland flooding. The effectiveness of these structures under highly nonlinear solitary waves—which mimic aspects of tsunamis and storm surges—remains a key focus in coastal engineering. Research suggests that understanding these interactions can improve infrastructure resilience, though challenges persist in predicting complex flow behaviors. Studies have extensively examined solitary wave interactions with coastal structures. Early work by Synolakis (1987) and Hall and Watts (1953) laid foundations for wave run-up and reflection on impermeable sloped or vertical walls. More recent analyses, such as Vinodh and Tanaka (2020) and Casella et al. (2022), have refined run-up formulas using experimental data on sloping beaches. Emerging research highlights the role of curved profiles: Anada et al. (2011) compared curved and vertical seawalls under regular waves, while Suba et al. (2019) assessed curvature effects on dike run-up. Castellino et al. (2018) numerically investigated impulsive forces on curved seawalls, revealing geometry's influence on wave loads. In the context of breaking waves, Kimmoun and Branger (2007) used particle image velocimetry (PIV) to study turbulent flows in surf zones over sloping beaches, providing insights into vortex formation and energy dissipation that align with solitary wave dynamics. Wave-induced loads are critical for structural design. Investigations like Lin et al. (2009), Luo et al. (2019), and Wang et al. (2023) have quantified forces and pressures on vertical and sloped walls. Wu (2022) focused on breaking solitary impacts on vertical seawalls, emphasizing turbulence. However, curved geometries are underexplored; Grilli et al. (2004), involving Kimmoun, combined numerical and experimental approaches to solitary wave shoaling and breaking, linking to potential scour. A systematic comparison across concave, convex, and sloped profiles—integrating loads, pressure, and turbulent structures—is lacking. Turbulence during impacts informs scour risks, with Turbulent Kinetic Energy (TKE) as a key metric. Lubin et al. (2011) and Lubin et al. (2019), building on Kimmoun's work, modeled breaking wave instabilities and vortices. Bona et al. (2023) and Stastna et al. (2022) from Water Waves journal analyzed solitary wave dynamics, offering theoretical support for nonlinear interactions. Yet, linking geometry to flow kinematics (e.g., vortices, jets) and scour potential remains incomplete. Extending prior numerical efforts (Ghassemizadeh and Ketabdari, 2016, 2022), this study uses FLOW-3D ® with RANS equations, k-ε RNG turbulence, and VOF method (Hirt and Nichols, 1981; Flow Science, Inc., 2011). Novel contributions include: ( 1 ) comparing hydrodynamic performance (run-up, reflection, overtopping) across geometries; ( 2 ) quantifying forces and pressures; ( 3 ) analyzing TKE evolution; and ( 4 ) connecting geometry to flows, loads, and scour, with a simplified economic assessment. Figure 1 illustrates wave run-up. This paper is structured as follows: Section 2 details the governing equations and numerical model setup. Section 3 presents the model validation against established experimental data and empirical formulas. Section 4 discusses the main results, including wave run-up, reflection, overtopping, wave loads, internal flow structures, and the economic assessment. Finally, Section 5 provides the main conclusions drawn from this investigation. 2. Numerical Model Setup This study employs a two-dimensional numerical model using the commercial CFD software FLOW-3D ® (v10.0.1) to simulate the interaction between solitary waves and coastal structures with different wall profiles. The model is based on solving the Reynolds-Averaged Navier-Stokes (RANS) equations coupled with a turbulence model and a free-surface tracking method. 2.1. Governing Equations The fluid motion is governed by the incompressible RANS equations, which, for a Newtonian fluid, consist of the continuity and momentum equations. In Cartesian coordinates, these are expressed as: Continuity Equation: $$\:\frac{{\partial\:u}_{i}}{{\partial\:x}_{i}}=0$$ 1 Momentum Equation: $$\:\frac{{\partial\:u}_{i}}{\partial\:\text{t}}+{u}_{j}\frac{{\partial\:u}_{i}}{{\partial\:x}_{j}}=-\frac{1}{\rho\:}\frac{\partial\:\text{p}}{{\partial\:x}_{i}}+\frac{\partial\:}{{\partial\:x}_{j}}\left[\left(\mu\:+{\mu\:}_{t}\right)+\left(\frac{{\partial\:u}_{i}}{{\partial\:x}_{j}}+\frac{{\partial\:u}_{j}}{{\partial\:x}_{i}}\right)\right]+{g}_{i}$$ 2 where u i and u j are the time-averaged velocity components in the x i and x j directions (with i,j = 1,2 for 2D flow, representing horizontal and vertical directions respectively), t is time, ρ is the fluid density, p is the mean pressure, ν is the kinematic viscosity of the fluid, ν t is the turbulent kinematic (eddy) viscosity, and g i is the gravitational acceleration component in the x i direction. The term ( ν + ν t ) represents the effective kinematic viscosity. The eddy viscosity ν t is determined by the turbulence model. 2.2. Turbulence Model (RNG k-ε) To account for the effects of turbulence, the Renormalization Group (RNG) k-ε turbulence model Yakhot et al. (1992) is employed. This model is known for its improved accuracy in simulating flows with high strain rates, streamline curvature, and regions of recirculation, which are characteristic of wave-structure interactions. The RNG k-ε model involves solving two additional transport equations for the turbulent kinetic energy ( k ) and its dissipation rate ( ε ): Turbulent Kinetic Energy ( k ) Equation: $$\:\frac{{\partial\:}_{k}}{\partial\:t}+{u}_{j}\frac{{\partial\:}_{k}}{{\partial\:x}_{j}}=\frac{\partial\:}{{\partial\:x}_{j}}\left({\alpha\:}_{k}{\nu\:}_{eff}\frac{{\partial\:}_{k}}{{\partial\:x}_{j}}\right)+{G}_{k}-\epsilon\:$$ 3 Turbulent Dissipation Rate ( ε ) Equation: $$\:\frac{{\partial\:}_{\epsilon\:}}{\partial\:t}+{u}_{j}\frac{{\partial\:}_{\epsilon\:}}{{\partial\:x}_{j}}=\frac{\partial\:}{{\partial\:x}_{j}}\left({\alpha\:}_{\epsilon\:}{\nu\:}_{eff}\frac{{\partial\:}_{\epsilon\:}}{{\partial\:x}_{j}}\right)+{{C}_{\epsilon\:1}\frac{\epsilon\:}{k}G}_{k}-{C}_{\epsilon\:2}\frac{{\epsilon\:}^{2}}{k}-{R}_{\epsilon\:}$$ 4 In these equations, G k is the generation of turbulent kinetic energy due to mean velocity gradients, ν eff is the effective turbulent viscosity (related to ν t ). C ε1 , C ε2 , α k , and α ε are model constants. The term R ε is an additional term in the RNG model that improves its performance for rapidly strained flows. The standard values for these constants as implemented in FLOW-3D ® are adopted for this study, as summarized in Table 1 . The eddy viscosity ν t is then calculated as \(\:{\nu\:}_{t}={C}_{\mu\:}\frac{{k}^{2}}{{\epsilon\:}}\) , where C µ is another model constant. Table 1 Constants for the RNG k-ε turbulence model Constants Amount c 3 0.0120 c µ 0.0850 c ε1 1.4200 c ε2 1.6800 α k 0.7194 α ε 0.7194 η 0 4.3800 2.3. Free Surface Tracking and Advection The Volume of Fluid (VOF) method, as proposed by Hirt & Nichols (1981), is used to track the interface between water and air. In this method, a scalar function F (volume of fluid fraction) is defined, representing the fractional volume of a computational cell occupied by water. F = 1 indicates a cell full of water, F = 0 indicates a cell full of air, and 0 < F < 1 indicates an interface cell. The advection of F is governed by: $$\:\frac{\partial\:F}{\partial\:t}+{u}_{j}\frac{\partial\:F}{{\partial\:x}_{j}}=0$$ 5 For the advection of the VOF function, the Split Lagrangian Advection method available in FLOW-3D ® is employed, which is suitable for maintaining a sharp interface in 2D interfacial flows and ensures mass conservation. Complex geometries and solid obstacles within the Cartesian mesh are represented using the Fractional Area/Volume Obstacle Representation (FAVOR™) method. 2.4. Numerical Discretization and Solution Algorithm The governing equations ( 1 – 5 ) are discretized using a Finite Volume Method (FVM) on a structured Cartesian mesh. FLOW-3D ® uses a staggered grid arrangement where scalar variables (like pressure) are stored at cell centers, and velocity components are stored at cell faces. For time integration, an implicit scheme is used. The pressure-velocity coupling is handled using a pressure correction method. The pressure equation, derived from the continuity and momentum equations, is solved using the Generalized Minimal Residual Solver (GMRES) algorithm, which is effective for non-symmetric linear systems and allows for numerically stable solutions with relatively larger time steps. First-order upwind differencing was used for the advection terms in the momentum and turbulence equations. 2.5. Computational Domain and Boundary Conditions The simulations were performed in a 2D numerical wave flume. The flume extended from X = -10.0 m to X = + 1.0 m (total length of 11.0 m) and had a height from Y = 0.0 m to Y = + 2.0 m. The seawall structure was located at the downstream end of the flume, starting at X = 0.0 m and extending to X = + 1.0 m (base length of 1.0 m). The crest elevation of all seawall structures was maintained at Z crest = 0.5 m from the flume bed ( Y = 0.0 m). The still water depth ( h₀ ) was set according to the specific test case (see Section 2.6 ). The boundary conditions applied to the computational domain were as follows (summarized in Table 2 ): Table 2 Boundary conditions employed in the numerical simulations Boundary Location Type of Boundary Condition Description Inlet (X= -10.0 m) Wave Solitary wave generation Outlet (X = + 1.0 m) Outflow Wave and overtopped water absorption, min. reflection Bottom (Y = 0.0 m) Wall (No-slip) Impermeable flume bed Top (Y = + 2.0 m) Pressure Atmospheric pressure, free air movement Structure Surfaces Wall (No-slip) Impermeable seawall surfaces Inlet (Left boundary, X = -10.0 m): A solitary wave was generated using the specific wave generation boundary condition in FLOW-3D ® . Outlet (Right boundary, X = + 1.0 m): An outflow boundary condition was applied to allow waves and any overtopped water to pass out of the domain with minimal reflection. Bottom (Bottom boundary, Y = 0.0 m): A no-slip wall boundary condition was applied, representing the impermeable bed of the flume. Top (Top boundary, Y = + 2.0 m): A pressure boundary condition was specified, representing atmospheric pressure, allowing air to move freely. Structure Surfaces: A no-slip wall boundary condition was applied to all seawall surfaces. 2.6. Solitary Wave Generation and Characteristics Solitary waves were generated at the inlet of the numerical flume using the built-in solitary wave generation feature in FLOW-3D®, which is based on Boussinesq-type approximations. The key input parameters for wave generation were the incident wave height ( H ) and the still water depth (h₀ ). Figure 2 illustrates the input parameters required by the software. The simulations in this study targeted two primary non-dimensional wave height-to-depth ratios: Target H/h₀ = 0.6: with an input H = 0.3 m and h₀ = 0.5 m. Target H/h₀ ≈ 0.8: with an input H = 0.45 m and h₀ = 0.55 m (resulting in an input H/h₀ ≈ 0.818). Analysis of the wave profiles propagating in the flume (prior to interaction with the structures, based on supplementary data from the simulations) indicated that the effective H/h₀ ratios were approximately 0.56 for the first target case ( H ≈ 0.28 m, h₀ = 0.5 m), and approximately 0.73 for the second target case ( H ≈ 0.40 m, h₀ = 0.55 m). These effective wave characteristics were confirmed to be largely non-breaking during their propagation in the 10.0 m long open flume section before reaching the structures. All subsequent analyses of wave run-up, reflection, and overtopping are based on these effective incident wave conditions. The initial position of the solitary wave crest was set at a distance of 0.5 L from the inlet, where L is the characteristic wavelength of the solitary wave, ensuring the wave was fully formed before reaching the structure. The domain was initialized with a quiescent water body (zero initial velocity). 2.7. Seawall Geometries Three main types of seawall front-face geometries were investigated, all with a crest elevation of 0.5 m above the flume bed: Concave Seawall: This geometry featured a circular arc curving inwards towards the landside. The radius of curvature was R c = 0.5 m. Convex Seawall: This geometry featured a circular arc curving outwards towards the seaside, also with a radius of curvature of R v = 0.5 m. Sloped Seawalls: Three different linear slopes were considered: Slope 1: gradient 1:0.577 (corresponding to an angle θ = 60° with the horizontal). Slope 2: gradient 1:1 (corresponding to an angle θ = 45° with the horizontal). Slope 3: gradient 1:1.732 (corresponding to an angle θ = 30° with the horizontal). A schematic depiction of these five seawall profiles is provided in Fig. 3 . 2.8. Mesh Configuration and Sensitivity A structured Cartesian mesh was employed for all simulations. A mesh sensitivity analysis was conducted prior to the main simulations to determine an optimal grid resolution that balances computational accuracy with efficiency. This analysis involved simulating wave run-up on a representative concave seawall (with an effective H/h₀ ≈ 0.56, based on an input H = 0.3 m and h₀ = 0.5 m) using various uniform mesh sizes ( Δx = Δz ) ranging from 0.06 m down to 0.005 m. The results of this sensitivity analysis are illustrated in Fig. 4 . It was observed that for mesh sizes smaller than 0.015 m, the computed maximum wave run-up tended to converge, showing minimal further changes with finer refinement. Therefore, a uniform mesh size of Δx = Δz = 0.015 m was selected and used for all subsequent simulations presented in this study, ensuring a good balance between accuracy and computational cost. 2.9. Data Extraction for Forces, Pressure, and TKE In addition to the primary hydrodynamic parameters (run-up, reflection, overtopping), the numerical model was used to extract detailed data on wave-induced loads and local turbulence intensity. Wave-Induced Forces: The total hydrodynamic forces exerted on the seawall, including the horizontal ( F h ) and vertical ( F v ) components, were calculated by the software at each time step. This calculation is based on the integration of both pressure ( p ) and viscous stresses ( τ ) over the wetted surface area ( A ) of the structure, conceptually represented as: $$\:F=\underset{A}{\overset{}{\int\:}}(-pn+\tau\:.n)dA$$ 6 where n is the unit normal vector to the surface. The time series of both horizontal and vertical force components were recorded for post-processing. Pressure and Turbulent Kinetic Energy (TKE): To analyze the pressure distribution and local turbulence intensity during wave impact, a series of numerical monitoring points (probes) were defined at specific, identical locations along the front face of each representative seawall geometry (concave, convex, and sloped 45°). These probes recorded the time series of dynamic pressure ( p ) and Turbulent Kinetic Energy (TKE, k ) at their respective coordinates. The locations of these probes are illustrated schematically in Fig. 5 . 3. Model Validation To ensure the accuracy and reliability of the numerical model for simulating solitary wave interactions with coastal structures, a series of validation tests were performed. The numerical results from FLOW-3D ® were compared against established experimental data and empirical/semi-empirical formulations from well-regarded studies. This validation process covered wave propagation, run-up on sloped structures, and focused on ensuring the model could replicate key physical phenomena before its application to the different seawall geometries. 3.1. Solitary Wave Profile and Propagation Firstly, the model's ability to generate and propagate a stable solitary wave was verified. The generated wave profile in the numerical flume (prior to interaction with any structure) was compared with the analytical solution for a solitary wave, often given by Boussinesq theory as: $$\:\eta\:\left(x,t\right)=H{sech}^{2}\left[\sqrt{\frac{3H}{4{h}_{0}^{3}}}(x-Ct)\right]$$ 7 where η is the free surface elevation, H is the wave height, h 0 is the still water depth, \(\:C=\sqrt{g({h}_{0}+H)}\) is the wave celerity, and g is the gravitational acceleration. For this validation, a wave with an effective H/h 0 ≈ 0.27 (specifically H = 0.056 m, h 0 = 0.21 m, as used in the experiments by Synolakis (1987)) was simulated. Figure 6 shows a comparison between the numerical wave profile at a specific time instance (e.g., t = 1.5 s after generation) and the theoretical profile, alongside experimental data points from Synolakis (1987). The comparison demonstrates good agreement in terms of wave shape, crest elevation, and width, confirming the model's capability to accurately represent a propagating solitary wave. The mesh size of Δx = Δz = 0.015 m, as determined from the sensitivity analysis (Section 2.8 ), was used. 3.2. Wave Run-up on Sloping Structures Next, the model was validated for solitary wave run-up on plane, impermeable slopes, which is a critical parameter in this study. The numerical results were compared against widely recognized experimental data and empirical relations. 3.2.1. Comparison with Synolakis (1987) Data The experimental work of Synolakis (1987) on solitary wave run-up on a slope of 1:19.85 (approximately 2.87°) is a common benchmark. Numerical simulations were conducted to replicate these experiments. For instance, for a case with an incident wave height H = 0.056 m and still water depth h 0 = 0.21 m (giving H/h 0 ≈ 0.27, which is a non-breaking wave condition), the computed wave run-up on the slope was compared. Figure 7 illustrates the comparison of the free surface profile during run-up between the numerical model and the experimental data from Synolakis (1987). The maximum run-up values ( R ) obtained from the simulation showed good agreement with the experimental findings, with deviations typically within an acceptable range (e.g., 3–6% as mentioned in the first manuscript), validating the model's performance for run-up predictions on gentle slopes. The empirical relation by Synolakis (1987) for non-breaking waves is often given as: $$\:\frac{R}{{h}_{0}}=2.831{\left(cot\theta\:\right)}^{0.5}{\left(\frac{H}{{h}_{0}}\right)}^{1.25}$$ 8 where θ is the beach slope angle. The numerical results were consistent with predictions from this type of formula for the given slope and wave conditions. 