Analyzing tandem effects in oscillating water column WEC arrays a numerical study of diverse geometries and inter-device gaps

2.1. Computational framework and equations

The study configures the hydrodynamic model to represent the viscous incompressible flow field surrounding OWCs

Figure 1. Iturrioz et al.’s experimental study setup (modified drawing from (Iturrioz et al., 2015)). and subject to the governing Reynolds-averaged NavierStokes (RANS) and continuity equations. The free surface is simulated using the volume of fluid (VOF) solver method inspired by Hirt & Nichols' (1981) study. The high-resolution interface capturing (HRIC) method was executed within a 3D computational model. Because the k-ω shear-stress transport (SST) model offers accurate predictions for simulating flow separation in adverse pressure gradients, turbulence in the flow field was represented using the k-ω SST equations. The turbulent kinetic energy (k) and specific dissipation rate (ω) are addressed through the dual-equation model established by Siemens Digital Industries Software (2020). The governing equations can be expressed as follows: (1)

where ui represents the velocity measured over time, p, ρ, and μ represent the pressure, density, and dynamic viscosity, respectively. The Reynolds stress tensor, which captures the impact of turbulence on the mean momentum, is contributed by the final element on the right-hand side of Eq. 2. The research employs numerical simulations utilizing the Eulerian multiphase model to simulate two distinct phases: air and water. The monitoring of free fluid surfaces was achieved through the Waves model within the Simcenter Star CCM+ software. The governing equations have been discretized using the finite volume method (FVM) to improve solution accuracy. A second-order scheme was employed for both the temporal and spatial discretization. The velocity-pressure coupling was achieved using the SIMPLE algorithm. The simulation duration was set at 90 s, with a time step of 0.01 s for each iteration. The properties of the numerical environment were characterized by the density and kinematic viscosity values for air (ρ0=1 kg⁄m3 , ν=1.48 × 10-5 m2/s) and water (ρw=1000 kg⁄m3 , ν=10-6 m2/s).

2.2. Creating the domain

The CFD model has been constructed based on the experimental parameters outlined by Iturrioz et al. (2015).

According to test tank set-up (see Iturrioz et al., 2014, 2015), wave flume length is 20.60 m, width is 0.68 m and height is 0.75 m. A beach located 10 m away from the wave-maker with a draught of 0.20 m serves to minimize wave reflections. Two pressure gauges (PG1 and PG2) are situated in the top section of the chamber. The schematic representation of the wave flume was adapted from Iturrioz et al. (2015) and is depicted in Figure 1. The NWT has been created in Simcenter Star CCM+. Iturrioz et al. (2015) explored various top slot scenarios, wave heights, and wave periods. The current study focuses on simulating Case C24 with a 9mm top slot scenario, as this demonstrates superior performance and achieves optimal efficiency. After validating the numerical setup of a single rectangular OWC, six unique array configurations have been systematically organized and evaluated for power efficiency.

2.3. Boundary conditions

The computations are carried out using a Cartesian coordinate framework, with the point of origin located at the intersection of the model. The positive x-axis and upwardpointing positive y-axis indicate the direction of the entering flow. The waves are propagating in the positive x direction. Carefully selecting appropriate boundary conditions is imperative for achieving accurate modeling of free surface flows. The created wave tank is 10 m long, so it ends right before the dissipative beach. The inlet and outlet boundaries are placed on the left and right sides of the tank, respectively. The OWC model is situated 5.47 m from the inlet boundary, and a uniform velocity is applied from this side. The outlet boundary is pressure, and the remaining walls are subject to a no-slip boundary constraint. The numerical wave tank sketch and model view of the OWC are shown in Figure 2.

2.4. Array configurations for the OWCs

This study employs three different geometries of OWC WECs, each evaluated at two different inter-WEC distances. As a result, a total of six unique configurations have been examined. The inter-device distances have been studied as 10 times and 5 times the width, which is represented as the diameter (D) of the WEC, and was held constant at 0.34 m for all cases. Table 1 provides an overview of each configuration.

Figure 2. Illustration of the numerical wave tank, and a single OWC model (Iturrioz et al., 2015) view in Simcenter Star CCM+. To accurately model the flow dynamics surrounding the OWCs, the arrays’ solution domain was meticulously constructed in accordance with the previous sections and Table 1. The sections and plan views are for all configurations, with Figure 3 showing an illustration of the array domain for A10 as an example.

2.5. Meshing

The mesh structure was developed using hexahedral grid elements, with volumetric controls implemented in specific regions to generate refinement zones. To enhance the accuracy of the model, particular attention was given to refining the computational mesh surrounding the free surface. Additionally, the mesh has precisely refined the flow around the OWCs to account for high velocity and pressure gradients. Notably, thorough computational modeling of the flow discharge has been achieved by utilizing very thin grid elements in the slot region. Figure 4 depicts the general structure of the volume mesh of an array. The total cell count in the computational array models ranges from 7.0 to 9.8 million cells and vary based on the distance between the geometries. The accurate representation of free surface waves necessitates high-resolution grid mapping. In cases involving extensive domain dimensions or significant wave heights, the total mesh count might become extremely large to achieve a fine mesh size. Hence, as the total mesh size expands, the required computational resources and solution times also rise. Adaptive mesh refinement (AMR) emerges as a feasible strategy for addressing this issue. AMR enables localized grid refinement in areas of interest by identifying a target scalar and refining the mesh solely in these areas. Additional details regarding the AMR technique can be found in the solver’s manual (Siemens Digital Industries Software, 2020).

