Wave transmission at low-crested structures

. This paper investigates the wave transmission through artificial reefs utilised as low-crested breakwaters to mitigate coastal erosion and reduce wave energy. The study utilises the DualSPHysics smoothed particle hydrodynamics (SPH) solver to evaluate the effectiveness of such structures. Wave transmission coefficients were computed for various reef configurations and wave conditions. Results indicate that the shifted reef configuration provides a higher level of wave height reduction for non-breaking waves, while there is no significant difference for breaking waves. The study further reveals that the efficiency of the breakwater decreases with an increase in wave steepness. However, the total wave energy significantly decreases, leading to a gradual reduction in wave height as it travels further from the breakwater. The outcomes of this research can aid in the optimisation and design of artificial reefs for coastal protection.


Introduction
Artificial reefs have been widely utilised as a coastal protection structure to reduce wave energy and protect shorelines from erosion.In particular, low-crested breakwaters (LCBs) have become a popular design solution for coastal protection, with artificial reefs serving as a viable option due to their cost-effectiveness and potential to enhance marine habitats.Numerous studies have been conducted to evaluate the performance of LCBs, including artificial reefs, under various wave conditions, water depths, and configurations.
One approach to studying the wave transmission characteristics of LCBs is through numerical simulation.In recent years, smoothed particle hydrodynamics (SPH) has emerged as a promising numerical method for simulating wave-structure interactions due to its ability to accurately capture free-surface flows and fluidstructure interactions.Among the available SPH solvers, DualSPHysics [1] has been widely used in coastal engineering research due to its versatility and accuracy in modelling wave-structure interactions.Dominguez et al. (2019) proposed an improved SPH model within DualSPHysics software that can generate, propagate, and break solitary waves.The study showed that the new model was capable of accurately reproducing experimental data, indicating the potential of SPH as a complete tool for coastal engineering.
Another study by Gonzalez-Cao et al. [2] investigated the accuracy of DualSPHysics, an SPH solver, in simulating violent collisions with coastal structures.The authors compared numerical results with experimental data and found that the model was able to *Corresponding author: osetyandito@binus.eduaccurately predict the impact loads and the response of the structures.
Overall, these studies demonstrate the potential of SPH as a powerful tool for coastal engineering, with the ability to accurately simulate complex wave phenomena and assess the performance of coastal protection structures.
In this paper, we aim to assess the wave transmission characteristics of artificial reefs used as low-crested breakwaters using DualSPHysics.The study will investigate the performance of the reef under varying wave conditions and reef configurations.Through numerical simulation, we will evaluate the wave attenuation and energy dissipation characteristics of the reef, with a specific focus on the wave transmission coefficient.The results of this study will provide insights into the effectiveness of artificial reefs as a lowcrested breakwater solution and demonstrate the utility of DualSPHysics as an effective SPH solver for coastal engineering applications.

Smoothed particles hydrodynamics (SPH)
The Smoothed Particle Hydrodynamics (SPH) method is a numerical technique that has been widely used in fluid dynamics to simulate the behaviour of fluid flows [3].The SPH method is a Lagrangian method, which means that the fluid is represented by a set of discrete particles that move with the fluid flow.The SPH method uses a kernel function to interpolate the properties of the fluid between the particles.This kernel function is used to calculate the forces acting on each particle and to update the velocity and position of each particle at each time step.
The SPH method is based on the Navier-Stokes equations, which describe the behaviour of fluid flows.The SPH method approximates the fluid properties by using a set of discrete particles.The properties of each particle, such as density, velocity, and pressure, are calculated by averaging the properties of the neighbouring particles using a kernel function, which in this study, the 5th-order Wendland kernel [4] was selected.
Here a brief weakly compressible SPH formulation is presented, however, more complete formulations and the latest features exist in DualSPHysics software are available in the work of Dominguez et al. [1] or its website: dual.sphysics.org/.
The momentum equation in SPH form can be written as Equation 1.
where  is the velocity,  is the pressure, Π  is the viscosity term and     is the gradient of the kernel function with respect to particle .Note that  is the particle of interest and  in the neighbouring particles.Lastly,  is the gravity acceleration.
The density () change rate is also calculated using the SPH summation from Equation 2.
where   is the density of particle ,   is the mass of neighbouring particle  and   =   −   is the relative velocity between particles  and .
The pressure of each particle is calculated using the Equation 3, state of the fluid.
where  =  0 2  0 ,  = 7,  0 is the reference density (in this case 1000 kg/m 3 and  0 = ( 0 ) is the speed of sound at the reference density.To prevent excessively small time steps during simulations, the fluid's compressibility is adjusted to ensure that the speed of sound remains within reasonable limits, thereby reducing computation costs.
Those variables ( and ) are integrated in time using the symplectic algorithm [5] with variable time step limited by the Courant-Friedrich-Lewy (CFL) number which is based on the force and viscous diffusion terms.

