Wave run-up characteristic on armour layer breakwater using the smoothed particle hydrodynamics (SPH) method

Coastal structures, such as breakwater, play a critical role in protecting shorelines from the erosive forces of waves and currents. Understanding the wave run-up phenomenon on these structures is essential for designing effective coastal defense systems. This research aims to investigate the wave run-up characteristics on an armour layer breakwater using the Smoothed Particle Hydrodynamics (SPH) method (Meshless particle method based on the Lagrangian formulation), a numerical method that captures complex fluid-structure interactions. The SPH method’s inherent ability to handle free-surface flows and dynamic interactions makes it particularly suited for simulating wave run-up. The research employs numerical testing and modelling of a non-overtopping breakwater using the smoothed particle hydrodynamics method simulated with the DualSPHysics program. The wave characteristic data used in this research are the waves on the south coast of Java. The results of this research indicate that the correlation between wave run-up characteristics (Ru/H) and the Iribarren number, both using the numerical SPH method and theory, tends to exhibit a similar pattern. Remarkably, the results from both approaches exhibit a striking similarity, revealing that as the Iribarren number increases, the incremental growth of Ru/H diminishes, eventually stabilizing at a critical threshold. The Coefficient of Determination (R2) between wave run-up characteristics (Ru/H) obtained from numerical simulations using the SPH method and Günbak’s theory is 0.8824.


Introduction
Coastal engineering plays a vital role in safeguarding coastal communities and infrastructure from the relentless forces of the ocean.One critical aspect of coastal engineering involves designing and evaluating protective structures, like breakwaters, that are intended to mitigate the impact of waves on shorelines.A breakwater is a structure designed to attenuate waves, creating a calm basin for the docking of ships during the loading and unloading goods and passengers.The breakwater absorbs a portion of wave energy, leading to a tranquil water area beyond the breakwater.It also serves to shield the docking basin from sediment deposits originating from river mouths around the harbor area.Breakwaters are frequently constructed for coastal protection purposes against erosion [1].The breakwater structure faces numerous wave attacks that endanger its stability.Wave attacks result in forces on the breakwater due to phenomena such as breaking, reflection, refraction, and the formation of rip currents.Breakwaters operate by dissipating wave energy.Consequently, the forces generated by wave attacks significantly impact the stability of the protective armour layer on the breakwater.Understanding the wave characteristics on the breakwater is essential as these phenomena affect coastal structures [2].
The stability of the breakwater structure, especially concerning the armour layer, is significantly influenced by wave characteristics.This study will utilize numerical methods in mathematical modeling to understand the wave characteristics on the armour layer of a non-overtopping breakwater.The numerical method employed in this research is the Smoothed Particle Hydrodynamics (SPH) method.Smoothed Particle Hydrodynamics (SPH) is a lagrangian meshless method that has been applied in various Computational Fluid Dynamics (CFD) applications, where fluid flow is represented by particles.This method can simulate how fluid flow interacts with structures and can also simulate various other cases in the fields of hydraulics and coastal engineering [3].
The program used to conduct the numerical approach with the SPH method is DualSPHysics.DualSPHysics is an SPH-based modeling code that is efficient and user-friendly for various applications in the fields of hydraulics and coastal engineering.This program was developed to study free-surface flow phenomena, in situations where Eulerian methods might be challenging to apply, such as wave impact on offshore structures [4].

Run-up and Iribarren Number
Wave run-up is the vertical distance reached by water above the calm water surface due to wave uprush on the beach or coastal structures.It is a critical design criterion for various types of coastal structures such as seawalls, breakwaters, and embankments.Wave run-up is influenced by several factors, including wave height, wave period, beach slope, and the presence of coastal structures.Wave run-up can result in significant damage to coastal structures, leading to erosion, flooding, and structural failures.Therefore, accurately predicting wave run-up is essential in the design and construction of coastal structures.Various methods, including empirical equations, numerical models, and physical model tests, can be employed to estimate wave run-up [5].Below are the equations used to calculate wave run-up.

Smoothed Particle Hydrodynamics (SPH) Formulation
According to Gomez-Gesteira et al. [6], Smoothed Particle Hydrodynamics (SPH) is a lagrangian meshless method that has been employed in various applications in the field of Computational Fluid Dynamics (CFD), where particles represent the flow, interact with structures, and exhibit significant deformations with moving boundaries.The SPH model has matured to a stage in CFD with continuous improvements and modifications, leading to an acceptable level of accuracy, stability, and model reliability for practical engineering applications.
In SPH, the fluid domain is discretized as a collection of computed fluid particles.This method has been employed to simulate various applications, including astrophysics, free-surface flows, and complex mixing problems in fluid dynamics.The removal of a fixed mesh in the SPH algorithm provides computational advantages compared to conventional methodologies [7].This method employs a collection of particles to represent continuous substances.When utilized to simulate fluid dynamics, the Navier-Stokes equations are integrated at each particle location based on the properties of neighboring particles.The set of neighboring particles is determined by a distance-based function, either a circle (2D) or a sphere (3D), with a characteristic length or smoothing length (often denoted as h).At each time step, new physical quantities are computed for each particle and they move based on the updated values [8].
Figure 1 The difference between mesh and particle discretization of continuous free-surface flow

DualSPHysics
DualSPHysics is an open-source code developed to solve problems related to free-surface flows.It is an SPH code accelerated by hardware that can be utilized to compute the movement of particles within a modeling domain.DualSPHysics is a solver designed to address real-life engineering problems.It is based on the SPHysics model and comprises a collection of C++ and CUDA codes.The SPH implementation in DualSPHysics focuses on three main consecutive tasks: Neighbour List (NL), Particle Interaction (PI), and System Update (SU) [8].When performing numerical simulations using the DualSPHysics program, there are several steps that need to be followed to obtain the output of the numerical simulation results.The workflow of the DualSPHysics program follows the diagram depicted in Figure 2.

