Analysis of overtopping wave energy converters with ramp angle variations using particle-based simulation

2.1 Kernel Function

Fluid analysis methods are broadly divided into Eulerian methods, which analyze fluid flow within a fixed spatial frame-work, and Lagrangian methods, which represent fluids as particles characterized by physical properties such as mass, density, pressure, and velocity. The SPH method is a fluid analysis technique based on a Lagrangian framework. In SPH, the physical properties of fluid particles are calculated based on their interactions with neighboring particles. These interactions are approximated using a kernel function, as expressed in Equation (1).

Here, r represents the physical quantity of a particle, and r^' represents the physical quantity of neighboring particles around r. The SPH method discretizes Equation (1) to obtain an approximate form, as shown in Equation (2).

Here, F(r_i) represents the physical quantity at particle r_i, and r_j denotes the individual particles surrounding particle r_i. m_j and ρ_j represent the mass and density of particle j, respectively. W(r_i-r_j,h) is the kernel function, which is defined based on smoothing length h and the distance between particles i and j. The kernel function is expressed as

In Equation (3), r denotes the distance between particle i and its neighboring particle j, while q=r/h represents the dimensionless distance between particles. The coefficient α_D is defined as 10/7πh² for two-dimensional cases and 1/πh³ for three-dimensional cases.

2.2 Momentum Equation

In the SPH method, the motion of the fluid is described by the Navier–Stokes equations. The Navier–Stokes equation with respect to time t is given as

Here, $$ -\frac{1}{\rho }\nabla P$$ is the pressure, v₀∇²v represents the viscous stress, and g denotes the gravitational acceleration. Artificial viscosity was applied to account for the viscous term that determines the fluid behavior in the Navier–Stokes equation. In this study, the artificial viscosity term proposed by Monaghan [8] was used and is expressed by the following equation:

In Equation (5), P and ρ represent the pressure and density at points i and j, respectively. The viscosity term ∏_ij is defined as follows:

Here, r_ij=r_i-r_j and v_ij=v_i-v_j denote the position and velocity differences between particles i and j, respectively. $$ \overline{c_{ij}}=0.5\left(c_i+c_j\right)$$ represents the average speed of sound, and α is a constant set to 0.01. The term μ_ij in Equation (6) is given as follows:

Here, η² is defined as the square of the smoothing length h, multiplied by 0.01.

In the SPH method, when analyzing incompressible fluids, the fluid is assumed to exhibit slight compressibility. The equation of state representing this compressibility was proposed by Monaghan et al. [9] and is expressed as follows:

In Equation (8), γ=7, $$ b=c_0^2{\rho }_0/\gamma $$ and ρ₀=1000kg/m³ represents the reference density. c₀ denotes the speed of sound at the reference density.

3.1 Model Conditions

Ulleungdo was selected as the study area based on the reference by An et al. [10]. The simulation scaling factor was set to 1/20 of the actual sea conditions to accommodate future comparisons with physical model experiments and align with the dimensions of a two-dimensional wave flume. The environmental parameters used in this study are listed in Table 1.

Table 1

Wave conditions

The model used in this study is identical to that used by An et al. [10]. The design and dimensions of the baseline model are illustrated in Figure 1 and detailed in Table 2. The ramp angle of the upper structure of the wave energy converter was chosen as a design variable, with the angles varying incrementally from 20° to 35° at 5° intervals. The geometries and specifications of the modified models are shown in Figure 2 and Table 3.

Table 2

Parameters of OWEC

Figure 1

Geometry of overtopping wave energy converter (OWEC)

Table 3

The parameters of change models

Figure 2

Designed models

3.2 2D-numerical water tank

A two-dimensional numerical water tank was generated and analyzed using the SPH method. The environmental and numerical parameters of the two-dimensional water tank are illustrated in Figure 3 and summarized in Table 4.

Figure 3

Geometry of 2D-numerical water tank

Table 4

Parameters of 2D-numerical water tank

A piston-type wave generator with a distance of 4 m between the wave generator and structure and a water depth of 0.9 m was employed. The motion of the piston-type wave generator is described as follows (Biesel and Suquet [11]):

The wave generator moves according to its displacement e(t) over time, where ω represents the wave frequency. δ is the initial phase, which is set to 0 in this study. S₀ denotes the stroke of the wave generator, which is related to the wave height H as expressed in Equation (10).