3.2.2. Comparison with Hall & Watts (1953) and Borthwick et al. (2005, 2006) For steeper slopes, the experimental data of Hall & Watts (1953) for a 1:1 slope (45°) were used for comparison. The normalized run-up ( R/h 0 ) versus the normalized incident wave height ( H/h 0 ) from the numerical simulations were plotted alongside the experimental data of Hall & Watts (1953) and the lower and upper asymptotic limits proposed by Borthwick et al. (2005, 2006), which were derived based on Synolakis's outputs and other data. Figure 8 presents this comparison. The numerical results generally fall within the range of the experimental data and align well with the established asymptotic trends, particularly for non-breaking or early-breaking wave conditions. The first manuscript noted an error within 2–4% for the same H/h 0 ratio, further supporting the model's reliability for these steeper slopes. It is acknowledged that differences in experimental flume materials versus the idealized smooth wall conditions in the numerical model can contribute to minor discrepancies. 3.2.3. Comparison with Vinodh & Tanaka (2020) Formulation To further validate the model against more recent work, the numerical run-up results for various slopes (30°, 45°, 60°, corresponding to the sloped seawalls in this study) were compared with the unified empirical formula proposed by Vinodh & Tanaka (2020). This formula predicts the maximum run-up of solitary waves on plane beaches across various slopes and wave conditions. The specific form of their unified empirical formula used for comparison in this study is: $$\:\frac{R}{H}=A{\left(\frac{H}{{h}_{0}}\right)}^{B}\bullet\:{tan}^{C}\left(\theta\:\right)$$ 9 where A, B, and C are empirically derived coefficients (as provided in Vinodh & Tanaka (2020), for specific conditions or a general form). Figure 9 , shows a comparison of the non-dimensional run-up R/h 0 against the non-dimensional wave height H/h 0 for the tested slopes. The numerical predictions were found to be in reasonable agreement with the empirical formula of Vinodh & Tanaka (2020) (Eq. 9 ) within the physically relevant range of H/h 0 (typically H/h 0 < 0.8 for stable solitary waves prior to breaking on a slope). This comparison further confirms the model's applicability across the range of slopes investigated in the main study. 3.3. Discussion on Validation Limitations and Applicability The validation exercises demonstrate that the numerical model, using the k-ε RNG turbulence model and VOF method with the selected mesh resolution, can reliably simulate solitary wave propagation and run-up on plane impermeable slopes across a range of conditions. However, it is important to note certain limitations: The validation cases primarily involved linear sloped beaches or structures. Direct experimental validation for solitary wave interaction with the specific concave and convex geometries investigated in this study was not available in the literature to the same extent. Therefore, the model's application to these arced structures relies on the fundamental accuracy of the RANS-VOF approach for complex free-surface flows and wave-boundary interactions, as demonstrated for sloped cases. The current study focuses on solitary waves whose effective H/h 0 ratios at the point of interaction with the structures were approximately 0.56 and 0.73 (as detailed in Section 2.6 ). While the higher ratio is close to the breaking limit for solitary waves (often cited around H/h 0 ≈ 0.78), visual inspection of the wave profiles during propagation in the flume prior to reaching the structures indicated that they maintained largely non-breaking forms. The validation cases also predominantly fall within this non-breaking or incipiently breaking regime. Scenarios involving intensely breaking waves might require different turbulence modeling approaches or finer mesh resolutions near the breaking zone. The numerical model assumes smooth, impermeable surfaces for the seawalls and flume bed. Effects of surface roughness or permeability, which can influence wave energy dissipation and run-up in real-world scenarios (e.g., Kobayashi et al. (1987)), are not included in this study. This was noted as a potential source of minor error when comparing with some experimental data. Despite these limitations, the comprehensive validation against various benchmarks for sloped geometries provides confidence in the model's capability to capture the essential hydrodynamics of solitary wave run-up, reflection, and overtopping for the comparative analysis of different seawall shapes presented in Section 4 . 4. Results and Discussion This section presents the numerical simulation results for solitary wave interaction with concave, convex, and various sloped seawall geometries. The analysis focuses on hydrodynamic parameters including wave run-up, reflection, and overtopping, as well as a detailed examination of the internal flow structures and their implications. The results correspond to incident solitary waves with effective H/h 0 ratios of approximately 0.56 and 0.73, as detailed in Section 2.6 . 4.1. General Hydrodynamic Behavior and Wave Transformation The interaction of a solitary wave with a coastal structure is a complex hydrodynamic process involving wave shoaling upon approach, impact, run-up on the structure face, reflection of wave energy, and potential overtopping at the crest. The specific geometry of the seawall front face plays a crucial role in dictating the nature and magnitude of these phenomena. Visual observations from the numerical simulations (representative snapshots shown in subsequent sections) indicate distinct differences in wave transformation for each seawall type. Upon encountering the structures, the solitary waves undergo significant deformation. For concave seawalls, the incident wave tends to be focused by the inward curvature, leading to a rapid increase in wave height near the structure and a strong upward jet during run-up. This often results in a more energetic reflection. For convex seawalls, the outward curvature appears to spread the incident wave energy more laterally (in a 3D scenario, though this is a 2D study) or dissipate it more gradually along the curved face, leading to a seemingly less aggressive run-up and reflection compared to the concave profile. Sloped seawalls allow the wave to propagate up their face, with the run-up length and the nature of wave breaking (if it occurs on the slope) being dependent on the slope angle and incident wave characteristics. Steeper slopes tend to cause more rapid shoaling and potentially earlier breaking or stronger reflection, while gentler slopes allow for more extended run-up and dissipation. The interaction process also generates complex internal flow patterns, including vortices and regions of high turbulence, particularly near the toe and crest of the structures, which are significantly influenced by the seawall geometry. These aspects will be explored in detail in the following subsections. Figure 10 provides a conceptual overview of these differing interaction behaviors which are quantitatively analyzed below. 4.2. Wave Run-up ( R ) and Reflection Coefficient ( C r ): The maximum vertical wave run-up ( R ) on the structure and the wave reflection coefficient ( C r ) are critical parameters for assessing seawall performance. These were quantified for each seawall geometry under the two effective incident solitary wave conditions (effective H/h 0 ≈ 0.56 and H/h 0 ≈ 0.73). The maximum run-up ( R ) was determined by tracking the highest point the water surface reached on the structure's face during the interaction. The reflection coefficient ( C r ) was calculated as the ratio of the reflected wave height ( H r ) to the incident wave height ( H i , being the effective wave height). The reflected wave height ( H r ) was estimated by analyzing the wave profile propagating away from the structure after the primary interaction, typically using a method like Goda's method for separating incident and reflected waves if sufficient data points are available, or by identifying the maximum amplitude of the primary reflected wave component at a fixed-point seaward of the structure. 4.2.1. Wave Run-up ( R/h 0 ) The normalized maximum wave run-up ( R/h 0 ) for the different seawall geometries as a function of the effective incident non-dimensional wave height ( H/h 0 ) is presented in Fig. 11 . The results indicate a clear dependence of run-up on both the incident wave energy (represented by H/h 0 ) and the seawall's front-face geometry. For both effective incident wave conditions, the concave seawall consistently exhibited the highest wave run-up values. For instance, at an effective H/h 0 ≈ 0.73, the R/h 0 for the concave wall was approximately 65% higher than that for the 1:1 sloped wall and 66% higher than for the convex wall. This enhanced run-up on the concave profile can be attributed to the focusing effect of the inward curvature, which concentrates the incoming wave energy and directs the flow upwards along the curved surface, leading to a more significant vertical excursion of the water. The convex seawall generally showed the lowest or among the lowest run-up values compared to the concave and steeper sloped walls. The outward curvature of the convex profile tends to radially disperse the incoming wave energy and promote a smoother, more gradual flow up the face, which can limit the maximum vertical reach of the water. For the sloped seawalls, run-up was, as expected, influenced by the slope angle. The steepest slope (1:0.58, 60°) resulted in run-up values that were generally lower than the concave wall but higher than the gentlest slope (1:1.73, 30°) for a given incident wave. The gentler slopes provide a longer path for wave energy dissipation through friction (though friction is minimal in these smooth-wall simulations) and breaking on the slope (if applicable), which can reduce the maximum vertical run-up height. 4.2.2. Wave Reflection Coefficient ( C r ) The wave reflection coefficient ( C r ) for the different seawall geometries is presented in Fig. 12 as a function of the effective H/h 0 . The results show that the concave seawall also produced the highest reflection coefficients among all tested geometries for both incident wave conditions. At an effective H/h 0 ≈ 0.73, the C r for the concave wall was approximately 225% higher than for the 1:1 sloped wall and 257% higher than for the convex wall. The inward-curving face of the concave structure acts to redirect a significant portion of the incident wave energy back towards the sea, minimizing energy transmission and dissipation on the structure itself, thus leading to higher reflection. This is consistent with its higher run-up, as more energy directed upwards and back (rather than dissipated or transmitted) contributes to both phenomena. Conversely, the convex seawall generally yielded the lowest reflection coefficients. The outward curvature likely promotes smoother flow deflection and some energy dissipation along its surface, reducing the amount of energy directly reflected. The sloped seawalls exhibited intermediate reflection characteristics, with steeper slopes (e.g., 60°) reflecting more energy than gentler slopes (e.g., 30°). This is expected, as gentler slopes allow for more gradual wave energy dissipation through processes like shoaling and breaking on the slope, thereby reducing reflection. The observed higher run-up and higher reflection for the concave wall are dynamically linked; the geometry that most effectively redirects incident wave momentum upwards and seawards will naturally score high on both metrics, assuming minimal energy dissipation through other means like breaking directly on the structure face or transmission. The subsequent analysis of internal flow structures (Section 4.4 ) will provide further insight into the mechanisms governing these differences. 4.3. Wave Overtopping Discharge ( q ) Wave overtopping, where water passes over the crest of a coastal structure, is a critical design consideration as it directly relates to potential flooding and damage to landward infrastructure. The overtopping discharge per unit length ( q ) was quantified for each seawall geometry under the two effective incident solitary wave conditions ( H/h 0 ≈ 0.56 and H/h 0 ≈ 0.73). This was achieved by utilizing a feature within the FLOW-3D ® software where a virtual flux surface was defined at the crest elevation of each seawall. The software then directly measured the net volumetric flow rate of water passing through this defined surface over time, providing the instantaneous overtopping discharge rate ( q , in m²/s, effectively m³/s per meter width). Figure 13 and Fig. 14 present the time histories of the overtopping discharge rate ( q ) for the concave, convex, and various sloped seawalls under the effective incident H/h 0 ≈ 0.56 and H/h 0 ≈ 0.73 conditions, respectively. The hydrographs in Figs. 13 and 14 illustrate both the peak discharge rate and the duration of the overtopping event for each seawall type. For a more direct comparison, the maximum overtopping discharge rates ( q max ) and the total overtopped volume per unit width ( V total ), calculated from the integration of the hydrographs from the new dataset, are summarized in Table 3 . Table 3 Summary of maximum overtopping discharge rate ( q max ) and total overtopped volume per unit width ( V total ) for different seawall geometries and effective incident wave conditions Seawall Geometry Effective Incident H/h 0 Maximum Overtopping Discharge Rate, q max (m²/s) Total Overtopped Volume per unit width, V total (m³/m) Concave ~ 0.56 (Target 0.6) 0.0073 0.0346 ~ 0.73 (Target 0.8) 0.0117 0.0343 Convex ~ 0.56 (Target 0.6) 0.0124 0.0705 ~ 0.73 (Target 0.8) 0.0199 0.0709 Sloped (1:1.73 or 30°) ~ 0.56 (Target 0.6) 0.0099 0.0798 ~ 0.73 (Target 0.8) 0.018 0.1028 Sloped (1:1 or 45°) ~ 0.56 (Target 0.6) 0.0097 0.0787 ~ 0.73 (Target 0.8) 0.0176 0.1067 Sloped (1:0.58 or 60°) ~ 0.56 (Target 0.6) 0.0093 0.0751 ~ 0.73 (Target 0.8) 0.0168 0.1016 Several key observations can be made from these results: Influence of Incident Wave Height: As expected, the overtopping discharge (both q max and V total ) significantly increases for all seawall types when the incident wave energy is higher (i.e., for effective H/h 0 ≈ 0.73 compared to H/h 0 ≈ 0.56). This is evident from the larger peak values and areas under the hydrographs in Fig. 14 compared to Fig. 12 , and quantified in Table 3 . Performance of Sloped Seawalls: Among the sloped seawalls, the gentlest slope (1:1.73 or 30°) generally experienced the highest overtopping discharge (e.g., for effective H/h 0 ≈ 0.73, V total for the 30° slope was 0.1028 m³/m). This is likely because the gentler slope allows the wave to run further up the face and maintain more of its coherent bulk as it approaches the crest. Conversely, the steepest slope (1:0.58 or 60°) tended to have slightly lower overtopping compared to the 30° slope for the higher wave condition (for effective H/h 0 ≈ 0.73, V total for the 60° slope was 0.1016 m³/m), possibly due to increased wave reflection and more significant disruption of the wave front by the abrupt change in geometry. Performance of Convex Seawalls: The convex seawall exhibited intermediate overtopping characteristics. For effective H/h 0 ≈ 0.73, the V total was 0.0709 m³/m. Its outward curvature might deflect some of the up-rushing water, but it does not appear to offer the same level of overtopping reduction as the concave profile under these solitary wave conditions. Performance of Concave Seawalls: A significant finding, consistent across both wave conditions in the new dataset, is that despite exhibiting the highest wave run-up (as discussed in Section 4.2.1 ), the concave seawall consistently resulted in the lowest overtopping discharges. For example, at an effective H/h 0 ≈ 0.73, the total overtopped volume for the concave wall was 0.0343 m³/m, which was approximately 66.6% less than that for the 1:1.73 sloped wall (0.1028 m³/m). This phenomenon, where high run-up does not directly translate to high overtopping for this specific geometry, suggests a distinct hydrodynamic mechanism at the crest. The inward curvature of the concave wall appears to direct a significant portion of the up-rushing water jet more vertically upwards. This upward momentum, combined with the re-directing effect of the sharp crest, likely causes a substantial part of the water to be thrown upwards and then fall back towards the seaside, rather than efficiently passing over the crest. This "jet deflection and return flow" mechanism, enhanced by the concave geometry, effectively reduces the net volume of water that overtops the structure. This explanation will be further supported by the analysis of flow velocity vectors near the crest in Section 4.4 . The specific shape of the overtopping hydrograph (Figs. 13 and 14 ) also varies with seawall geometry. Sloped walls, particularly gentler ones, tend to show a broader hydrograph with a longer duration of overtopping, indicating a more prolonged but potentially less intense overtopping event. In contrast, the concave and convex walls appear to produce sharper, more concentrated overtopping pulses, though the magnitude for the concave wall is notably lower. These differences in overtopping behavior have significant implications for the design of crest and landward protection measures, as both the peak rate and total volume of overtopping are critical. 4.4. Wave-Induced Forces and Pressures Beyond the hydrodynamic performance, understanding the wave-induced loads is essential for the structural stability and design of seawalls. This section presents a comparative analysis of the horizontal and vertical forces, as well as the pressure distribution, exerted by the solitary waves on the three representative seawall geometries: concave, convex, and sloped 45°. 