Simulation setup
The simulation domain is configured as shown in a sketch in Fig. 1 showing the lateral view of a 100 m long flume equipped with a piston-type wavemaker.The wavemaker board is located 0.5 m away from the left wall of the flume to accommodate the back-and-forth movement.The last 25 m of the other end of the flume is set to be a damping zone to avoid wave reflection.Note that the damping process is done numerically following the formulation described in Altomare et al.The wave is generated by imposing a motion to the wave paddle mimicking the piston-type wavemaker in the physical laboratory.The time series of the piston movement is assigned following either first-order or second-order wave theory from the given wave height (), wave period () and water depth ().More comprehensive validation on wave generation is available in Altomare et al. [6].In this work, we consider using the second-order wave generation to better represent the real water wave.

Artificial reef geometry
The present study employed a synthetic reef composed of several cube-shaped reef units located 45 m away from the wave paddle.The detailed geometry of a single unit, measuring 1x1x1 m, constructed solely from its edges, is presented in Fig. 2. The edges themselves measure 15x15 cm.In actual implementation, the reef would be constructed using reinforced concrete casted as a singular entity.The breakwater structure in this study was formed by arranging the reef units in a specific pattern, as illustrated in Fig. 3.The units were assumed to be interconnected in a rigid manner, although in reality, they may be linked using a kind of durable rope, allowing for slight movement upon impact by waves while maintaining contact.The structure movement is constrained in this simulation.
To assess the impact of different arrangements on wave transmission, two configurations were tested.The first configuration, depicted in Fig. 3(a), involved arranging the units in a pattern coloured in blue, with subsequent units in the same row following the same pattern.In the second configuration (Fig. 3(b)), the cube row located behind the red pattern was shifted forward by 0.5 m.This shifted grid arrangement resulted in smaller cavities within the structure while maintaining the same number of unit cubes, which can be visually evidenced by the side views facing the incoming waves.

Numerical parameters
Table 1 presents the complete list of SPH numerical parameters employed in these simulations.The parameter coefh represents the ratio of smoothing length to the initial inter-particle distance (Dp).The second-order symplectic position Verlet integration scheme [5] was employed for time integration, with a variable time step size computed based on the maximum velocity and acceleration at each step.The 5th-order Wendland kernel [4] was selected as the smoothing kernel function.Artificial viscosity [7] was incorporated to reduce the numerical instability with a tuning value of α = 0.05, while the boundary particles were excluded from the viscosity term to prevent excessive boundary friction.The force computation included the density diffusion term (DensityDT) proposed by Fourtakas et al. [8] to achieve a smoother pressure field.The shifting algorithm [9] was activated with a coefficient of -2 and a free-surface threshold value of 2.75 to maintain a consistent particle distribution and ensure the accuracy of the SPH interpolation.Lastly, TimeMax refers to the overall duration of the simulation which is 40 seconds.This time is equivalent to generating waves eight times.

Wave parameters
Two different wave heights () were tested in this study: 0.5 m and 1.5 m, with a fixed wave period () of 8 s and a water depth () of 3 m.Both waves are classified as intermediate depth waves and are best described at least by the second order wave theory, as shown in Fig. 4. Therefore, the piston motion was determined in accordance with the second order wave theory.The wave height of 1.5 m is close to the breaking wave limit, and in fact, the simulation demonstrated a spilling-type breaking condition, which will be presented later.As the reef was utilised as a breakwater structure, the breaking wave condition was used to evaluate the effectiveness of the reef in reducing highenergy waves.Furthermore, a non-breaking wave with =0.5 m was also tested to provide a more comprehensive analysis.

Numerical performance
This section briefly explains the numerical performance of DualSPHysics software used in this study.All simulations were performed on a personal computer (PC) with: • Intel® Core™ i7-9700K @ 3.60GHz CPU, and The PC was utilised to simulate all six cases, as listed in Table 2, which included two different wave heights tested against two reef configurations along with a water-only case for benchmarking purposes.A fixed inter-particle distance of Dp=0.075 m was adopted for all simulations, resulting in approximately 2.65 million particles generated per case.This configuration produced output files consisting of postprocessing data totalling almost 160 Gigabytes (GB).It is important to note that the output file size depends on how frequently the data is printed during simulation time intervals.
Although the number of particles used in all cases is nearly the same, simulations with higher wave heights (=1.5 m) required a greater number of time steps to complete the 40-second simulation, resulting in longer computation times.The higher wave height produces greater individual particle velocities and accelerations, which necessitates smaller time step sizes according to the CFL (Courant-Friedrich-Lewy) condition.