The current investigation employs an AMR technique to enhance the accuracy of free surface wave representations. Figure 5 highlights the grid structure and AMR visualization. AMR techniques are used to automatically identify the free surface interface and refine the grid around the water-air interface. The refined regions move in conjunction with the free surface every 5 timesteps as the waves progress.

2.6. Energy extraction and efficiency

The average power absorbed by a regular wave is calculated by analyzing the numerical data in the time domain. POWC is the pneumatic power extracted on a time-averaged basis (Morris-Thomas et al., 2007). Pinc refers to the energy flux associated with the incident wave and represents the available power within the wave. The formulations are given as follows: (3)  ΔP(t) is the differential air pressure and q(t) is the air flow rate, and (4)

where Ai is the amplitude of incident wave, and g is the acceleration of gravity. Cg represents group velocity and is formulated as follows: (5)

Based on Equations 4 and 5, Equation 6 can be obtained as: (6)

where g, k, and h are gravitational acceleration, wave number, and fixed water depth, respectively. The dispersion relation is calculated as shown in Equation 7. ω2=gk tanh (kh)(7)

Table 1. Information about wave characteristics and array cases Array name A5 A10 B5 B10 C5 C10

Rectangular Rectangular Square Square Cylindrical Cylindrical

3.1. The single OWC model

The slot in an OWC system serves as the interface through which air enters and exits the chamber as the water column oscillates, with narrower slots restrict the flow of air through it more. As a result, the velocity of the air passing through the slot increases, leading to higher shear forces and increased viscous damping. Therefore, an increase in damping is observed as the slot of an OWC system becomes thinner. Iturrioz et al.’s (2015) study also highlighted this by comparing chambers’ mean pressure oscillation with different top orifice sizes (e.g., 50mm, 9mm, 4.5mm). The current study has chosen the 9mm slot size because of its greater hydrodynamic efficiency. All the experimental data (Iturrioz et al., 2015) about case C24 and its 9mm top slot have been obtained from the wave and pressure gauges. Because the air velocity along the z-axis is the primary driver, it is one of the values utilized to validate the model. Figure 6 presents a comparison of the velocity values, and the well-matched crests and troughs show strong consistency between the air velocities from both the computational model and physical measurements.

Figure 4. Numerical domain view and mesh generation of array A10.

Figure 5. Section view of the grid system and AMR at the free surface. AMR: Adaptive Mesh Refinement.

Figure 6. Numerical and experimental comparison of air velocity (Iturrioz et al., 2015) (C24 with 9mm top slot). CFD: Computational fluid dynamics.

The current numerical model has accurately captured the effective pressure. Figure 7 demonstrates the comparison of ΔP (Ptotal-Patm = Peffective) between the experimental and numerical results, showing strong agreement. The oscillating water column wave energy converter is characterized by a repetitive process controlled by the rise and fall of water levels in a chamber. Figures 8a-h show this cycle over a duration of one wave period. When an ocean wave crest approaches the device, the increasing water level in the chamber compresses the air trapped above it, leading to a rise in pressure. This increased air pressure causes air to flow through an orifice (i.e., slot) that directs the compressed air. Subsequently, as the wave trough passes and the water level in the chamber decreases, the air pressure diminishes, permitting external air to enter the chamber through the opening and initiating a new cycle. Throughout the analysis, the outward (positive) velocity (i.e., pressure) within the chamber was consistently observed to significantly exceed the inward (negative) velocity (i.e., pressure). This phenomenon can primarily be attributed to the placement of the measuring volume above the opening.

3.2. Impact of shape and spacing on array models

This research examines the numerical analysis of six array layout models. Figure 9 presents a comparative overview of the air velocities observed across various OWCs within the array. The results indicate the shape of the OWC to significantly influence air velocity regardless of distance, with square-shaped OWCs demonstrating superior performance compared to cylindrical OWCs. Upon initial evaluation of the OWC at a distance of 10D, a 27% discrepancy in velocity is noted between the square and cylindrical OWCs located in the first row. This discrepancy further increases as the rows progress. Specifically, air movement within square OWCs is observed to be approximately 39%, 45%, and 47% faster than cylindrical OWCs in the 2nd, 3rd, and 4th rows, respectively. When exploring the 5D distance, it becomes evident that while speed may lower in the 3rd and 4th columns, the most significant contrast between the square and cylinder occurs in the 3rd column. The rectangular OWCs exhibit intermediate velocity characteristics compared to the square and cylindrical variants. However, velocity dynamics change when considering arrays with a 5D distance, displaying a different pattern from those with a 10D distance. The square-shaped OWCs in the 2nd, 3rd, and 4th rows maintain velocity advantages of nearly 23%, 44%, and 31% over their respective cylindrical OWCs. Table 2 also summarizes the velocity values and the differences. Beyond shape considerations, the optimal spacing between OWCs is row-dependent rather than being uniformly advantageous for both 5D and 10D distances. Figure 10 illustrates that a 5D distance is preferable for the 2nd row but less so for the 3rd row, only to become advantageous again for the 4th row. This observation aligns with average power generation, where the 1st, 2nd, and 4th rows of OWCs in a 5D array outperform those in a 10D array. Figure 10 depicts how a pattern occurs irrespective of distance length. The 5D distance exhibits a positive tandem effect between adjacent OWCs, increasing power