4.4.1. Time History of Wave Forces The time series of the total horizontal force ( F h ) and total vertical force ( F v ) acting on the wetted surface of each representative seawall are presented in Fig. 15 and Fig. 16 , respectively, for the higher energy wave condition (effective H/h 0 ≈ 0.73). The horizontal force history in Fig. 15 shows significant differences in loading characteristics. The concave seawall experiences the highest peak horizontal force ( F h , max ), which occurs as a sharp, high-intensity impact over a very short duration. This is characteristic of a "slamming" load. In contrast, the sloped seawall exhibits a much slower build-up of force and a lower, broader peak, indicating a more quasi-static or "surging" type of loading. The convex wall shows an intermediate behavior. The vertical force history in Fig. 16 primarily represents the uplift force on the structure. Again, the concave wall is subjected to the highest peak uplift force (most negative F v ). A summary of the peak force values for the representative geometries and both wave conditions is provided in Table 4 . Table 4 Summary of maximum horizontal ( F h,max ) and maximum vertical uplift ( F v,max ) forces for the representative seawall geometries Seawall Geometry Effective Incident H/h 0 Maximum Horizontal Force, F h,max (kN/m) Maximum Horizontal Force, F v,max (kN/m) Concave ~ 0.56 (Target 0.6) 1862.71 -740.84 ~ 0.73 (Target 0.8) 2519.25 -994.10 Convex ~ 0.56 (Target 0.6) 1849.83 -278.04 ~ 0.73 (Target 0.8) 2446.85 -504.66 Sloped (45°) ~ 0.56 (Target 0.6) 1830.50 -509.96 ~ 0.73 (Target 0.8) 2348.65 -726.22 4.4.2. Pressure Distribution at Peak Impact To understand the reason behind the differences in peak forces, the distribution of dynamic pressure along the face of the three representative seawalls was analyzed at the exact moment of maximum horizontal force for the H/h 0 ≈ 0.73 wave condition. The pressure values were extracted from the numerical probes whose locations are defined in Fig. 5 . The pressure profiles in Fig. 17 reveal the underlying loading mechanism for each geometry. The curve for the concave seawall exhibits a highly non-linear pressure distribution. Unlike a typical hydrostatic profile, it shows a significant pressure concentration on its upper curved section (around Y = 0.5 m). This high pressure is caused by the direct impact of the focused, up-rushing water jet, which quantitatively explains the sharp, high-magnitude "slamming" force observed for this geometry in Fig. 15 . In contrast, the curve for the sloped seawall shows a much more linear, quasi-hydrostatic pressure distribution, with pressure being highest at the toe and decreasing steadily with elevation. The convex wall shows an intermediate distribution, with pressure values at all points being generally lower than those of the concave profile. This analysis confirms that the concave geometry, while effective for other purposes, must be designed to withstand significantly higher localized impact pressures on its upper face. This focused loading is the primary reason for the larger total horizontal force experienced by the concave structure compared to conventional sloped designs. It is noteworthy that the peak pressures at individual locations are not simultaneous, reflecting the upward propagation of the pressure wave along with the water jet. For instance, while the pressure at point P4 for concave and convex face was zero at the moment of maximum total force, it later reached its absolute maximum of 22.51 and 1.32 kPa at t = 4.3 s respectively as the wave crest passed its elevation. The pressure distribution shown in Fig. 17 , however, represents the simultaneous loading across the structure at the single most critical instant for overall horizontal stability. 4.5. Analysis of Flow Structures, TKE, and Scour Potential To establish a complete physical understanding of the seawalls' performance, this section links the macroscopic findings (run-up, overtopping, forces) to the underlying hydrodynamics. The analysis focuses on the evolution of Turbulent Kinetic Energy (TKE) and the detailed kinematic flow fields for the three representative geometries under the higher energy wave condition (effective H/h 0 ≈ 0.73). 4.5.1. Turbulent Kinetic Energy (TKE) Evolution at the Seawall Toe Turbulent Kinetic Energy (TKE, k ) provides a direct quantitative measure of the turbulence intensity. The time history of TKE was analyzed at the numerical probe located nearest to the structure's toe (Probe P1, locations shown in Fig. 5 ), a critical area for scour development. Figure 18 compares the TKE time series at this location for the three representative geometries. The results in Fig. 18 reveal that while the peak TKE values are of a similar order of magnitude for all three geometries, the temporal evolution and duration of the turbulence differ significantly. The peak TKE values for the concave, convex, and sloped walls are approximately 0.00067 m²/s², 0.00063 m²/s², and 0.00063 m²/s², respectively. However, the crucial difference is that the TKE for the concave seawall, after reaching its peak, remains significantly elevated for a longer duration during the wave drawdown phase compared to the other two profiles. The turbulence generated by the convex and sloped walls, while momentarily intense, decays much more rapidly after the initial impact. This period of sustained, high-intensity turbulence at the toe of the concave wall corresponds directly to the formation and persistence of the large, organized vortex at its base (which will be shown in Section 4.5.2 ). Therefore, it is this prolonged exposure to high turbulence, rather than the peak value alone, that is the key indicator of a higher and more significant scour potential for the concave seawall. 4.5.2. Detailed Flow Kinematics and Governing Mechanisms The detailed velocity fields reveal the physical mechanisms responsible for the observed results. At the moment of maximum run-up, Fig. 18 shows that the concave seawall creates a powerful upward jet with vertical velocities reaching 2.27 m/s, explaining its low overtopping. During the drawdown phase, Fig. 20 shows the formation of a large, coherent vortex at the toe of the concave seawall. The X-velocity contours quantify this structure, revealing strong offshore flow near the bed (min Vx = − 1.36 m/s) beneath a region of onshore flow (max Vx = 0.30 m/s). This stable vortex, not present in the other cases, is the mechanism that drives the sustained turbulence shown in Fig. 18 . 4.5.3. Synthesis: Turbulent Viscosity and Scour Potential Finally, Fig. 21 links the vortex structure to scour potential using dynamic viscosity contours. For the concave seawall (Fig. 21 a), a region of elevated dynamic viscosity is observed to be perfectly co-located with the energetic vortex from Fig. 20 a. This direct correlation demonstrates that the vortex generates intense, sustained turbulence at the foundation, confirming a high potential for local scour. The other geometries show more benign conditions. This comprehensive analysis highlights the critical engineering trade-off in the concave seawall design: its superior overtopping performance is achieved at the cost of creating aggressive hydrodynamic conditions at its foundation. 5. Conclusion This study presented a systematic numerical investigation into the hydrodynamic interaction of solitary waves with coastal seawalls of concave, convex, and sloped geometries. Using a validated RANS-VOF model, the research provided a comprehensive comparative analysis of hydrodynamic performance (run-up, reflection, overtopping), wave-induced loads (forces, pressures), and local turbulence characteristics (TKE, flow structures). The primary findings are summarized as follows: Concave Seawall Performance and Trade-offs: The concave geometry demonstrated a distinct and complex performance profile. It generated the highest wave run-up and reflection coefficients, as well as the largest, sharpest impact forces, both horizontally and vertically. The high forces were explained by a significant pressure concentration on the upper curved part of the structure due to a focused, up-rushing water jet. Paradoxically, this same jet, with vertical velocities reaching up to 2.27 m/s, was responsible for the concave wall exhibiting the lowest overtopping discharge, as it effectively deflected the flow upwards and back towards the sea. Vortex Generation and Scour Potential: A critical finding was the formation of a large, stable, and energetic vortex at the toe of the concave seawall during the wave drawdown phase. This vortex was quantitatively identified through velocity fields and was shown to be the source of sustained, high-intensity Turbulent Kinetic Energy (TKE) and a co-located region of high dynamic viscosity. This indicates that the concave profile, while superior in overtopping protection, presents a significant potential for local scour at its foundation. Convex and Sloped Seawall Performance: The convex and sloped seawalls exhibited more conventional and moderate behavior. They were subjected to lower wave loads and generated significantly less organized turbulence at their toe compared to the concave wall, suggesting a lower scour risk. However, they were also less effective at reducing wave overtopping. These findings have direct engineering implications. The concave seawall design presents an interesting trade-off: it offers superior performance in terms of overtopping reduction, which is a primary goal for coastal protection, but its tendency to generate aggressive local hydrodynamics at its base may necessitate costly toe protection measures (e.g., scour aprons or riprap). This must be considered in the overall lifecycle cost and design optimization of such structures. This study reveals a critical engineering trade-off inherent in the concave seawall's design: its excellent hydraulic performance in minimizing overtopping comes at the cost of higher structural loads and a significant risk of foundation instability due to scour. These factors must be carefully weighed in any practical application, likely requiring a more robust structural design and costly toe protection measures to be viable. The limitations of this study include its two-dimensional nature, the assumption of smooth, impermeable boundaries, a focus on non-breaking solitary waves, and the lack of direct experimental validation for the curved geometries. Future research should be directed towards 3D simulations to investigate lateral effects, physical modeling to validate the complex flow structures and forces observed, analysis of different wave types such as breaking and irregular waves, and the design of effective scour protection for concave seawall profiles. Declarations Declarations of generative AI and AI-assisted technologies in the writing process Author Contribution Author Contributions StatementS.M.G. contributed to all aspects of the study, including Conceptualization, Data curation, Formal Analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, and Writing – review & editing.F.S.B., A.M., R.S.B., A.G., and E.T.A. contributed to Writing – review & editing.All authors reviewed the manuscript. Acknowledgement I gratefully acknowledge the late Professor Mohammad Javad Ketabdari of the Department of Maritime Engineering, Amirkabir University of Technology, whose collaborative insight and guidance laid the foundation for this research. Tragically, Dr. Ketabdari passed away from a heart attack on Monday, November 14, 2022. I honor his memory and dedication to the advancement of coastal engineering. References Synolakis, C. E. 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Cite Share Download PDF Status: Published Journal Publication published 03 Feb, 2026 Read the published version in Water Waves → Version 1 posted Editorial decision: Revision requested 13 Oct, 2025 Reviews received at journal 11 Oct, 2025 Reviewers agreed at journal 07 Oct, 2025 Reviews received at journal 07 Oct, 2025 Reviewers agreed at journal 01 Oct, 2025 Reviewers invited by journal 12 Sep, 2025 Editor assigned by journal 05 Sep, 2025 Submission checks completed at journal 25 Aug, 2025 First submitted to journal 24 Aug, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7445438","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":514143025,"identity":"9a4a73cb-aa8f-408c-afac-9b24e2d3e766","order_by":0,"name":"SeyedMahmood Ghassemizadeh","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+UlEQVRIiWNgGAWjYDACCQjFA8QGzAwVMGEDorWcgbCI0gJWxszYBtOCB8hHNz/++KNim4x8++GNjwvn2SXuZ2B++IGh4B5OLYZ3jplJ85y5zcPYk1ZsPHNbcmIPA5uxBINBMW4tMxLMgO65zcPMkGMmzbuNGaiFwQzoyAQ8WtI/f/z57zYPG/8b89+8c+qBWti/4dUiL5FjIMHbcJuHRyLHjJm34TBQCw9+WwwkcsqkeY7d5pGQeFYMZBw37jnMUyyRgM+WGembP/6ouW0v35+88TNPTbVse3v7xg8f/uCx5QCGEDMQ49YAtKUBj+QoGAWjYBSMAjAAAA5JS8+j2T6sAAAAAElFTkSuQmCC","orcid":"","institution":"Amirkabir University of Technology","correspondingAuthor":true,"prefix":"","firstName":"SeyedMahmood","middleName":"","lastName":"Ghassemizadeh","suffix":""},{"id":514143026,"identity":"554a0b29-11f9-4f29-b942-f9a1213cb516","order_by":1,"name":"Farzad ShojaeeBaghdar","email":"","orcid":"","institution":"Ferdowsi University","correspondingAuthor":false,"prefix":"","firstName":"Farzad","middleName":"","lastName":"ShojaeeBaghdar","suffix":""},{"id":514143027,"identity":"7c06934c-2f0a-43b1-81fd-cee09f21c0d3","order_by":2,"name":"Amin Mazaherizaveh","email":"","orcid":"","institution":"MA Graduate, Royal Roads University","correspondingAuthor":false,"prefix":"","firstName":"Amin","middleName":"","lastName":"Mazaherizaveh","suffix":""},{"id":514143028,"identity":"547b8cd6-7b8e-49da-a923-e0cebda92dbe","order_by":3,"name":"Reza BaghdarShojaee","email":"","orcid":"","institution":"Islamic Azad University","correspondingAuthor":false,"prefix":"","firstName":"Reza","middleName":"","lastName":"BaghdarShojaee","suffix":""},{"id":514143029,"identity":"73115783-a6ad-4a48-acde-4132b7d5dc3b","order_by":4,"name":"Ali Ghasemizadeh","email":"","orcid":"","institution":"Mahar-Ab Consulting Engineers","correspondingAuthor":false,"prefix":"","firstName":"Ali","middleName":"","lastName":"Ghasemizadeh","suffix":""},{"id":514143030,"identity":"5387293a-edc8-4c6f-bbd9-390dabece694","order_by":5,"name":"Esmaeel Talesh-Alikhani","email":"","orcid":"","institution":"National Energy Oil and Gas","correspondingAuthor":false,"prefix":"","firstName":"Esmaeel","middleName":"","lastName":"Talesh-Alikhani","suffix":""}],"badges":[],"createdAt":"2025-08-24 09:53:19","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7445438/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7445438/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s42286-026-00131-2","type":"published","date":"2026-02-03T15:59:17+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":91702019,"identity":"96463c21-174f-4d64-811d-6961082ee2cf","added_by":"auto","created_at":"2025-09-19 10:42:29","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":47389,"visible":true,"origin":"","legend":"\u003cp\u003eDefinition sketch of solitary wave run-up on a generic coastal structure, indicating key parameters: incident wave height (\u003cem\u003eH\u003c/em\u003e), still water depth (\u003cem\u003eh₀\u003c/em\u003e), maximum vertical run-up (\u003cem\u003eR\u003c/em\u003e), and structure slope angle (\u003cem\u003eθ\u003c/em\u003e)\u003c/p\u003e","description":"","filename":"1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/5b189cd3ba0341f0c9c9f430.jpg"},{"id":91702349,"identity":"537657e3-9dc0-4664-92fc-7ebdd3135794","added_by":"auto","created_at":"2025-09-19 10:50:29","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":53501,"visible":true,"origin":"","legend":"\u003cp\u003eInput parameters for solitary wave generation in FLOW-3D\u003csup\u003e®\u003c/sup\u003e, including Wave Height, Undisturbed Water Depth, and Current Velocity (set to zero for quiescent conditions)\u003c/p\u003e","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/87da6ee0881a6b7a8c49f39f.jpg"},{"id":91702021,"identity":"a089dcb3-4e03-41ee-9cb0-21323607acf9","added_by":"auto","created_at":"2025-09-19 10:42:29","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":41941,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic profiles of the five seawall geometries investigated: (a) Concave face (R=0.5m), (b) Convex face (R=0.5m), (c) Sloped (1:0.58 or 60°), (d) Sloped (1:1 or 45°), (e) Sloped (1:1.73 or 30°). All structures have a crest height of 0.5m and a base length of 1.0 m\u003c/p\u003e","description":"","filename":"3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/336a3fac017954c0b1a43114.jpg"},{"id":91702350,"identity":"b8c5fd83-efac-48f2-a9f4-3223379050d6","added_by":"auto","created_at":"2025-09-19 10:50:29","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":47078,"visible":true,"origin":"","legend":"\u003cp\u003eMesh sensitivity analysis showing the variation of computed maximum wave run-up on a concave seawall with different mesh sizes (target \u003cem\u003eH\u003c/em\u003e= 0.3 m, \u003cem\u003eh₀\u003c/em\u003e= 0.5 m; effective \u003cem\u003eH/h₀\u003c/em\u003e≈ 0.56)\u003c/p\u003e","description":"","filename":"4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/3cb75bca94e57c3ec79afaa8.jpg"},{"id":91702351,"identity":"2e32d4b3-df34-4518-a9a4-1406024b15fa","added_by":"auto","created_at":"2025-09-19 10:50:29","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":126612,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram showing the locations of the numerical pressure and TKE monitoring points (probes) on the (a) Concave, (b) Convex, and (d) Sloped 45° seawall profiles\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/1e54709efcd08e0345c8f30a.png"},{"id":91703881,"identity":"e0797679-c71b-4b5c-866b-efd7ac080c6b","added_by":"auto","created_at":"2025-09-19 11:06:29","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":76014,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of the numerically generated solitary wave profile (mesh size 0.015m) with the analytical solution (e.g., Munk (1949), or appropriate Boussinesq solution reference) and experimental data from Synolakis (1987) for \u003cem\u003eH/h₀\u003c/em\u003e≈ 0.27 (\u003cem\u003eH\u003c/em\u003e= 0.056 m, \u003cem\u003eh₀\u003c/em\u003e= 0.21 m) at \u003cem\u003et\u003c/em\u003e= 1.5 s\u003c/p\u003e","description":"","filename":"6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/194acb4d90cc07c0c58b9b69.jpg"},{"id":91703551,"identity":"84809c31-e255-411e-9280-c2e0154b4023","added_by":"auto","created_at":"2025-09-19 10:58:29","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":46213,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of the simulated solitary wave profile during run-up on a 1:19.85 slope with experimental data from Synolakis (1987) for \u003cem\u003eH/h₀\u003c/em\u003e≈ 0.27\u003c/p\u003e","description":"","filename":"7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/c836dbd7a23393a48d2a8ef7.jpg"},{"id":91703547,"identity":"d027a8a8-0e98-4870-807a-4507aeaa7663","added_by":"auto","created_at":"2025-09-19 10:58:29","extension":"jpg","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":70627,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of normalized wave run-up (\u003cem\u003eR/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e) versus normalized wave height (\u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e) for a 1:1 sloped wall: current study results against experimental data of Hall \u0026amp; Watts (1953) and asymptotic limits by Borthwick et al. (2005)\u003c/p\u003e","description":"","filename":"8.