Effect on wave profile
This section presents the analysis of breakwater performance in dissipating wave energy by comparing the water surface elevation after passing through the reef structure with the simulation without the structure.The water surface elevation is computed using "MeasureTool", a post-processing tool included in the DualSPHysics package, which sums the scattered fluid particles by means of SPH interpolation.Fig. 5 shows the time history of water surface elevation at the wave gauge position (60 m from the wave paddle) as shown in Fig. 1.The first graph compares the water surface without a structure, with regular reef, and shifted reef configurations for nonbreaking waves with  = 0.5 m.The wave height reduction can be observed, with the shifted reef configuration yielding a higher reduction level.However, for waves with  = 1.5 m, the wave crest does not show significant reduction for both regular and shifted reef configurations, except for the wave trough, which is due to the wave creeping over the reef structure.
We define a wave transmission coefficient, where   and   are the incoming and transmitted wave heights, respectively.The coefficient is plotted against the non-dimensional parameter    2 , which represents the wave steepness in Fig. 6.The graph shows that the shifted reef configuration yields a lower   than the regular formation for both wave conditions.Additionally, the breakwater becomes less efficient as the wave becomes steeper, contrary to the findings of Armono [11], who showed decreasing   values as the wave became steeper.This difference in behaviour is due to the assumption of a low-crested breakwater condition in this study, whereas Armono's work was submerged.Further studies are needed to confirm this statement with a wider range of wave criteria.Fig. 8 illustrates the breaking wave approaching the reef structure in more detail.The spilling-type breaking wave is clearly visible, starting to hit the structure at =30.2 s.As shown in Fig. 8(a), the splashy wave tip gradually creeps over the reef crest, demonstrating the capability of SPH in simulating free surface flow.By =30.8s (Fig. 8(b)), approximately half of the wave body has passed the structure.It is evident that the flow velocity under the reef crest is significantly reduced by the structure, with the shifted reef configuration again showing better performance.However, the overtopping wave velocity remains high and is not much affected.After a few seconds (=33.6 s), the velocity field becomes similar to that of the wave without structure, albeit still with lower velocities.Ultimately, the shifted reef configuration successfully removes the breaking wave, resulting in a normal wave.It is worth noting that the artificial reef effectively reduces both wave height and velocity, as shown previously.However, further testing under varying wave conditions is required to fully assess its effectiveness.As the wave period and wavelength in this study were relatively long, the reduction of long wave transmission by the artificial reef was not as significant as expected.Nonetheless, the shifted reef configuration demonstrated better performance in reducing both wave height and velocity, while utilising the same amount of reef material, making it an encouraging option for sustainable development.

Conclusions
In conclusion, this study has presented a numerical investigation of the wave transmission through artificial reefs used as low-crested breakwaters.The effectiveness of the shifted reef configuration in reducing wave energy has been evaluated using an SPH solver, specifically the DualSPHysics code.The wave transmission coefficients were calculated for different wave conditions and reef configurations.The results show that the shifted reef configuration provides a higher reduction level for non-breaking waves, but no significant difference in wave height reduction for breaking waves behind the breakwater.Furthermore, the efficiency of the breakwater decreases as the wave steepness increases.However, the total wave energy did significantly decrease, hence the wave height will gradually decrease as it travels further away behind the breakwater.
Overall, the artificial reef does reduce the wave height and velocity, with the shifted reef configuration giving better results in reducing the wave height and velocity with the same amount of reef material.However, its effectiveness still needs to be tested with various wave conditions.The present study is important for the design and optimisation of artificial reefs as coastal protection structures, particularly in developing sustainable solutions for coastal erosion.It is recommended to conduct further studies to explore different configurations of artificial reefs and assess their performance under different wave conditions.The numerical simulations, coupled with experimental data, could provide more insights into the behaviour of artificial reefs as coastal protection structures.
[6].This feature is readily available in the DualSPHysics software package.The flume is filled with water up to 3 m depth, equal to the breakwater crest elevation.Both side wall of the flume is represented by periodic boundary conditions with a domain width of 3.85 m, equal to 4 layers of the typical artificial reef blocks.The detailed geometry of the reef is described in section 3.1.

Fig. 1 .
Fig. 1.The lateral view of the flume for wave simulation.

Fig. 2 .
Fig. 2. The dimension of the artificial reef unit.

Fig. 3 .
Fig. 3.The different arrangements of the reef units composing the breakwater structures: (a) regular grid and (b) shifted grid.

Fig. 4 .
Fig. 4. The wave conditions used in the simulation were indicated by red marks on the Le Méhauté diagram [10].

Fig. 7 (
Fig.7(a) depicts the simulation results for a 0.5 m wave height at time =32 s, which corresponds to the moment when the wave has just passed through the breakwater.It can be observed from the velocity field that the presence of the artificial reef has caused an overall reduction in the fluid velocity.Additionally, the flow through the regular reef is relatively faster compared to the shifted reef owing to the larger void within the latter.At =34 s (Fig.7(b)), the velocity of the transmitted wave is inherently reduced, with the shifted reef providing a slightly better reduction in the fluid velocity field.Fig. 8 illustrates the breaking wave approaching the reef structure in more detail.The spilling-type breaking wave is clearly visible, starting to hit the structure at =30.2 s.As shown in Fig. 8(a), the splashy wave tip gradually creeps over the reef crest, demonstrating the capability of SPH in simulating free surface flow.By =30.8s (Fig. 8(b)), approximately half of the wave

Table 2 .
Computational performance of the SPH simulation.