Data analysis
In this study, there are two parameters that will be varied in the research modeling.These parameters are wave height (H) and wave period (T), both of which will be combined to create several variations in this study.Table 1 shows are the modeling variations data for this research.The breakwater to be used in this study is a non-overtopping breakwater with a tetrapod-type armor layer.The tetrapods used as the armor layer in this study have a unit weight of 6 tons.The wave characteristics utilized in this research correspond to the wave characteristics of the South Java and Bali coasts, with a high-water spring of 2.6 meters.Figure 3 illustrates a cross-section of the breakwater employed in this study.

Wave Run-up Based on Günbak's Theory.
The theory to be used for analyzing the wave run-up height occurring on the non-overtopping breakwater structure is Günbak's theory (Equation 1).The equation proposed by Günbak is more relevant and suitable for the conducted research.The results of this analysis will serve as the foundation to validate the wave run-up based on the Smoothed Particle Hydrodynamics (SPH) method.Here is an example calculation of wave run-up for Variation 16, beginning with the computation of wavelength (L) using Equation 3 and the Iribarren number using Equation 2. In Table 2, the calculated wave run-up results for each variation in this study against the non-overtopping breakwater structure will be summarized.

Wave Run-up Based on SPH method.
This study will utilize numerical modeling using the Smoothed Particle Hydrodynamics (SPH) method, simulated through the DualSPHysics program.The following is an example of a numerical simulation result demonstrating the wave run-up height (Variation 16).The wave run-up height is determined based on the coordinates of the highest fluid particle when the wave arrives and interacts with the breakwater structure.The coordinates in Figure 4 depict the positions of particles along the x, y, and z axes.Therefore, following the wave run-up definition, the run-up height will be reduced by the water depth.For Variation 16, the coordinate of the highest fluid particle along the z-axis is 12.75 meters.Hence, the wave run-up height for Variation 16 is calculated as follows: Ru = 12.75m -8.86m = 3.89 m.

Validation Based on Previous Theory.
In this section, the results of numerical simulations using the Smoothed Particle Hydrodynamics (SPH) method with the DualSPHysics program will be validated against previous research.In Figure 5, the wave run-up (Ru/H) obtained from the numerical simulations using the SPH method will be plotted on a Ru/H graph against the Iribarren number, based on previous studies.In this graph, the green plot, or the plot for the tetrapod armor layer studied by Jackson, will serve as a reference for validating the wave run-up results in this research.

Figure 5 Wave run-up graph against Iribarren number
In Figure 5, it can be observed that the numerical simulation results display several data points that align with Jackson's presented graph, while some have higher Ru/H values than the reference plot.However, the data points with higher Ru/H values tend to level off at a certain point.This pattern reflects a similar trend as the graph presented by Jackson when studying this phenomenon for the tetrapod armor layer type.Therefore, it can be stated that the results of this research are consistent with previous studies.Consequently, the conducted numerical method simulation is valid and can be used to assess the stability of the armor layer on the non-overtopping breakwater.Figure 6 illustrates the relationship between wave run-up characteristics (Ru/H) and the Iribarren number.

Figure 6
The relationship between wave run-up characteristics (Ru/H) and the Iribarren number  In Figure 6, it can be observed that the relationship between wave run-up characteristics (Ru/H) and the Iribarren number, both with the numerical SPH method and the theory, tends to exhibit a similar pattern.Through Figure 6, it becomes apparent that as the Iribarren number increases, the incremental growth of Ru/H becomes smaller, and it even tends to level off at a certain point.The results of numerical simulations using the Smoothed Particle Hydrodynamics (SPH) method will also be validated against the prior calculations from Günbak's theory.Figure 7 illustrates the relationship between the numerical simulation results using the SPH method and the theoretical calculations from Günbak.

Figure 7 Relationship between Numerical Simulation Results (SPH) and Theory
In Figure 7, the relationship between wave run-up characteristics (Ru/H) based on the results of numerical simulations using the Smoothed Particle Hydrodynamics (SPH) method and the theory from Günbak can be observed.In Figure 7, the coefficient of determination, R2 = 0.8824, is visible, where a value closer to 1 indicates similarity between the two variables.The coefficient of determination value in the graph is quite high, approaching 1, which means that the two variables, numerical simulation (SPH) and Günbak's theory, tend to yield similar results.

Conclusion
Based on the results and discussions of the stability analysis of the armor layer on the non-overtopping breakwater using the Smoothed Particle Hydrodynamics (SPH) method with the DualSPHysics program, the following conclusions can be drawn: a.The relationship between wave run-up characteristics (Ru/H) and the Iribarren number, both through the numerical SPH method and the theory, tends to exhibit a similar pattern.As the Iribarren number increases, the increase in Ru/H becomes smaller, and it even levels off at a certain point.b.The Coefficient of Determination (R2) between the wave run-up characteristics (Ru/H) obtained from numerical simulations using the SPH method and Günbak's theory is 0.8824.A value closer to 1 signifies a similarity between the two variables.The achieved coefficient of determination value is highly satisfactory, approaching 1, which indicates a significant resemblance in outcomes between the two variables, the numerical simulation (SPH) and Günbak's theory.

Figure 2
Figure 2 Workflow of dualSPHysics program

Figure 4
Figure 4 Wave Run-Up Height on Non-Overtopping Breakwater

Table 2
Wave Run-up Height (m) Based on Theory

Table 3
Wave Run-Up Height (m) Based on Numerical Simulation