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/37aaaaef73b3077ee3990bac.jpg"},{"id":91702352,"identity":"3662bd9c-5a7f-4c42-b83b-ec37db6bde98","added_by":"auto","created_at":"2025-09-19 10:50:29","extension":"jpg","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":75757,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of normalized run-up (\u003cem\u003eR/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e) against normalized wave height (\u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e) for various slopes (e.g., 1:1.732, 1:1, 1:0.577), comparing current study results with the empirical formulation by Vinodh \u0026amp; Tanaka (2020) (Eq. (9)). The \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e axis is limited to reflect physically realistic pre-breaking solitary wave conditions\u003c/p\u003e","description":"","filename":"9.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/1bfcdb455fc6df4d5aa039ce.jpg"},{"id":91702042,"identity":"5cfdf5b0-e9aa-4c00-be4d-06ff35b7dbfb","added_by":"auto","created_at":"2025-09-19 10:42:29","extension":"jpg","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":198781,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of simulated solitary wave profiles illustrating key interaction phenomena (run-up, reflection, and overtopping) for representative concave, convex, and sloped seawall geometries under two effective incident wave conditions (\u003cem\u003eH/h₀\u003c/em\u003e≈ 0.56 and \u003cem\u003eH/h₀\u003c/em\u003e≈ 0.73)\u003c/p\u003e","description":"","filename":"10.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/fc0159a82aa65887b7e82d66.jpg"},{"id":91704613,"identity":"7dfd4705-7012-4df5-9df1-46c88c0496e7","added_by":"auto","created_at":"2025-09-19 11:14:29","extension":"jpg","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":60256,"visible":true,"origin":"","legend":"\u003cp\u003eNormalized maximum wave run-up (\u003cem\u003eR/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e) versus effective normalized incident wave height (\u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e) for concave, convex, and various sloped seawalls for effective \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e≈ 0.56 and \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e≈ 0.73\u003c/p\u003e","description":"","filename":"11.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/abe54ec51293c72780e42983.jpg"},{"id":91703882,"identity":"d7a2a457-8087-406d-aa41-fb8a2f15ef32","added_by":"auto","created_at":"2025-09-19 11:06:29","extension":"jpg","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":58309,"visible":true,"origin":"","legend":"\u003cp\u003eWave reflection coefficient (\u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e) versus effective normalized incident wave height (\u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e) for concave, convex, and various sloped seawalls for effective \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e≈ 0.56 and \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e≈ 0.73\u003c/p\u003e","description":"","filename":"12.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/dc08cd7ab0aabe6186e8b45c.jpg"},{"id":91702034,"identity":"477d185c-b938-4587-bedc-c3895fced294","added_by":"auto","created_at":"2025-09-19 10:42:29","extension":"jpg","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":70460,"visible":true,"origin":"","legend":"\u003cp\u003eOvertopping discharge rate vs. time for various seawall geometries - Effective incident \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e≈ 0.56\u003c/p\u003e","description":"","filename":"13.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/2254a501fd4ab80095bf99d6.jpg"},{"id":91702038,"identity":"a6e5b553-c80a-4b79-818b-8942536042b1","added_by":"auto","created_at":"2025-09-19 10:42:29","extension":"jpg","order_by":14,"title":"Figure 14","display":"","copyAsset":false,"role":"figure","size":71547,"visible":true,"origin":"","legend":"\u003cp\u003eOvertopping discharge rate vs. time for various seawall geometries - Effective incident \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e≈ 0.73\u003c/p\u003e","description":"","filename":"14.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/4e282a0b1fe5cdacb183d046.jpg"},{"id":91702049,"identity":"3425f41a-cdba-4e41-8f0f-d8de152f3265","added_by":"auto","created_at":"2025-09-19 10:42:29","extension":"jpg","order_by":15,"title":"Figure 15","display":"","copyAsset":false,"role":"figure","size":52282,"visible":true,"origin":"","legend":"\u003cp\u003eTime history of the horizontal wave force (\u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003eh\u003c/em\u003e\u003c/sub\u003e) on the three representative seawall geometries (Concave, Convex, Sloped 45°) for the effective incident wave condition \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e≈ 0.73\u003c/p\u003e","description":"","filename":"15.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/94f55010486f4459e02ad617.jpg"},{"id":91703553,"identity":"43455056-2b78-4719-ae45-514aeb208648","added_by":"auto","created_at":"2025-09-19 10:58:29","extension":"jpg","order_by":16,"title":"Figure 16","display":"","copyAsset":false,"role":"figure","size":53989,"visible":true,"origin":"","legend":"\u003cp\u003eTime history of the horizontal wave force (\u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003ev\u003c/em\u003e\u003c/sub\u003e) on the three representative seawall geometries (Concave, Convex, Sloped 45°) for the effective incident wave condition \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e≈ 0.73\u003c/p\u003e","description":"","filename":"16.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/3bb082821c35c27e09fadfa7.jpg"},{"id":91702057,"identity":"00c94dee-6d73-4eb6-8352-8666f30f2131","added_by":"auto","created_at":"2025-09-19 10:42:30","extension":"jpg","order_by":17,"title":"Figure 17","display":"","copyAsset":false,"role":"figure","size":58160,"visible":true,"origin":"","legend":"\u003cp\u003eDynamic pressure distribution along the face of the (a) Concave, (b) Convex, and (c) Sloped 45° seawalls at the instant of maximum horizontal force (\u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003eh,max\u003c/em\u003e\u003c/sub\u003e) for the effective incident wave condition \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e≈ 0.73\u003c/p\u003e","description":"","filename":"17.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/be08490f9a475cd022268f7b.jpg"},{"id":91703555,"identity":"11081937-129e-43e3-88a0-cb036e60a1a8","added_by":"auto","created_at":"2025-09-19 10:58:29","extension":"jpg","order_by":18,"title":"Figure 18","display":"","copyAsset":false,"role":"figure","size":55462,"visible":true,"origin":"","legend":"\u003cp\u003eTime history of Turbulent Kinetic Energy (TKE) at the probe point nearest the toe (P1) for the concave, convex, and sloped 45° seawalls for the effective incident wave condition \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e≈ 0.73\u003c/p\u003e","description":"","filename":"18.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/96656905a5a33b285087e6b1.jpg"},{"id":91702071,"identity":"c91980a8-0762-49c6-a3bf-88d24d7bcba8","added_by":"auto","created_at":"2025-09-19 10:42:30","extension":"jpg","order_by":19,"title":"Figure 19","display":"","copyAsset":false,"role":"figure","size":145053,"visible":true,"origin":"","legend":"\u003cp\u003eFlow kinematics near the seawall crest at the time of maximum run-up for the three representative geometries. Each row shows two panels: (i) Velocity Vectors and (ii) Z-Velocity Contours \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e≈ 0.73\u003c/p\u003e","description":"","filename":"19.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/d21f2b8190a9135841946d50.jpg"},{"id":91702048,"identity":"51694e22-99c0-4890-b59f-8719aa3c09d4","added_by":"auto","created_at":"2025-09-19 10:42:29","extension":"jpg","order_by":20,"title":"Figure 20","display":"","copyAsset":false,"role":"figure","size":183437,"visible":true,"origin":"","legend":"\u003cp\u003eFlow kinematics near the seawall toe at the time of maximum drawdown for the three representative geometries. Each row shows two panels: (i) Velocity Vectors and (ii) X-Velocity Contours \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e≈ 0.73\u003c/p\u003e","description":"","filename":"20.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/aef442782724251dfd16a7a9.jpg"},{"id":91702056,"identity":"920e0bf3-1e27-4372-bafe-baa9ec1a3d20","added_by":"auto","created_at":"2025-09-19 10:42:30","extension":"jpg","order_by":21,"title":"Figure 21","display":"","copyAsset":false,"role":"figure","size":175689,"visible":true,"origin":"","legend":"\u003cp\u003eContours of dynamic viscosity near the toe of the three representative geometries at the time of maximum drawdown for the effective incident \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e≈ 0.73\u003c/p\u003e","description":"","filename":"21.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/70a5ce48a3907a0dae2dac15.jpg"},{"id":102234802,"identity":"8efd23bf-8963-4222-8059-7772ef620a27","added_by":"auto","created_at":"2026-02-09 16:13:21","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3353630,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7445438/v1/87b6c6ca-69cc-4df2-8d78-a9a709cd3a70.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Comparative Numerical Study of Solitary Wave Interaction with Concave, Convex, and Sloped Seawalls: Hydrodynamics, Wave Loads, and Turbulent Flow Analysis","fulltext":[{"header":"Highlights","content":"\u003cp\u003e1. A comparative analysis links seawall geometry to wave loads, hydrodynamics, and turbulence.\u003c/p\u003e\u003cp\u003e2. Concave seawalls amplify wave reflection and impact loads, creating intense toe vortices.\u003c/p\u003e\u003cp\u003e3. Convex profiles reduce wave loads and near-bed turbulence by promoting smoother flow separation.\u003c/p\u003e\u003cp\u003e4. Sloped geometry is most effective at dissipating energy but shows the highest overtopping risk.\u003c/p\u003e\u003cp\u003e5. Seawall curvature is identified as a critical parameter for optimizing structural stability and hydrodynamic performance.\u003c/p\u003e"},{"header":"1. Introduction","content":"\u003cp\u003eCoastal structures, such as seawalls and dikes, serve as essential defenses against wave attacks, erosion, and flooding. Seawalls, often built as vertical or sloped barriers along shorelines, primarily reflect wave energy, while dikes, typically earthen embankments, aim to contain water and prevent inland flooding. The effectiveness of these structures under highly nonlinear solitary waves\u0026mdash;which mimic aspects of tsunamis and storm surges\u0026mdash;remains a key focus in coastal engineering. Research suggests that understanding these interactions can improve infrastructure resilience, though challenges persist in predicting complex flow behaviors.\u003c/p\u003e\u003cp\u003eStudies have extensively examined solitary wave interactions with coastal structures. Early work by Synolakis (1987) and Hall and Watts (1953) laid foundations for wave run-up and reflection on impermeable sloped or vertical walls. More recent analyses, such as Vinodh and Tanaka (2020) and Casella et al. (2022), have refined run-up formulas using experimental data on sloping beaches. Emerging research highlights the role of curved profiles: Anada et al. (2011) compared curved and vertical seawalls under regular waves, while Suba et al. (2019) assessed curvature effects on dike run-up. Castellino et al. (2018) numerically investigated impulsive forces on curved seawalls, revealing geometry's influence on wave loads. In the context of breaking waves, Kimmoun and Branger (2007) used particle image velocimetry (PIV) to study turbulent flows in surf zones over sloping beaches, providing insights into vortex formation and energy dissipation that align with solitary wave dynamics.\u003c/p\u003e\u003cp\u003eWave-induced loads are critical for structural design. Investigations like Lin et al. (2009), Luo et al. (2019), and Wang et al. (2023) have quantified forces and pressures on vertical and sloped walls. Wu (2022) focused on breaking solitary impacts on vertical seawalls, emphasizing turbulence. However, curved geometries are underexplored; Grilli et al. (2004), involving Kimmoun, combined numerical and experimental approaches to solitary wave shoaling and breaking, linking to potential scour. A systematic comparison across concave, convex, and sloped profiles\u0026mdash;integrating loads, pressure, and turbulent structures\u0026mdash;is lacking.\u003c/p\u003e\u003cp\u003eTurbulence during impacts informs scour risks, with Turbulent Kinetic Energy (TKE) as a key metric. Lubin et al. (2011) and Lubin et al. (2019), building on Kimmoun's work, modeled breaking wave instabilities and vortices. Bona et al. (2023) and Stastna et al. (2022) from Water Waves journal analyzed solitary wave dynamics, offering theoretical support for nonlinear interactions. Yet, linking geometry to flow kinematics (e.g., vortices, jets) and scour potential remains incomplete.\u003c/p\u003e\u003cp\u003eExtending prior numerical efforts (Ghassemizadeh and Ketabdari, 2016, 2022), this study uses FLOW-3D\u003csup\u003e\u0026reg;\u003c/sup\u003e with RANS equations, k-ε RNG turbulence, and VOF method (Hirt and Nichols, 1981; Flow Science, Inc., 2011). Novel contributions include: (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e) comparing hydrodynamic performance (run-up, reflection, overtopping) across geometries; (\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e) quantifying forces and pressures; (\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e) analyzing TKE evolution; and (\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e) connecting geometry to flows, loads, and scour, with a simplified economic assessment. Figure\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e illustrates wave run-up.\u003c/p\u003e\u003cp\u003eThis paper is structured as follows: Section \u003cspan refid=\"Sec2\" class=\"InternalRef\"\u003e2\u003c/span\u003e details the governing equations and numerical model setup. Section \u003cspan refid=\"Sec12\" class=\"InternalRef\"\u003e3\u003c/span\u003e presents the model validation against established experimental data and empirical formulas. Section \u003cspan refid=\"Sec19\" class=\"InternalRef\"\u003e4\u003c/span\u003e discusses the main results, including wave run-up, reflection, overtopping, wave loads, internal flow structures, and the economic assessment. Finally, Section \u003cspan refid=\"Sec32\" class=\"InternalRef\"\u003e5\u003c/span\u003e provides the main conclusions drawn from this investigation.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e"},{"header":"2. Numerical Model Setup","content":"\u003cp\u003eThis study employs a two-dimensional numerical model using the commercial CFD software FLOW-3D\u003csup\u003e\u0026reg;\u003c/sup\u003e (v10.0.1) to simulate the interaction between solitary waves and coastal structures with different wall profiles. The model is based on solving the Reynolds-Averaged Navier-Stokes (RANS) equations coupled with a turbulence model and a free-surface tracking method.\u003c/p\u003e\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\u003ch2\u003e2.1. Governing Equations\u003c/h2\u003e\u003cp\u003eThe fluid motion is governed by the incompressible RANS equations, which, for a Newtonian fluid, consist of the continuity and momentum equations. In Cartesian coordinates, these are expressed as:\u003c/p\u003e\u003cp\u003eContinuity Equation:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:\\frac{{\\partial\\:u}_{i}}{{\\partial\\:x}_{i}}=0$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eMomentum Equation:\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:\\frac{{\\partial\\:u}_{i}}{\\partial\\:\\text{t}}+{u}_{j}\\frac{{\\partial\\:u}_{i}}{{\\partial\\:x}_{j}}=-\\frac{1}{\\rho\\:}\\frac{\\partial\\:\\text{p}}{{\\partial\\:x}_{i}}+\\frac{\\partial\\:}{{\\partial\\:x}_{j}}\\left[\\left(\\mu\\:+{\\mu\\:}_{t}\\right)+\\left(\\frac{{\\partial\\:u}_{i}}{{\\partial\\:x}_{j}}+\\frac{{\\partial\\:u}_{j}}{{\\partial\\:x}_{i}}\\right)\\right]+{g}_{i}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cem\u003eu\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eu\u003c/em\u003e\u003csub\u003e\u003cem\u003ej\u003c/em\u003e\u003c/sub\u003e are the time-averaged velocity components in the \u003cem\u003ex\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003ex\u003c/em\u003e\u003csub\u003e\u003cem\u003ej\u003c/em\u003e\u003c/sub\u003e directions (with \u003cem\u003ei,j\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1,2 for 2D flow, representing horizontal and vertical directions respectively), \u003cem\u003et\u003c/em\u003e is time, \u003cem\u003eρ\u003c/em\u003e is the fluid density, \u003cem\u003ep\u003c/em\u003e is the mean pressure, \u003cem\u003eν\u003c/em\u003e is the kinematic viscosity of the fluid, \u003cem\u003eν\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e is the turbulent kinematic (eddy) viscosity, and \u003cem\u003eg\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e is the gravitational acceleration component in the \u003cem\u003ex\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e direction. The term (\u003cem\u003eν\u0026thinsp;+\u0026thinsp;ν\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e) represents the effective kinematic viscosity. The eddy viscosity \u003cem\u003eν\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e is determined by the turbulence model.