The wave frequency ω is determined using the dispersion relationship for shallow water, as shown in Equation (11).

where d represents the water depth, and k denotes the wave number.

4.1 Hydraulic Efficiency

The performance of the OWEC was analyzed by comparing its overtopping flow rate and energy efficiency. In this study, we utilized hydraulic efficiency, which represents the energy efficiency related to overtopping, as proposed by Margheritini et al. [12]. The equation for the hydraulic efficiency is as follows:

where P_crest denotes the potential energy of the overtopped fluid. For a multistage OWEC, this is expressed as the sum of the potential energies of the fluid overtopped into each reservoir.

In this equation, ρ is the density of the fluid, g is the gravitational acceleration, q_j represents the overtopping flow rate into the j-th reservoir, and R_c,j is the height from the sea level to the inlet of the j-th reservoir. The flow rate was measured based on the wave period used in the analysis, and the average value was adopted. P_wave represents the kinetic energy of the incident wave approaching the OWEC.

where c is the phase velocity of the wave, H is the wave height, d is the water depth, and k is the number of waves.

4.2 Analysis Results

This study aims to analyze the differences in hydraulic efficiency according to changes in the ramp angle of an OWEC. The analysis results for the models with varying ramp angles are presented in Table 5 and Figure 4 and Figure 5. Additionally, Figure 6 illustrates the flow process in each reservoir over a single-wave period. Figure 6 shows that the overtopped waves enter the reservoir through two processes: ascending along the ramp and descending along the ramp.

Table 5

Hydraulic efficiency of models

Figure 4

Hydraulic efficiency distribution across model

Figure 5

Hydraulic efficiency for each reservoir

Figure 4 shows the hydraulic efficiency of each reservoir in the models, and Figure 5 highlights the trend in the overtopping efficiency of each reservoir. As shown in Table 5, the hydraulic efficiencies of the models ranges from 30% to 37%. Overall hydraulic efficiency is highest in RAC 4, and it increases with ramp angles. Examining the changes in hydraulic efficiency by stage reveals that the efficiency of the 1st and 2nd reservoirs increases as the ramp angle becomes steeper, while the hydraulic efficiency of the 3rd and 4th reservoirs decreases with increasing ramp angle.

Although RAC 1 exhibits the lowest overall hydraulic efficiency, it demonstrates the highest hydraulic efficiency at the 4th reservoir inlet among the four models. Figure 6 shows that in the RAC 1 model, the wave’s kinetic energy in the direction of the structure is transmitted along the ramp to the 4th reservoir inlet. For RAC 1, waves primarily enter the reservoir during the ascending motion. The RAC 2 model shows a similar trend to RAC 1.

For RAC 3 and RAC 4 models with ramp angles of 30° or higher, most of the hydraulic efficiency was concentrated in the 1st and 2nd reservoirs. As shown in Figure 6, the overtopping waves in RAC 3 and RAC 4 ascend along the ramp to the 4th reservoir inlet. However, owing to the steep angles, more wave energy enters the reservoirs during the descending motion than during the ascending motion. This indicates that as the ramp angle increases, the descending waves have a greater influence on hydraulic efficiency.

The analysis results indicate that the RAC 4, with a ramp angle of 35 degrees, demonstrated the highest performance in the simulation domain. This superior performance is attributed to the increased ramp angle, which effectively concentrated wave energy into the 1st and 2nd reservoirs, maximizing energy efficiency. Notably, the steeper angle enhanced the descending wave motion, resulting in an increased inflow of water into the reservoirs. Conversely, the efficiency of the 3rd and 4th reservoirs decreased as the ramp angle increased. However, the high efficiency of the first and second reservoirs offset this decline, leading to an overall improvement in performance.

Author Contributions

Conceptualization, S.-H. An and J.-H. Lee; Methodology, S.-H. An; Software, S.-H. An; Validation, S.-H. An and J.-H. Lee; Formal Analysis, S.-H. An; Investigation, S.-H. An; Resources, S.-H. An; Data Curation, S.-H. An; Writing—Original Draft Preparation, S.-H. An; Writing—Review & Editing, J.-H. Lee; Visualization, S.-H. An; Supervision, J.-H. Lee; Project Administration, J.-H. Lee; Funding Acquisition, J.-H. Lee.