Numerical simulation of water wave propagation using DualSPHysics

In this study, the study of water waves was carried out using one of major the mesh-free CFD i.e. Smoothed Particle Hydrodynamics (SPH). DualSPHysics ver 5.0 which is an open-source SPH solver based on Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) was used to reproduce three dimension of water waves in large and medium numerical wave tanks (NWT). This study analyses water wave profiles and the accuracy of DualSPHysics for regular waves with period 1.15 second and two NWT were used, i.e., 35 and 15 m. In addition, advanced post-processing using VisualSPHysics was used to mimic water waves in real conditions. Single water depth was used, i.e. 0.4 meters for large and medium NWT. The results showed SPH has limitations for reproducing large NWT. However medium NWT successfully reproduce water wave with advanced post-processing using VisualSPHysics.


Introduction
Water Water wave propagation is a demanding problem in marine and coastal engineering field.One of the difficulties of free-surface flow problems is generating intended nonlinear water waves and predicting complicated interactions between water waves and fixed or floating objects, which is necessary for the utilization and development of further marine problems.Experimental and numerical methods have been used in several investigations.Because experiments are more expensive and costly, researchers typically utilize computer models instead.The finite element method (FEM), the finite difference method (FDM), the finite volume method (FVM), and smoothed particle hydrodynamics (SPH) are well-known numerical techniques to perform numerical models of water waves [1,2].
Most researchers use the SPH method in analyzing nonlinear free-surface flow problems and the interactions on solid structures.Such as research on breaking wave modelling and the interaction of coastal structures with water waves [3,4], sloshing in the prismatic tank [5][6][7][8], and the impact of water waves on offshore structures [9].These studies showed the potential of using SPH method in marine and coastal engineering applications and application for actual problem in engineering field.
DualSPHysics is an SPH numerical simulation software based on the Smoothed Particle Hydrodynamics (SPH) method.It is used for modeling fluid flows, especially those involving free surfaces and complex interactions [10,11], and engineering problems in coastal areas [12].The SPH method discretises the water domain into many small particles.Each particle has specific physical properties such as mass, position, velocity and density.The momentum and continuity equations govern the motion of each SPH particle.Reviews regarding the SPH method and its application can be seen in reference [13].Trimulyono and Hashimoto [4].In previous research, three-dimensional wave simulation in large numerical wave tanks (NWT) experienced a decrease in accuracy compared to two-dimensional wave simulation and had yet to carry out advanced post-processing in the research.In this study, the authors conducted further parameter analysis to increase the level of accuracy, which aims to test the ability of the SPH method to carry out three-dimensional wave simulations.In addition, threedimensional wave simulation in medium numerical wave tanks (NWT) was carried out with advanced post-processing using VisualSPHysics.

Methods
Smoothed Particle Hydrodynamics (SPH) is a computational method for solving fluid dynamics problems using the Lagrangian approach.SPH method was originally used by Gingold and Monaghan in the field of astrophysics field [14].Then, free-surface simulation using the SPH method was introduced by Monaghan [15] and began to develop in problems in the field of fluid dynamics.
The basis of the SPH formulation is integral representation or what is commonly called the kernel approximation.The integral representation is estimated by adding up the values of the closest neighboring particles which produce a particle approximation of the function at a point or a discrete particle whose results can be seen in equation (1) where  is kernel function and  is vector position.
Equation ( 1) can be approximated into discrete form by replacing the integral as number of neighboring particles, where  is the particle and  is the position.So, the particle equation in equation ( 2) becomes equation ( 3), where m is mass, ρ is density and v is velocity.
The momentum equation (Navier-Stokes) is used to set the motion of the particle due to the influence of the pressure generated by the neighboring particle.The form of the Lagrangian formula which forms the basis of the Navier-Stokes equation is as follows: The kernel function in this study uses the Wendland kernel function for computational efficiency, as stated by Cao et al [16].This function can be seen in equation (7).
The momentum equation in the water phase is described in equation ( 8) DualSPHysics is based on weakly compressible smoothed particle hydrodynamics (WCSPH) by treating fluids as weakly compressed.Thus, the equation of state is used in equation (10) which is used to find the pressure field based on the density of the particles.
Where  =  0 2  0  γ,  0 =   ( 0 ) = √   | 0, and  = 7 with  0 ,  0 and  are the speed of sound at the reference density, the reference density and polytropic constant, respectively.This study used two main models to be analyzed on reproducing the observed water waves in the experiment that conducted by Trimulyono and Hashimoto [4], i.e., large numerical wave tank (NWT) and medium numerical wave tank (NWT).The large NWT has dimensions of 35 meters in length, 0.5 meter in width, 1.5 meters in depth and a wave probe distance of 24.6 meters with a simulation time of 25 seconds and a piston-type wave generator as shown in Figure 1.Medium NWT has dimensions of 15 meters in length, 1 meter in width, 1.5 meters in depth and a wave probe distance of 6 meters with a simulation time of 30 seconds and a piston-type wave generator as show in Figure 2.
The parameters used in DualSPHysics were set as shown in Table 1.Coefh is defined as ℎ/√2 where dp is particle spacing.Artificial viscosity affects on proper dissipation in water wave propagation problem.For this subject problem the optimal value of artificial viscosity is 0.001.Coefsound is defined as /√.  where  and   are the speed of sound and the height of water, respectively.