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\u003ch2\u003e2.2. Turbulence Model (RNG k-ε)\u003c/h2\u003e\u003cp\u003eTo account for the effects of turbulence, the Renormalization Group (RNG) k-ε turbulence model Yakhot et al. (1992) is employed. This model is known for its improved accuracy in simulating flows with high strain rates, streamline curvature, and regions of recirculation, which are characteristic of wave-structure interactions. The RNG k-ε model involves solving two additional transport equations for the turbulent kinetic energy (\u003cem\u003ek\u003c/em\u003e) and its dissipation rate (\u003cem\u003eε\u003c/em\u003e):\u003c/p\u003e\u003cp\u003eTurbulent Kinetic Energy (\u003cem\u003ek\u003c/em\u003e) Equation:\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:\\frac{{\\partial\\:}_{k}}{\\partial\\:t}+{u}_{j}\\frac{{\\partial\\:}_{k}}{{\\partial\\:x}_{j}}=\\frac{\\partial\\:}{{\\partial\\:x}_{j}}\\left({\\alpha\\:}_{k}{\\nu\\:}_{eff}\\frac{{\\partial\\:}_{k}}{{\\partial\\:x}_{j}}\\right)+{G}_{k}-\\epsilon\\:$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eTurbulent Dissipation Rate (\u003cem\u003eε\u003c/em\u003e) Equation:\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\:\\frac{{\\partial\\:}_{\\epsilon\\:}}{\\partial\\:t}+{u}_{j}\\frac{{\\partial\\:}_{\\epsilon\\:}}{{\\partial\\:x}_{j}}=\\frac{\\partial\\:}{{\\partial\\:x}_{j}}\\left({\\alpha\\:}_{\\epsilon\\:}{\\nu\\:}_{eff}\\frac{{\\partial\\:}_{\\epsilon\\:}}{{\\partial\\:x}_{j}}\\right)+{{C}_{\\epsilon\\:1}\\frac{\\epsilon\\:}{k}G}_{k}-{C}_{\\epsilon\\:2}\\frac{{\\epsilon\\:}^{2}}{k}-{R}_{\\epsilon\\:}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eIn these equations, \u003cem\u003eG\u003c/em\u003e\u003csub\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e is the generation of turbulent kinetic energy due to mean velocity gradients, \u003cem\u003eν\u003c/em\u003e\u003csub\u003e\u003cem\u003eeff\u003c/em\u003e\u003c/sub\u003e is the effective turbulent viscosity (related to \u003cem\u003eν\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e). \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003eε1\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003eε2\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003eα\u003c/em\u003e\u003csub\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e, and \u003cem\u003eα\u003c/em\u003e\u003csub\u003e\u003cem\u003eε\u003c/em\u003e\u003c/sub\u003e are model constants. The term \u003cem\u003eR\u003c/em\u003e\u003csub\u003e\u003cem\u003eε\u003c/em\u003e\u003c/sub\u003e is an additional term in the RNG model that improves its performance for rapidly strained flows. The standard values for these constants as implemented in FLOW-3D\u003csup\u003e\u0026reg;\u003c/sup\u003e are adopted for this study, as summarized in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. The eddy viscosity \u003cem\u003eν\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e is then calculated as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\nu\\:}_{t}={C}_{\\mu\\:}\\frac{{k}^{2}}{{\\epsilon\\:}}\\)\u003c/span\u003e\u003c/span\u003e, where \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003e\u0026micro;\u003c/em\u003e\u003c/sub\u003e is another model constant.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eConstants for the RNG k-ε turbulence model\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"2\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eConstants\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eAmount\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003ec\u003csub\u003e3\u003c/sub\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.0120\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003ec\u003csub\u003e\u0026micro;\u003c/sub\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.0850\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003ec\u003csub\u003eε1\u003c/sub\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e1.4200\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003ec\u003csub\u003eε2\u003c/sub\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e1.6800\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eα\u003csub\u003ek\u003c/sub\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.7194\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eα\u003csub\u003eε\u003c/sub\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.7194\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eη\u003csub\u003e0\u003c/sub\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e4.3800\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\u003ch2\u003e2.3. Free Surface Tracking and Advection\u003c/h2\u003e\u003cp\u003eThe Volume of Fluid (VOF) method, as proposed by Hirt \u0026amp; Nichols (1981), is used to track the interface between water and air. In this method, a scalar function F (volume of fluid fraction) is defined, representing the fractional volume of a computational cell occupied by water. \u003cem\u003eF\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1 indicates a cell full of water, \u003cem\u003eF\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0 indicates a cell full of air, and 0\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eF\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;1 indicates an interface cell. The advection of \u003cem\u003eF\u003c/em\u003e is governed by:\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:F}{\\partial\\:t}+{u}_{j}\\frac{\\partial\\:F}{{\\partial\\:x}_{j}}=0$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eFor the advection of the VOF function, the Split Lagrangian Advection method available in FLOW-3D\u003csup\u003e\u0026reg;\u003c/sup\u003e is employed, which is suitable for maintaining a sharp interface in 2D interfacial flows and ensures mass conservation. Complex geometries and solid obstacles within the Cartesian mesh are represented using the Fractional Area/Volume Obstacle Representation (FAVOR\u0026trade;) method.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\u003ch2\u003e2.4. Numerical Discretization and Solution Algorithm\u003c/h2\u003e\u003cp\u003eThe governing equations (\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan refid=\"Equ5\" class=\"InternalRef\"\u003e5\u003c/span\u003e) are discretized using a Finite Volume Method (FVM) on a structured Cartesian mesh. FLOW-3D\u003csup\u003e\u0026reg;\u003c/sup\u003e uses a staggered grid arrangement where scalar variables (like pressure) are stored at cell centers, and velocity components are stored at cell faces. For time integration, an implicit scheme is used. The pressure-velocity coupling is handled using a pressure correction method. The pressure equation, derived from the continuity and momentum equations, is solved using the Generalized Minimal Residual Solver (GMRES) algorithm, which is effective for non-symmetric linear systems and allows for numerically stable solutions with relatively larger time steps. First-order upwind differencing was used for the advection terms in the momentum and turbulence equations.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e\u003ch2\u003e2.5. Computational Domain and Boundary Conditions\u003c/h2\u003e\u003cp\u003eThe simulations were performed in a 2D numerical wave flume. The flume extended from \u003cem\u003eX\u003c/em\u003e= -10.0 m to \u003cem\u003eX\u003c/em\u003e\u0026thinsp;=\u0026thinsp;+\u0026thinsp;1.0 m (total length of 11.0 m) and had a height from \u003cem\u003eY\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.0 m to \u003cem\u003eY\u003c/em\u003e\u0026thinsp;=\u0026thinsp;+\u0026thinsp;2.0 m. The seawall structure was located at the downstream end of the flume, starting at \u003cem\u003eX\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.0 m and extending to \u003cem\u003eX\u003c/em\u003e\u0026thinsp;=\u0026thinsp;+\u0026thinsp;1.0 m (base length of 1.0 m). The crest elevation of all seawall structures was maintained at \u003cem\u003eZ\u003c/em\u003e\u003csub\u003e\u003cem\u003ecrest\u003c/em\u003e\u003c/sub\u003e= 0.5 m from the flume bed (\u003cem\u003eY\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.0 m). The still water depth (\u003cem\u003eh₀\u003c/em\u003e) was set according to the specific test case (see Section \u003cspan refid=\"Sec8\" class=\"InternalRef\"\u003e2.6\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eThe boundary conditions applied to the computational domain were as follows (summarized in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e):\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eBoundary conditions employed in the numerical simulations\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"3\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eBoundary Location\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eType of Boundary Condition\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eDescription\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eInlet (X= -10.0 m)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eWave\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eSolitary wave generation\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eOutlet (X\u0026thinsp;=\u0026thinsp;+\u0026thinsp;1.0 m)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eOutflow\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eWave and overtopped water absorption, min. reflection\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eBottom (Y\u0026thinsp;=\u0026thinsp;0.0 m)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eWall (No-slip)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eImpermeable flume bed\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eTop (Y\u0026thinsp;=\u0026thinsp;+\u0026thinsp;2.0 m)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003ePressure\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eAtmospheric pressure, free air movement\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eStructure Surfaces\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eWall (No-slip)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eImpermeable seawall surfaces\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003eInlet (Left boundary, \u003cem\u003eX\u003c/em\u003e= -10.0 m): A solitary wave was generated using the specific wave generation boundary condition in FLOW-3D\u003csup\u003e\u0026reg;\u003c/sup\u003e.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eOutlet (Right boundary, \u003cem\u003eX\u003c/em\u003e\u0026thinsp;=\u0026thinsp;+\u0026thinsp;1.0 m): An outflow boundary condition was applied to allow waves and any overtopped water to pass out of the domain with minimal reflection.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eBottom (Bottom boundary, \u003cem\u003eY\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.0 m): A no-slip wall boundary condition was applied, representing the impermeable bed of the flume.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eTop (Top boundary, \u003cem\u003eY\u003c/em\u003e\u0026thinsp;=\u0026thinsp;+\u0026thinsp;2.0 m): A pressure boundary condition was specified, representing atmospheric pressure, allowing air to move freely.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eStructure Surfaces: A no-slip wall boundary condition was applied to all seawall surfaces.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\u003ch2\u003e2.6. Solitary Wave Generation and Characteristics\u003c/h2\u003e\u003cp\u003eSolitary waves were generated at the inlet of the numerical flume using the built-in solitary wave generation feature in FLOW-3D\u0026reg;, which is based on Boussinesq-type approximations. The key input parameters for wave generation were the incident wave height (\u003cem\u003eH\u003c/em\u003e) and the still water depth \u003cem\u003e(h₀\u003c/em\u003e). Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e illustrates the input parameters required by the software. The simulations in this study targeted two primary non-dimensional wave height-to-depth ratios:\u003c/p\u003e\u003cp\u003e\u003col\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eTarget \u003cem\u003eH/h₀\u003c/em\u003e = 0.6: with an input \u003cem\u003eH\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.3 m and \u003cem\u003eh₀\u003c/em\u003e= 0.5 m.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eTarget \u003cem\u003eH/h₀\u003c/em\u003e \u0026asymp; 0.8: with an input \u003cem\u003eH\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.45 m and \u003cem\u003eh₀\u003c/em\u003e= 0.55 m (resulting in an input \u003cem\u003eH/h₀\u003c/em\u003e \u0026asymp; 0.818).\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003c/ol\u003e\u003c/p\u003e\u003cp\u003eAnalysis of the wave profiles propagating in the flume (prior to interaction with the structures, based on supplementary data from the simulations) indicated that \u003cem\u003ethe\u003c/em\u003e effective \u003cem\u003eH/h₀\u003c/em\u003e ratios were approximately 0.56 for the first target case (\u003cem\u003eH\u003c/em\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.28 m, \u003cem\u003eh₀\u003c/em\u003e= 0.5 m), and approximately 0.73 for the second target case (\u003cem\u003eH\u003c/em\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.40 m, \u003cem\u003eh₀\u003c/em\u003e= 0.55 m). These effective wave characteristics were confirmed to be largely non-breaking during their propagation in the 10.0 m long open flume section before reaching the structures. All subsequent analyses of wave run-up, reflection, and overtopping are based on these effective incident wave conditions. The initial position of the solitary wave crest was set at a distance of 0.5\u003cem\u003eL\u003c/em\u003e from the inlet, where \u003cem\u003eL\u003c/em\u003e is the characteristic wavelength of the solitary wave, ensuring the wave was fully formed before reaching the structure. The domain was initialized with a quiescent water body (zero initial velocity).\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e\u003ch2\u003e2.7. Seawall Geometries\u003c/h2\u003e\u003cp\u003eThree main types of seawall front-face geometries were investigated, all with a crest elevation of 0.5 m above the flume bed:\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003eConcave Seawall: This geometry featured a circular arc curving inwards towards the landside. The radius of curvature was R\u003csub\u003ec\u003c/sub\u003e= 0.5 m.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eConvex Seawall: This geometry featured a circular arc curving outwards towards the seaside, also with a radius of curvature of R\u003csub\u003ev\u003c/sub\u003e= 0.5 m.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eSloped Seawalls: Three different linear slopes were considered:\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003eSlope 1: gradient 1:0.577 (corresponding to an angle θ\u0026thinsp;=\u0026thinsp;60\u0026deg; with the horizontal).\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eSlope 2: gradient 1:1 (corresponding to an angle θ\u0026thinsp;=\u0026thinsp;45\u0026deg; with the horizontal).\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eSlope 3: gradient 1:1.732 (corresponding to an angle θ\u0026thinsp;=\u0026thinsp;30\u0026deg; with the horizontal).\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003eA schematic depiction of these five seawall profiles is provided in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec10\" class=\"Section2\"\u003e\u003ch2\u003e2.8. Mesh Configuration and Sensitivity\u003c/h2\u003e\u003cp\u003eA structured Cartesian mesh was employed for all simulations. A mesh sensitivity analysis was conducted prior to the main simulations to determine an optimal grid resolution that balances computational accuracy with efficiency. This analysis involved simulating wave run-up on a representative concave seawall (with an effective \u003cem\u003eH/h₀\u003c/em\u003e\u0026asymp; 0.56, based on an input \u003cem\u003eH\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.3 m and \u003cem\u003eh₀\u003c/em\u003e= 0.5 m) using various uniform mesh sizes (\u003cem\u003eΔx\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eΔz\u003c/em\u003e) ranging from 0.06 m down to 0.005 m. The results of this sensitivity analysis are illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. It was observed that for mesh sizes smaller than 0.015 m, the computed maximum wave run-up tended to converge, showing minimal further changes with finer refinement. Therefore, a uniform mesh size of \u003cem\u003eΔx\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eΔz\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.015 m was selected and used for all subsequent simulations presented in this study, ensuring a good balance between accuracy and computational cost.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e\u003ch2\u003e2.9. Data Extraction for Forces, Pressure, and TKE\u003c/h2\u003e\u003cp\u003eIn addition to the primary hydrodynamic parameters (run-up, reflection, overtopping), the numerical model was used to extract detailed data on wave-induced loads and local turbulence intensity.\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003eWave-Induced Forces:\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003eThe total hydrodynamic forces exerted on the seawall, including the horizontal (\u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003eh\u003c/em\u003e\u003c/sub\u003e) and vertical (\u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003ev\u003c/em\u003e\u003c/sub\u003e) components, were calculated by the software at each time step. This calculation is based on the integration of both pressure (\u003cem\u003ep\u003c/em\u003e) and viscous stresses (\u003cem\u003eτ\u003c/em\u003e) over the wetted surface area (\u003cem\u003eA\u003c/em\u003e) of the structure, conceptually represented as:\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\:F=\\underset{A}{\\overset{}{\\int\\:}}(-pn+\\tau\\:.n)dA$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cem\u003en\u003c/em\u003e is the unit normal vector to the surface. The time series of both horizontal and vertical force components were recorded for post-processing.