Figure 7. Numerical and experimental comparison of OWC chamber air pressure on a timeline, with the experimental study (Iturrioz et al., 2015) values being for Case 24_9mm top slot. PG: Pressure gauge; CFD: Computational fluid dynamics; OWC: Oscillating Water Column.

Figure 8. Snapshots (a-h) illustrate air movement of wave period duration through the slot. generation. In contrast, the impact of a 10D distance varies significantly, with a nearly 40% energy loss observed in the 2nd row, while the 3rd and 4th rows yield comparable average power outputs. The study demonstrates the rectangular-shaped oscillating water column to generate nearly twice the power of the square-type, resulting in a doubling of chamber volume. However, when considering the average power output per unit volume, the square OWC exhibits an almost 10% higher efficiency compared to the rectangular design.

Figure 11 illustrates the hydrodynamic efficiency (η) as a function of dimensionless wave number (kh) for each type of OWC. Peak efficiency values consistently occur at a kh value of approximately 1.40. Among the OWCs located at a distance of 5D from each other, the 2nd row exhibits the highest efficiency. Specifically, the square type has an efficiency of 0.29, the rectangular type has an efficiency of

0.24. and the cylindrical type has an efficiency of 0.22. All of

these forms exhibit efficiencies that are 40-55% higher than those achieved in a 10D space.

Figure 9. Comparing the velocity values. CYL: Cylinder; SQ: Square; REC: Rectangle; OWC: Oscillating Water Column.

At a distance of 10D, the most effective devices are located in the initial rows regardless of their shape, as previously indicated. Specifically, square, rectangular, and cylindrical devices exhibit efficiencies of η =

0.27, η = 0.23, and η = 0.21, respectively. After the 1strow OWC device, efficiency diminishes and remains relatively consistent across the array. The findings are also tabulated in Tables 3 and 4.

Table 2. Summarization of air velocity values for square and cylindrical shapes OWC type

1.72. 1.93 0.74 1.4

Figure 10. Comparing the mean power for each OWC. OWC: Oscillating Water Column; WEC: Wave Energy Converter; SQ: Square; CYL: Cylinder; REC: Rectangle.

Figure 11. Hydrodynamic efficiency versus dimensionless wave number for all OWCs. CWR: Capture width ratio; OWC: Oscillating Water Column.

0.19. 0.22 0.09 0.15

Table 4. Summarizing the efficiency data for representing the effects of OWC distances OWC type

η Difference % Square Square Cylindrical Cylindrical Rectangular Rectangular

0.23. 0.14 0.14 0.14

used the air velocity in the z direction and pressure difference value at the orifice region, with all visual representations indicating a significant correlation to have been achieved. The shape of a device is already known to be important for generating power from waves. The working principle of OWC is based on the movement of air and water, so the design of the chamber in which fluids are flowing is crucial. This article has investigated the effect different geometric configurations have on the hydrodynamic effectiveness of the OWC WEC. According to the conducted research, a square shape is shown to be more effective than rectangular or cylindrical shapes, regardless of inter-device distance. Another important aim of this study has been to observe the relation between the spaces in the layout and generated power. In addition to mean power, the study has also explored efficiency, choosing to investigate inter-device lengths of 5D and 10D. Neither 5D nor 10D can easily be said to perform better, because their performances changed alongside other parameters. However, the OWC in the 2nd row was observed to perform better than the OWC in the first row. This means the OWC in the 2nd row is not affected negatively by the surrounding devices. Meanwhile, the OWC in the 3rd row was greatly affected and generated much less power. The last OWC in the 4th row was again not affected negatively. As demonstrated by the results from the 10D distance, this distance is noted to be large enough not to be affected by the other devices, because after the 1st row, the OWCs in 2nd, 3rd, and 4th rows produced practically the same power. The outcome may not have been optimal under

the particular conditions of this study. However, should this unaffectedness continue under real-world sea conditions, then this means any shape could be implemented for OWCs at this distance. Therefore, this distance could be beneficial for maintaining a consistent level of power generation. The goals and objectives of the research study have been effectively achieved based on the obtained outcomes. This study recommends upcoming research to be able to further refine the design configurations and distances. Future investigations may also involve incorporating actual sea conditions into their analyses.