Grid Study
A grid study was conducted to investigate the sensitivity of the particle spacing to the numerical accuracy.Figure 3 shows several particle spacing used for grid study on the medium NWT model with water depth d = 0.4 m.It shows that the value of particle spacing (dp) will affect the wave elevation and phase.Particle spacing 0.006 m was chosen over the others because it shows the best accordance in the wave elevation with the experiment.Decreasing particle spacing causes the number of particles to increase, but it also will increase computation times and the total number of particles.

Wave Propagation on Large Numerical Wave Tank (NWT)
Wave propagation simulation on large NWT with water depth d = 0.4 m uses the parameters in Table 1. Figure 4 shows the comparison results of wave elevation at 24.6 meters distance from the wave generator between the experiment and SPH simulation of the large NWT, which shows that the phase difference is quite large.The SPH simulated wave elevation is not in accordance with the experimental wave elevation.It shows that the wave generated from the SPH simulation for large NWT has poor accuracy, and further parameter improvements are needed.Figure 5 shows the three-dimensional wave profile of the large numerical wave tank (NWT).

Wave Propagation on Medium Numerical Wave Tank (NWT)
Wave propagation simulation on medium NWT with water depth d = 0.4 m uses the parameters in Table 1. Figure 6 shows the comparison results of wave elevation at 6 meters distance from the wave generator between the experiment and SPH simulation of the medium NWT, which indicates that the SPH simulated wave has no phase difference compared to the experimental results.The wave elevation from the SPH simulation is following the experiment.It shows that the waves generated from the SPH simulation have good accuracy.Figure 7 shows the three-dimensional wave profile of the medium numerical wave tank (NWT).Figure 8 shows the result of the post-processing application in Paraview and Blender software that is visualized from VTK files generated from DualSPHysics.Paraview shows simple visualization from wave propagation.Blender offers complex and realistic visualization that can represent simulation results more realistically..The particle spacing (dp) affects the wave profile both in terms of height and the wave phase, where decreasing particle spacing will cause the number of particles to increase.Still, it also will increase computation times and the total number of particles.The numerical simulations can be visualized through post-processing using Paraview software, which supports simple scientific visualization, and Blender software, which supports complex and realistic visualization.
This research has promising results, however, further studies are needed, especially for other parameter adjustments to produce a numerical simulation of three-dimensional regular waves in large numerical wave tanks (NWT).

Figure 1 .Figure 2 .
Figure 1.Schematic view of large numerical wave tank

Figure 3 .
Figure 3. Grid study on medium NWT with d = 0.4 m

5 Figure 4 .Figure 5 .
Figure 4. Comparison results of wave elevation between experiment and SPH simulation

Figure 8 .
Figure 8. Post-processing results of wave propagation simulation of medium NWT with water depth d = 0.4 m in Paraview (a) and Blender (b) This study uses the DualSPHysics application to carry out further development of previous research conducted by 2

Table 1 .
Numerical setting in DualSPHysics