\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003ePressure and Turbulent Kinetic Energy (TKE):\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003eTo analyze the pressure distribution and local turbulence intensity during wave impact, a series of numerical monitoring points (probes) were defined at specific, identical locations along the front face of each representative seawall geometry (concave, convex, and sloped 45\u0026deg;). These probes recorded the time series of dynamic pressure (\u003cem\u003ep\u003c/em\u003e) and Turbulent Kinetic Energy (TKE, \u003cem\u003ek\u003c/em\u003e) at their respective coordinates. The locations of these probes are illustrated schematically in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e"},{"header":"3. Model Validation","content":"\u003cp\u003eTo ensure the accuracy and reliability of the numerical model for simulating solitary wave interactions with coastal structures, a series of validation tests were performed. The numerical results from FLOW-3D\u003csup\u003e\u0026reg;\u003c/sup\u003e were compared against established experimental data and empirical/semi-empirical formulations from well-regarded studies. This validation process covered wave propagation, run-up on sloped structures, and focused on ensuring the model could replicate key physical phenomena before its application to the different seawall geometries.\u003c/p\u003e\u003cdiv id=\"Sec13\" class=\"Section2\"\u003e\u003ch2\u003e3.1. Solitary Wave Profile and Propagation\u003c/h2\u003e\u003cp\u003eFirstly, the model's ability to generate and propagate a stable solitary wave was verified. The generated wave profile in the numerical flume (prior to interaction with any structure) was compared with the analytical solution for a solitary wave, often given by Boussinesq theory as:\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$\\:\\eta\\:\\left(x,t\\right)=H{sech}^{2}\\left[\\sqrt{\\frac{3H}{4{h}_{0}^{3}}}(x-Ct)\\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cem\u003eη\u003c/em\u003e is the free surface elevation, \u003cem\u003eH\u003c/em\u003e is the wave height, \u003cem\u003eh\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e is the still water depth, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:C=\\sqrt{g({h}_{0}+H)}\\)\u003c/span\u003e\u003c/span\u003e is the wave celerity, and \u003cem\u003eg\u003c/em\u003e is the gravitational acceleration. For this validation, a wave with an effective \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026asymp; 0.27 (specifically \u003cem\u003eH\u003c/em\u003e= 0.056 m, \u003cem\u003eh\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e= 0.21 m, as used in the experiments by Synolakis (1987)) was simulated. Figure\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e shows a comparison between the numerical wave profile at a specific time instance (e.g., \u003cem\u003et\u003c/em\u003e= 1.5 s after generation) and the theoretical profile, alongside experimental data points from Synolakis (1987). The comparison demonstrates good agreement in terms of wave shape, crest elevation, and width, confirming the model's capability to accurately represent a propagating solitary wave. The mesh size of \u003cem\u003eΔx\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eΔz\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.015 m, as determined from the sensitivity analysis (Section \u003cspan refid=\"Sec10\" class=\"InternalRef\"\u003e2.8\u003c/span\u003e), was used.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec14\" class=\"Section2\"\u003e\u003ch2\u003e3.2. Wave Run-up on Sloping Structures\u003c/h2\u003e\u003cp\u003eNext, the model was validated for solitary wave run-up on plane, impermeable slopes, which is a critical parameter in this study. The numerical results were compared against widely recognized experimental data and empirical relations.\u003c/p\u003e\u003cdiv id=\"Sec15\" class=\"Section3\"\u003e\u003ch2\u003e3.2.1. Comparison with Synolakis (1987) Data\u003c/h2\u003e\u003cp\u003eThe experimental work of Synolakis (1987) on solitary wave run-up on a slope of 1:19.85 (approximately 2.87\u0026deg;) is a common benchmark. Numerical simulations were conducted to replicate these experiments. For instance, for a case with an incident wave height H\u0026thinsp;=\u0026thinsp;0.056 m and still water depth \u003cem\u003eh\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;0.21 m (giving \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.27, which is a non-breaking wave condition), the computed wave run-up on the slope was compared. Figure\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e illustrates the comparison of the free surface profile during run-up between the numerical model and the experimental data from Synolakis (1987). The maximum run-up values (\u003cem\u003eR\u003c/em\u003e) obtained from the simulation showed good agreement with the experimental findings, with deviations typically within an acceptable range (e.g., 3\u0026ndash;6% as mentioned in the first manuscript), validating the model's performance for run-up predictions on gentle slopes. The empirical relation by Synolakis (1987) for non-breaking waves is often given as:\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$$\\:\\frac{R}{{h}_{0}}=2.831{\\left(cot\\theta\\:\\right)}^{0.5}{\\left(\\frac{H}{{h}_{0}}\\right)}^{1.25}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cem\u003eθ\u003c/em\u003e is the beach slope angle. The numerical results were consistent with predictions from this type of formula for the given slope and wave conditions.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec16\" class=\"Section3\"\u003e\u003ch2\u003e3.2.2. Comparison with Hall \u0026amp; Watts (1953) and Borthwick et al. (2005, 2006)\u003c/h2\u003e\u003cp\u003eFor steeper slopes, the experimental data of Hall \u0026amp; Watts (1953) for a 1:1 slope (45\u0026deg;) were used for comparison. The normalized run-up (\u003cem\u003eR/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e) versus the normalized incident wave height (\u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e) from the numerical simulations were plotted alongside the experimental data of Hall \u0026amp; Watts (1953) and the lower and upper asymptotic limits proposed by Borthwick et al. (2005, 2006), which were derived based on Synolakis's outputs and other data. Figure\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e presents this comparison. The numerical results generally fall within the range of the experimental data and align well with the established asymptotic trends, particularly for non-breaking or early-breaking wave conditions. The first manuscript noted an error within 2\u0026ndash;4% for the same \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e ratio, further supporting the model's reliability for these steeper slopes. It is acknowledged that differences in experimental flume materials versus the idealized smooth wall conditions in the numerical model can contribute to minor discrepancies.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec17\" class=\"Section3\"\u003e\u003ch2\u003e3.2.3. Comparison with Vinodh \u0026amp; Tanaka (2020) Formulation\u003c/h2\u003e\u003cp\u003eTo further validate the model against more recent work, the numerical run-up results for various slopes (30\u0026deg;, 45\u0026deg;, 60\u0026deg;, corresponding to the sloped seawalls in this study) were compared with the unified empirical formula proposed by Vinodh \u0026amp; Tanaka (2020). This formula predicts the maximum run-up of solitary waves on plane beaches across various slopes and wave conditions. The specific form of their unified empirical formula used for comparison in this study is:\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$$\\:\\frac{R}{H}=A{\\left(\\frac{H}{{h}_{0}}\\right)}^{B}\\bullet\\:{tan}^{C}\\left(\\theta\\:\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ewhere A, B, and C are empirically derived coefficients (as provided in Vinodh \u0026amp; Tanaka (2020), for specific conditions or a general form). Figure\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e, shows a comparison of the non-dimensional run-up \u003cem\u003eR/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e against the non-dimensional wave height \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e for the tested slopes. The numerical predictions were found to be in reasonable agreement with the empirical formula of Vinodh \u0026amp; Tanaka (2020) (Eq.\u0026nbsp;\u003cspan refid=\"Equ9\" class=\"InternalRef\"\u003e9\u003c/span\u003e) within the physically relevant range of \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e (typically \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u0026lt;\u0026thinsp;0.8 for stable solitary waves prior to breaking on a slope). This comparison further confirms the model's applicability across the range of slopes investigated in the main study.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Sec18\" class=\"Section2\"\u003e\u003ch2\u003e3.3. Discussion on Validation Limitations and Applicability\u003c/h2\u003e\u003cp\u003eThe validation exercises demonstrate that the numerical model, using the k-ε RNG turbulence model and VOF method with the selected mesh resolution, can reliably simulate solitary wave propagation and run-up on plane impermeable slopes across a range of conditions.\u003c/p\u003e\u003cp\u003eHowever, it is important to note certain limitations:\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003eThe validation cases primarily involved linear sloped beaches or structures. Direct experimental validation for solitary wave interaction with the specific concave and convex geometries investigated in this study was not available in the literature to the same extent. Therefore, the model's application to these arced structures relies on the fundamental accuracy of the RANS-VOF approach for complex free-surface flows and wave-boundary interactions, as demonstrated for sloped cases.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eThe current study focuses on solitary waves whose effective \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e ratios at the point of interaction with the structures were approximately 0.56 and 0.73 (as detailed in Section \u003cspan refid=\"Sec8\" class=\"InternalRef\"\u003e2.6\u003c/span\u003e). While the higher ratio is close to the breaking limit for solitary waves (often cited around \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.78), visual inspection of the wave profiles during propagation in the flume prior to reaching the structures indicated that they maintained largely non-breaking forms. The validation cases also predominantly fall within this non-breaking or incipiently breaking regime. Scenarios involving intensely breaking waves might require different turbulence modeling approaches or finer mesh resolutions near the breaking zone.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eThe numerical model assumes smooth, impermeable surfaces for the seawalls and flume bed. Effects of surface roughness or permeability, which can influence wave energy dissipation and run-up in real-world scenarios (e.g., Kobayashi et al. (1987)), are not included in this study. This was noted as a potential source of minor error when comparing with some experimental data.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003eDespite these limitations, the comprehensive validation against various benchmarks for sloped geometries provides confidence in the model's capability to capture the essential hydrodynamics of solitary wave run-up, reflection, and overtopping for the comparative analysis of different seawall shapes presented in Section \u003cspan refid=\"Sec19\" class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e\u003c/div\u003e"},{"header":"4. Results and Discussion","content":"\u003cp\u003eThis section presents the numerical simulation results for solitary wave interaction with concave, convex, and various sloped seawall geometries. The analysis focuses on hydrodynamic parameters including wave run-up, reflection, and overtopping, as well as a detailed examination of the internal flow structures and their implications. The results correspond to incident solitary waves with effective \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e ratios of approximately 0.56 and 0.73, as detailed in Section \u003cspan refid=\"Sec8\" class=\"InternalRef\"\u003e2.6\u003c/span\u003e.\u003c/p\u003e\u003cdiv id=\"Sec20\" class=\"Section2\"\u003e\u003ch2\u003e4.1. General Hydrodynamic Behavior and Wave Transformation\u003c/h2\u003e\u003cp\u003eThe interaction of a solitary wave with a coastal structure is a complex hydrodynamic process involving wave shoaling upon approach, impact, run-up on the structure face, reflection of wave energy, and potential overtopping at the crest. The specific geometry of the seawall front face plays a crucial role in dictating the nature and magnitude of these phenomena.\u003c/p\u003e\u003cp\u003eVisual observations from the numerical simulations (representative snapshots shown in subsequent sections) indicate distinct differences in wave transformation for each seawall type. Upon encountering the structures, the solitary waves undergo significant deformation. For concave seawalls, the incident wave tends to be focused by the inward curvature, leading to a rapid increase in wave height near the structure and a strong upward jet during run-up. This often results in a more energetic reflection. For convex seawalls, the outward curvature appears to spread the incident wave energy more laterally (in a 3D scenario, though this is a 2D study) or dissipate it more gradually along the curved face, leading to a seemingly less aggressive run-up and reflection compared to the concave profile. Sloped seawalls allow the wave to propagate up their face, with the run-up length and the nature of wave breaking (if it occurs on the slope) being dependent on the slope angle and incident wave characteristics. Steeper slopes tend to cause more rapid shoaling and potentially earlier breaking or stronger reflection, while gentler slopes allow for more extended run-up and dissipation.\u003c/p\u003e\u003cp\u003eThe interaction process also generates complex internal flow patterns, including vortices and regions of high turbulence, particularly near the toe and crest of the structures, which are significantly influenced by the seawall geometry. These aspects will be explored in detail in the following subsections. Figure\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e provides a conceptual overview of these differing interaction behaviors which are quantitatively analyzed below.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec21\" class=\"Section2\"\u003e\u003ch2\u003e4.2. Wave Run-up (\u003cem\u003eR\u003c/em\u003e) and Reflection Coefficient (\u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e):\u003c/h2\u003e\u003cp\u003eThe maximum vertical wave run-up (\u003cem\u003eR\u003c/em\u003e) on the structure and the wave reflection coefficient (\u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e) are critical parameters for assessing seawall performance. These were quantified for each seawall geometry under the two effective incident solitary wave conditions (effective \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.56 and \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.73). The maximum run-up (\u003cem\u003eR\u003c/em\u003e) was determined by tracking the highest point the water surface reached on the structure's face during the interaction. The reflection coefficient (\u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e) was calculated as the ratio of the reflected wave height (\u003cem\u003eH\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e) to the incident wave height (\u003cem\u003eH\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e, being the effective wave height). The reflected wave height (\u003cem\u003eH\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e) was estimated by analyzing the wave profile propagating away from the structure after the primary interaction, typically using a method like Goda's method for separating incident and reflected waves if sufficient data points are available, or by identifying the maximum amplitude of the primary reflected wave component at a fixed-point seaward of the structure.\u003c/p\u003e\u003cdiv id=\"Sec22\" class=\"Section3\"\u003e\u003ch2\u003e4.2.1. Wave Run-up (\u003cem\u003eR/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e)\u003c/h2\u003e\u003cp\u003eThe normalized maximum wave run-up (\u003cem\u003eR/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e) for the different seawall geometries as a function of the effective incident non-dimensional wave height (\u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e) is presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e. The results indicate a clear dependence of run-up on both the incident wave energy (represented by \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e) and the seawall's front-face geometry.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFor both effective incident wave conditions, the concave seawall consistently exhibited the highest wave run-up values. For instance, at an effective \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.73, the \u003cem\u003eR/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e for the concave wall was approximately 65% higher than that for the 1:1 sloped wall and 66% higher than for the convex wall. This enhanced run-up on the concave profile can be attributed to the focusing effect of the inward curvature, which concentrates the incoming wave energy and directs the flow upwards along the curved surface, leading to a more significant vertical excursion of the water.\u003c/p\u003e\u003cp\u003eThe convex seawall generally showed the lowest or among the lowest run-up values compared to the concave and steeper sloped walls. The outward curvature of the convex profile tends to radially disperse the incoming wave energy and promote a smoother, more gradual flow up the face, which can limit the maximum vertical reach of the water.\u003c/p\u003e\u003cp\u003eFor the sloped seawalls, run-up was, as expected, influenced by the slope angle. The steepest slope (1:0.58, 60\u0026deg;) resulted in run-up values that were generally lower than the concave wall but higher than the gentlest slope (1:1.73, 30\u0026deg;) for a given incident wave. The gentler slopes provide a longer path for wave energy dissipation through friction (though friction is minimal in these smooth-wall simulations) and breaking on the slope (if applicable), which can reduce the maximum vertical run-up height.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec23\" class=\"Section3\"\u003e\u003ch2\u003e4.2.2. Wave Reflection Coefficient (\u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e)\u003c/h2\u003e\u003cp\u003eThe wave reflection coefficient (\u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e) for the different seawall geometries is presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e as a function of the effective \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe results show that the concave seawall also produced the highest reflection coefficients among all tested geometries for both incident wave conditions. At an effective \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.73, the \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e for the concave wall was approximately 225% higher than for the 1:1 sloped wall and 257% higher than for the convex wall. The inward-curving face of the concave structure acts to redirect a significant portion of the incident wave energy back towards the sea, minimizing energy transmission and dissipation on the structure itself, thus leading to higher reflection. This is consistent with its higher run-up, as more energy directed upwards and back (rather than dissipated or transmitted) contributes to both phenomena.\u003c/p\u003e\u003cp\u003eConversely, the convex seawall generally yielded the lowest reflection coefficients. The outward curvature likely promotes smoother flow deflection and some energy dissipation along its surface, reducing the amount of energy directly reflected. The sloped seawalls exhibited intermediate reflection characteristics, with steeper slopes (e.g., 60\u0026deg;) reflecting more energy than gentler slopes (e.g., 30\u0026deg;). This is expected, as gentler slopes allow for more gradual wave energy dissipation through processes like shoaling and breaking on the slope, thereby reducing reflection.\u003c/p\u003e\u003cp\u003eThe observed higher run-up and higher reflection for the concave wall are dynamically linked; the geometry that most effectively redirects incident wave momentum upwards and seawards will naturally score high on both metrics, assuming minimal energy dissipation through other means like breaking directly on the structure face or transmission. The subsequent analysis of internal flow structures (Section \u003cspan refid=\"Sec25\" class=\"InternalRef\"\u003e4.4\u003c/span\u003e) will provide further insight into the mechanisms governing these differences.\u003c/p\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Sec24\" class=\"Section2\"\u003e\u003ch2\u003e4.3. Wave Overtopping Discharge (\u003cem\u003eq\u003c/em\u003e)\u003c/h2\u003e\u003cp\u003eWave overtopping, where water passes over the crest of a coastal structure, is a critical design consideration as it directly relates to potential flooding and damage to landward infrastructure. The overtopping discharge per unit length (\u003cem\u003eq\u003c/em\u003e) was quantified for each seawall geometry under the two effective incident solitary wave conditions (\u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.56 and \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.73). This was achieved by utilizing a feature within the FLOW-3D\u003csup\u003e\u0026reg;\u003c/sup\u003e software where a virtual flux surface was defined at the crest elevation of each seawall. The software then directly measured the net volumetric flow rate of water passing through this defined surface over time, providing the instantaneous overtopping discharge rate (\u003cem\u003eq\u003c/em\u003e, in m\u0026sup2;/s, effectively m\u0026sup3;/s per meter width).\u003c/p\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan refid=\"Fig14\" class=\"InternalRef\"\u003e14\u003c/span\u003e present the time histories of the overtopping discharge rate (\u003cem\u003eq\u003c/em\u003e) for the concave, convex, and various sloped seawalls under the effective incident \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.56 and \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.73 conditions, respectively.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe hydrographs in Figs.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e and \u003cspan refid=\"Fig14\" class=\"InternalRef\"\u003e14\u003c/span\u003e illustrate both the peak discharge rate and the duration of the overtopping event for each seawall type. For a more direct comparison, the maximum overtopping discharge rates (\u003cem\u003eq\u003c/em\u003e\u003csub\u003e\u003cem\u003emax\u003c/em\u003e\u003c/sub\u003e) and the total overtopped volume per unit width (\u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003etotal\u003c/em\u003e\u003c/sub\u003e), calculated from the integration of the hydrographs from the new dataset, are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eSummary of maximum overtopping discharge rate (\u003cem\u003eq\u003c/em\u003e\u003csub\u003e\u003cem\u003emax\u003c/em\u003e\u003c/sub\u003e) and total overtopped volume per unit width (\u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003etotal\u003c/em\u003e\u003c/sub\u003e) for different seawall geometries and effective incident wave conditions\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"4\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSeawall Geometry\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eEffective Incident \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eMaximum Overtopping Discharge Rate, \u003cem\u003eq\u003c/em\u003e\u003csub\u003e\u003cem\u003emax\u003c/em\u003e\u003c/sub\u003e (m\u0026sup2;/s)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eTotal Overtopped Volume per unit width, \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003etotal\u003c/em\u003e\u003c/sub\u003e (m\u0026sup3;/m)\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e\u003cp\u003eConcave\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e~\u0026thinsp;0.56 (Target 0.6)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.0073\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.0346\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e~\u0026thinsp;0.73 (Target 0.8)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.0117\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.0343\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e\u003cp\u003eConvex\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e~\u0026thinsp;0.56 (Target 0.6)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.0124\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.0705\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e~\u0026thinsp;0.73 (Target 0.8)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.0199\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.0709\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e\u003cp\u003eSloped (1:1.73 or 30\u0026deg;)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e~\u0026thinsp;0.56 (Target 0.6)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.0099\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.0798\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e~\u0026thinsp;0.73 (Target 0.8)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.018\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.1028\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e\u003cp\u003eSloped (1:1 or 45\u0026deg;)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e~\u0026thinsp;0.56 (Target 0.6)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.0097\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.0787\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e~\u0026thinsp;0.73 (Target 0.8)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.0176\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.1067\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e\u003cp\u003eSloped (1:0.58 or 60\u0026deg;)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e~\u0026thinsp;0.56 (Target 0.6)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.0093\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.0751\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e~\u0026thinsp;0.73 (Target 0.8)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.0168\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.1016\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eSeveral key observations can be made from these results:\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003eInfluence of Incident Wave Height: As expected, the overtopping discharge (both \u003cem\u003eq\u003c/em\u003e\u003csub\u003e\u003cem\u003emax\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003etotal\u003c/em\u003e\u003c/sub\u003e) significantly increases for all seawall types when the incident wave energy is higher (i.e., for effective \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.73 compared to \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.56). This is evident from the larger peak values and areas under the hydrographs in Fig.\u0026nbsp;\u003cspan refid=\"Fig14\" class=\"InternalRef\"\u003e14\u003c/span\u003e compared to Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e, and quantified in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003ePerformance of Sloped Seawalls: Among the sloped seawalls, the gentlest slope (1:1.73 or 30\u0026deg;) generally experienced the highest overtopping discharge (e.g., for effective \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.73, \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003etotal\u003c/em\u003e\u003c/sub\u003e for the 30\u0026deg; slope was 0.1028 m\u0026sup3;/m). This is likely because the gentler slope allows the wave to run further up the face and maintain more of its coherent bulk as it approaches the crest. Conversely, the steepest slope (1:0.58 or 60\u0026deg;) tended to have slightly lower overtopping compared to the 30\u0026deg; slope for the higher wave condition (for effective \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.73, \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003etotal\u003c/em\u003e\u003c/sub\u003e for the 60\u0026deg; slope was 0.1016 m\u0026sup3;/m), possibly due to increased wave reflection and more significant disruption of the wave front by the abrupt change in geometry.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003ePerformance of Convex Seawalls: The convex seawall exhibited intermediate overtopping characteristics. For effective \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.73, the \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003etotal\u003c/em\u003e\u003c/sub\u003e was 0.0709 m\u0026sup3;/m. Its outward curvature might deflect some of the up-rushing water, but it does not appear to offer the same level of overtopping reduction as the concave profile under these solitary wave conditions.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003ePerformance of Concave Seawalls: A significant finding, consistent across both wave conditions in the new dataset, is that despite exhibiting the highest wave run-up (as discussed in Section \u003cspan refid=\"Sec22\" class=\"InternalRef\"\u003e4.2.1\u003c/span\u003e), the concave seawall consistently resulted in the lowest overtopping discharges. For example, at an effective \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.73, the total overtopped volume for the concave wall was 0.0343 m\u0026sup3;/m, which was approximately 66.6% less than that for the 1:1.73 sloped wall (0.1028 m\u0026sup3;/m). This phenomenon, where high run-up does not directly translate to high overtopping for this specific geometry, suggests a distinct hydrodynamic mechanism at the crest. The inward curvature of the concave wall appears to direct a significant portion of the up-rushing water jet more vertically upwards. This upward momentum, combined with the re-directing effect of the sharp crest, likely causes a substantial part of the water to be thrown upwards and then fall back towards the seaside, rather than efficiently passing over the crest. This \"jet deflection and return flow\" mechanism, enhanced by the concave geometry, effectively reduces the net volume of water that overtops the structure. This explanation will be further supported by the analysis of flow velocity vectors near the crest in Section \u003cspan refid=\"Sec25\" class=\"InternalRef\"\u003e4.4\u003c/span\u003e.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003eThe specific shape of the overtopping hydrograph (Figs.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e and \u003cspan refid=\"Fig14\" class=\"InternalRef\"\u003e14\u003c/span\u003e) also varies with seawall geometry. Sloped walls, particularly gentler ones, tend to show a broader hydrograph with a longer duration of overtopping, indicating a more prolonged but potentially less intense overtopping event. In contrast, the concave and convex walls appear to produce sharper, more concentrated overtopping pulses, though the magnitude for the concave wall is notably lower. These differences in overtopping behavior have significant implications for the design of crest and landward protection measures, as both the peak rate and total volume of overtopping are critical.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec25\" class=\"Section2\"\u003e\u003ch2\u003e4.4. Wave-Induced Forces and Pressures\u003c/h2\u003e\u003cp\u003eBeyond the hydrodynamic performance, understanding the wave-induced loads is essential for the structural stability and design of seawalls. This section presents a comparative analysis of the horizontal and vertical forces, as well as the pressure distribution, exerted by the solitary waves on the three representative seawall geometries: concave, convex, and sloped 45\u0026deg;.\u003c/p\u003e\u003cdiv id=\"Sec26\" class=\"Section3\"\u003e\u003ch2\u003e4.4.1. Time History of Wave Forces\u003c/h2\u003e\u003cp\u003eThe time series of the total horizontal force (\u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003eh\u003c/em\u003e\u003c/sub\u003e) and total vertical force (\u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003ev\u003c/em\u003e\u003c/sub\u003e) acting on the wetted surface of each representative seawall are presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig15\" class=\"InternalRef\"\u003e15\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan refid=\"Fig16\" class=\"InternalRef\"\u003e16\u003c/span\u003e, respectively, for the higher energy wave condition (effective \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.73).\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe horizontal force history in Fig.\u0026nbsp;\u003cspan refid=\"Fig15\" class=\"InternalRef\"\u003e15\u003c/span\u003e shows significant differences in loading characteristics. The concave seawall experiences the highest peak horizontal force (\u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003eh\u003c/em\u003e\u003c/sub\u003e,\u003csub\u003emax\u003c/sub\u003e), which occurs as a sharp, high-intensity impact over a very short duration. This is characteristic of a \"slamming\" load. In contrast, the sloped seawall exhibits a much slower build-up of force and a lower, broader peak, indicating a more quasi-static or \"surging\" type of loading. The convex wall shows an intermediate behavior. The vertical force history in Fig.\u0026nbsp;\u003cspan refid=\"Fig16\" class=\"InternalRef\"\u003e16\u003c/span\u003e primarily represents the uplift force on the structure. Again, the concave wall is subjected to the highest peak uplift force (most negative \u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003ev\u003c/em\u003e\u003c/sub\u003e). A summary of the peak force values for the representative geometries and both wave conditions is provided in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eSummary of maximum horizontal (\u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003eh,max\u003c/em\u003e\u003c/sub\u003e) and maximum vertical uplift (\u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003ev,max\u003c/em\u003e\u003c/sub\u003e) forces for the representative seawall geometries\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"4\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSeawall Geometry\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eEffective Incident \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eMaximum Horizontal Force, \u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003eh,max\u003c/em\u003e\u003c/sub\u003e (kN/m)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eMaximum Horizontal Force, \u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003ev,max\u003c/em\u003e\u003c/sub\u003e (kN/m)\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e\u003cp\u003eConcave\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e~\u0026thinsp;0.56 (Target 0.6)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e1862.71\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e-740.84\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e~\u0026thinsp;0.73 (Target 0.8)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e2519.25\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e-994.10\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e\u003cp\u003eConvex\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e~\u0026thinsp;0.56 (Target 0.6)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e1849.83\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e-278.04\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e~\u0026thinsp;0.73 (Target 0.8)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e2446.85\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e-504.66\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e\u003cp\u003eSloped (45\u0026deg;)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e~\u0026thinsp;0.56 (Target 0.6)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e1830.50\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e-509.96\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e~\u0026thinsp;0.73 (Target 0.8)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e2348.65\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e-726.22\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec27\" class=\"Section3\"\u003e\u003ch2\u003e4.4.2. Pressure Distribution at Peak Impact\u003c/h2\u003e\u003cp\u003eTo understand the reason behind the differences in peak forces, the distribution of dynamic pressure along the face of the three representative seawalls was analyzed at the exact moment of maximum horizontal force for the \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.73 wave condition. The pressure values were extracted from the numerical probes whose locations are defined in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe pressure profiles in Fig.\u0026nbsp;\u003cspan refid=\"Fig17\" class=\"InternalRef\"\u003e17\u003c/span\u003e reveal the underlying loading mechanism for each geometry. The curve for the concave seawall exhibits a highly non-linear pressure distribution. Unlike a typical hydrostatic profile, it shows a significant pressure concentration on its upper curved section (around \u003cem\u003eY\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.5 m). This high pressure is caused by the direct impact of the focused, up-rushing water jet, which quantitatively explains the sharp, high-magnitude \"slamming\" force observed for this geometry in Fig.\u0026nbsp;\u003cspan refid=\"Fig15\" class=\"InternalRef\"\u003e15\u003c/span\u003e. In contrast, the curve for the sloped seawall shows a much more linear, quasi-hydrostatic pressure distribution, with pressure being highest at the toe and decreasing steadily with elevation. The convex wall shows an intermediate distribution, with pressure values at all points being generally lower than those of the concave profile. This analysis confirms that the concave geometry, while effective for other purposes, must be designed to withstand significantly higher localized impact pressures on its upper face. This focused loading is the primary reason for the larger total horizontal force experienced by the concave structure compared to conventional sloped designs.\u003c/p\u003e\u003cp\u003eIt is noteworthy that the peak pressures at individual locations are not simultaneous, reflecting the upward propagation of the pressure wave along with the water jet. For instance, while the pressure at point P4 for concave and convex face was zero at the moment of maximum total force, it later reached its absolute maximum of 22.51 and 1.32 kPa at \u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;4.3 \u003cem\u003es\u003c/em\u003e respectively as the wave crest passed its elevation. The pressure distribution shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig17\" class=\"InternalRef\"\u003e17\u003c/span\u003e, however, represents the simultaneous loading across the structure at the single most critical instant for overall horizontal stability.\u003c/p\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Sec28\" class=\"Section2\"\u003e\u003ch2\u003e4.5. Analysis of Flow Structures, TKE, and Scour Potential\u003c/h2\u003e\u003cp\u003eTo establish a complete physical understanding of the seawalls' performance, this section links the macroscopic findings (run-up, overtopping, forces) to the underlying hydrodynamics. The analysis focuses on the evolution of Turbulent Kinetic Energy (TKE) and the detailed kinematic flow fields for the three representative geometries under the higher energy wave condition (effective \u003cem\u003eH/h\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u0026asymp;\u0026thinsp;0.73).\u003c/p\u003e\u003cdiv id=\"Sec29\" class=\"Section3\"\u003e\u003ch2\u003e4.5.1. Turbulent Kinetic Energy (TKE) Evolution at the Seawall Toe\u003c/h2\u003e\u003cp\u003eTurbulent Kinetic Energy (TKE, \u003cem\u003ek\u003c/em\u003e) provides a direct quantitative measure of the turbulence intensity. The time history of TKE was analyzed at the numerical probe located nearest to the structure's toe (Probe P1, locations shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e), a critical area for scour development. Figure\u0026nbsp;\u003cspan refid=\"Fig18\" class=\"InternalRef\"\u003e18\u003c/span\u003e compares the TKE time series at this location for the three representative geometries.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe results in Fig.\u0026nbsp;\u003cspan refid=\"Fig18\" class=\"InternalRef\"\u003e18\u003c/span\u003e reveal that while the peak TKE values are of a similar order of magnitude for all three geometries, the temporal evolution and duration of the turbulence differ significantly. The peak TKE values for the concave, convex, and sloped walls are approximately 0.00067 m\u0026sup2;/s\u0026sup2;, 0.00063 m\u0026sup2;/s\u0026sup2;, and 0.00063 m\u0026sup2;/s\u0026sup2;, respectively.\u003c/p\u003e\u003cp\u003eHowever, the crucial difference is that the TKE for the concave seawall, after reaching its peak, remains significantly elevated for a longer duration during the wave drawdown phase compared to the other two profiles. The turbulence generated by the convex and sloped walls, while momentarily intense, decays much more rapidly after the initial impact. This period of sustained, high-intensity turbulence at the toe of the concave wall corresponds directly to the formation and persistence of the large, organized vortex at its base (which will be shown in Section \u003cspan refid=\"Sec30\" class=\"InternalRef\"\u003e4.5.2\u003c/span\u003e). Therefore, it is this prolonged exposure to high turbulence, rather than the peak value alone, that is the key indicator of a higher and more significant scour potential for the concave seawall.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec30\" class=\"Section3\"\u003e\u003ch2\u003e4.5.2. Detailed Flow Kinematics and Governing Mechanisms\u003c/h2\u003e\u003cp\u003eThe detailed velocity fields reveal the physical mechanisms responsible for the observed results. At the moment of maximum run-up, Fig.\u0026nbsp;\u003cspan refid=\"Fig18\" class=\"InternalRef\"\u003e18\u003c/span\u003e shows that the concave seawall creates a powerful upward jet with vertical velocities reaching 2.27 m/s, explaining its low overtopping.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eDuring the drawdown phase, Fig.\u0026nbsp;\u003cspan refid=\"Fig20\" class=\"InternalRef\"\u003e20\u003c/span\u003e shows the formation of a large, coherent vortex at the toe of the concave seawall. The X-velocity contours quantify this structure, revealing strong offshore flow near the bed (min Vx\u0026thinsp;=\u0026thinsp;\u0026minus;\u0026thinsp;1.36 m/s) beneath a region of onshore flow (max Vx\u0026thinsp;=\u0026thinsp;0.30 m/s). This stable vortex, not present in the other cases, is the mechanism that drives the sustained turbulence shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig18\" class=\"InternalRef\"\u003e18\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec31\" class=\"Section3\"\u003e\u003ch2\u003e4.5.3. Synthesis: Turbulent Viscosity and Scour Potential\u003c/h2\u003e\u003cp\u003eFinally, Fig.\u0026nbsp;\u003cspan refid=\"Fig21\" class=\"InternalRef\"\u003e21\u003c/span\u003e links the vortex structure to scour potential using dynamic viscosity contours. For the concave seawall (Fig.\u0026nbsp;\u003cspan refid=\"Fig21\" class=\"InternalRef\"\u003e21\u003c/span\u003ea), a region of elevated dynamic viscosity is observed to be perfectly co-located with the energetic vortex from Fig.\u0026nbsp;\u003cspan refid=\"Fig20\" class=\"InternalRef\"\u003e20\u003c/span\u003ea. This direct correlation demonstrates that the vortex generates intense, sustained turbulence at the foundation, confirming a high potential for local scour. The other geometries show more benign conditions. This comprehensive analysis highlights the critical engineering trade-off in the concave seawall design: its superior overtopping performance is achieved at the cost of creating aggressive hydrodynamic conditions at its foundation.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003c/div\u003e"},{"header":"5. Conclusion","content":"\u003cp\u003eThis study presented a systematic numerical investigation into the hydrodynamic interaction of solitary waves with coastal seawalls of concave, convex, and sloped geometries. Using a validated RANS-VOF model, the research provided a comprehensive comparative analysis of hydrodynamic performance (run-up, reflection, overtopping), wave-induced loads (forces, pressures), and local turbulence characteristics (TKE, flow structures). The primary findings are summarized as follows:\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003eConcave Seawall Performance and Trade-offs: The concave geometry demonstrated a distinct and complex performance profile. It generated the highest wave run-up and reflection coefficients, as well as the largest, sharpest impact forces, both horizontally and vertically. The high forces were explained by a significant pressure concentration on the upper curved part of the structure due to a focused, up-rushing water jet. Paradoxically, this same jet, with vertical velocities reaching up to 2.27 m/s, was responsible for the concave wall exhibiting the lowest overtopping discharge, as it effectively deflected the flow upwards and back towards the sea.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eVortex Generation and Scour Potential: A critical finding was the formation of a large, stable, and energetic vortex at the toe of the concave seawall during the wave drawdown phase. This vortex was quantitatively identified through velocity fields and was shown to be the source of sustained, high-intensity Turbulent Kinetic Energy (TKE) and a co-located region of high dynamic viscosity. This indicates that the concave profile, while superior in overtopping protection, presents a significant potential for local scour at its foundation.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eConvex and Sloped Seawall Performance: The convex and sloped seawalls exhibited more conventional and moderate behavior. They were subjected to lower wave loads and generated significantly less organized turbulence at their toe compared to the concave wall, suggesting a lower scour risk. However, they were also less effective at reducing wave overtopping.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eThese findings have direct engineering implications. The concave seawall design presents an interesting trade-off: it offers superior performance in terms of overtopping reduction, which is a primary goal for coastal protection, but its tendency to generate aggressive local hydrodynamics at its base may necessitate costly toe protection measures (e.g., scour aprons or riprap). This must be considered in the overall lifecycle cost and design optimization of such structures.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003eThis study reveals a critical engineering trade-off inherent in the concave seawall's design: its excellent hydraulic performance in minimizing overtopping comes at the cost of higher structural loads and a significant risk of foundation instability due to scour. These factors must be carefully weighed in any practical application, likely requiring a more robust structural design and costly toe protection measures to be viable.\u003c/p\u003e\u003cp\u003eThe limitations of this study include its two-dimensional nature, the assumption of smooth, impermeable boundaries, a focus on non-breaking solitary waves, and the lack of direct experimental validation for the curved geometries. Future research should be directed towards 3D simulations to investigate lateral effects, physical modeling to validate the complex flow structures and forces observed, analysis of different wave types such as breaking and irregular waves, and the design of effective scour protection for concave seawall profiles.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cb\u003eDeclarations of generative AI and AI-assisted technologies in the writing process\u003c/b\u003e\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eAuthor Contributions StatementS.M.G. contributed to all aspects of the study, including Conceptualization, Data curation, Formal Analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing \u0026ndash; original draft, and Writing \u0026ndash; review \u0026amp; editing.F.S.B., A.M., R.S.B., A.G., and E.T.A. contributed to Writing \u0026ndash; review \u0026amp; editing.All authors reviewed the manuscript.\u003c/p\u003e\u003ch2\u003eAcknowledgement\u003c/h2\u003e\u003cp\u003eI gratefully acknowledge the late Professor Mohammad Javad Ketabdari of the Department of Maritime Engineering, Amirkabir University of Technology, whose collaborative insight and guidance laid the foundation for this research. Tragically, Dr. Ketabdari passed away from a heart attack on Monday, November 14, 2022. I honor his memory and dedication to the advancement of coastal engineering.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eSynolakis, C. E. (1987). The run-up of solitary waves. Journal of Fluid Mechanics, 185, 523-545. https://doi.org/10.1017/S002211208700329X\u003c/li\u003e\n\u003cli\u003eHall, J. V., \u0026amp; Watts, G. M. (1953). Laboratory investigation of the vertical rise of solitary waves on impermeable slopes (Technical Memorandum No. 33). Beach Erosion Board, U.S. Army Corps of Engineers.\u003c/li\u003e\n\u003cli\u003eVinodh, T. L. C., \u0026amp; Tanaka, N. (2020). A unified runup formula for solitary waves on a plane beach. Ocean Engineering, 216, 108038. https://doi.org/10.1016/j.oceaneng.2020.108038\u003c/li\u003e\n\u003cli\u003eAnada, K. V., Sundar, V., \u0026amp; Sannasiraj, S. A. (2011). 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Wave reflection and run-up on rough slopes. \u003cem\u003eJournal of Waterway, Port, Coastal, and Ocean Engineering\u003c/em\u003e, \u003cem\u003e113\u003c/em\u003e(3), 282-298. https://doi.org/10.1061/(ASCE)0733-950X(1987)113:3(282)\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"water-waves","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"wawa","sideBox":"Learn more about [Water Waves](https://link.springer.com/journal/42286)","snPcode":"42286","submissionUrl":"https://submission.nature.com/new-submission/42286/3","title":"Water Waves","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"solitary wave, seawall geometry, wave-structure interaction, wave loads, turbulent flow, numerical modeling","lastPublishedDoi":"10.21203/rs.3.rs-7445438/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7445438/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eSeawalls are critical for coastal protection, yet a comprehensive understanding of how their geometry affects hydrodynamic loads, performance, and local turbulence remains incomplete. While previous studies have investigated solitary wave forces on conventional vertical or sloped structures, a systematic comparative analysis of concave, convex, and sloped seawalls one that holistically links hydrodynamic performance (run-up, reflection), wave-induced loads, and the resulting turbulent flow structures has been a notable gap in the literature. This study addresses this gap by conducting a detailed numerical investigation using a validated RANS model coupled with a k-ε RNG turbulence scheme and the Volume of Fluid (VOF) method. We analyze the interaction of highly nonlinear solitary waves with these distinct geometries. The results demonstrate that seawall curvature is a critical design parameter. Concave seawalls significantly increase wave reflection and generate concentrated, high-energy vortices at the structure's toe, leading to amplified wave loads and a higher potential for local scour. In contrast, convex profiles promote smoother flow separation, resulting in reduced wave forces and diminished turbulence near the bed. Sloped seawalls are effective at dissipating energy along the structure's face, thereby minimizing loads but increasing the likelihood of significant overtopping. By providing an integrated analysis of pressure distribution, impact forces, and Turbulent Kinetic Energy (TKE), this research offers crucial insights for the optimal design of coastal defenses, enabling engineers to balance structural stability, hydrodynamic efficiency, and scour mitigation.\u003c/p\u003e","manuscriptTitle":"A Comparative Numerical Study of Solitary Wave Interaction with Concave, Convex, and Sloped Seawalls: Hydrodynamics, Wave Loads, and Turbulent Flow Analysis","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-09-19 10:42:24","doi":"10.21203/rs.3.rs-7445438/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-10-13T08:31:13+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-10-11T06:30:55+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"72114572012146618005974437286182855937","date":"2025-10-08T01:25:50+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-10-07T11:36:30+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"13640095943478360467276880062872992621","date":"2025-10-01T09:05:35+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-09-12T09:19:07+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-09-05T16:42:33+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-08-25T15:16:51+00:00","index":"","fulltext":""},{"type":"submitted","content":"Water Waves","date":"2025-08-24T09:49:39+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"water-waves","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"wawa","sideBox":"Learn more about [Water Waves](https://link.springer.com/journal/42286)","snPcode":"42286","submissionUrl":"https://submission.nature.com/new-submission/42286/3","title":"Water Waves","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"92cb739e-45d7-4185-bf47-58cb2b5c03a3","owner":[],"postedDate":"September 19th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2026-02-09T16:08:02+00:00","versionOfRecord":{"articleIdentity":"rs-7445438","link":"https://doi.org/10.1007/s42286-026-00131-2","journal":{"identity":"water-waves","isVorOnly":false,"title":"Water Waves"},"publishedOn":"2026-02-03 15:59:17","publishedOnDateReadable":"February 3rd, 2026"},"versionCreatedAt":"2025-09-19 10:42:24","video":"","vorDoi":"10.1007/s42286-026-00131-2","vorDoiUrl":"https://doi.org/10.1007/s42286-026-00131-2","workflowStages":[]},"version":"v1","identity":"rs-7445438","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7445438","identity":"rs-7445438","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}