Wave Energy Convertor for Bilateral Offshore Wave Flows: A Computational Fluid Dynamics (CFD) Study

Abstract

Human activities have adversely affected the Earth’s habitable environment. Carbon emissions and other greenhouse gas emissions are the primary cause of climate change and ozone layer depletion. In addition, the exponential decay of fossil fuel resources has resulted in the rising demand for renewable and environmentally-friendly energy sources. Wave energy is the most consistent of all intermittent renewable energy sources and offers a promising solution to our energy needs. This study focuses on harnessing offshore wave energy resources, specifically targeting offshore conditions with the highest energy density. A novel direct drive cross-flow turbine with an improved augmentation channel shape was designed and analyzed using commercial computational fluid dynamics software. The turbine’s base model reached a maximum efficiency of 54.3%, with 33.4 kW of power output at 35 rev/min and a 3.0 m head. Bidirectional flow simulations were carried out, and the peak cyclic efficiency was recorded at 56.8% with a 36.4 kW average power output. The nozzle entry arc angle of 150 degrees was found to be the most efficient, and the numerical simulation’s fully developed solution computed the flow behavior through the runner and nozzle under steady-state conditions.

1. Introduction

The energy consumption of the modern world is increasing at an exponential rate, creating a high demand for energy. As a result, and due to the rapid depletion of fossil fuel sources, there is a growing need for systems that generate power from green and renewable energy sources. According to OPEC (Organization of the Petroleum Exporting Countries), the demand for fossil fuels has grown rapidly in recent decades and is expected to continue increasing, reaching an estimated usage of 1.3 million barrels per day in 2016 [1]. While wind and solar energy have been harnessed to provide energy to national power grids, the potential of ocean energy, another green energy source, is yet to be fully explored. The oceans cover more than 71% of the Earth’s surface and contain 97% of the Earth’s water [2]. While many countries have attempted to harness energy from the ocean, most remain unsuccessful due to a lack of experimental data and research in the ocean energy sector [3]. However, successful projects producing green and renewable wave energy have been developed, mainly focusing on the oscillating moment of ocean waters. Ocean Renewable energy is the term used for all marine renewable energy fields available to harness from the ocean waters [2,4]. The fundamental energy forms include potential energy, kinetic energy, thermal energy, and chemical energy of ocean water [4,5]. Ocean waves, tidal streams and tidal systems, oceanic currents, ocean salinity gradients, and ocean thermal gradients require different technological approaches to harness energy. The available useful energy from ocean energy sources can be converted into a useful form of power, such as electricity for the grid or hydrogen production [6]. Similar to wind and solar power, offshore wave power has the potential to play a valuable part in the energy transition.

This research focuses on designing a higher-efficiency cross-flow turbine to be utilized as a Wave Energy Conversion (WEC) Turbine. The CFT was originally developed as a hydro-turbine by the Australian engineer A.G.M Mitchell in 1903. In the original theoretical design and experimental results, the efficiency reached 80%. The popularity of the turbine increased due to its ability to be efficient at the low head with a wide range of flow. The cross-flow turbine mostly operates in the hydro industry due to its simple design and ease of manufacturing. In a CFT, a water jet of rectangular cross-section passes twice through the rotor blades, which are arranged at the periphery of the cylindrical rotor, perpendicular to the rotor shaft. Water flows through the blades from the periphery towards the center, then crosses the open space inside the runner from the inside outwards. Energy conversion occurs twice: first upon impingement of water on the blades upon entry and then when water strikes the blades upon exiting the runner. Ref. [7] studied the use of two working stages and found that it provides no advantage, except that it is a highly effective and simple means of discharging the water from the runner. The main parts of a typical CFT in its parametric form are illustrated in Figure 1 [8].

The dual water passing illustrates the turbomachinery velocity triangle (Figure 2). The design of the turbine becomes complex if in a 3D plane. In the first stage, the maximum amount of energy is transferred to the blades, comprising about 75% of the transfer energy, while the second stage imparts approximately 25% of energy to the blades [10]. One-Dimensional (1-D) Fluid Machinery theory is the basis for the construction of CTF.

Most studies aiming to improve the performance of the CFT have concentrated on optimizing its geometric design parameters, such as the number of blades, as well as the runner diameter, width, nozzle, and nozzle entry arc, using laboratory-based experiments [11]. Other studies focused on performance improvement with regard to the turbine flow profile (Masuda et al., 1987) [12]. When the efficiency of the turbine was compared with the flow rate of other turbine types in Figure 3, the CFT had a flatter efficiency curve. Constant efficiency can be expected regardless of flow rate fluctuations. Due to unidirectional rotational capability, most wave power convertors use pneumatic turbines such as a well turbine. However, CFTs are also competitive candidates for wave energy applications, possessing the same capability to maintain unidirectional rotation regardless of the flow direction. CFT was utilized as a WEC; Y-H Lee et al. 2010 [11] and several other researchers conducted numerical and experimental studies to increase the turbine efficiency as a wave energy convertor in many cases [4,5,12,13,14,15,16,17,18,19]. A similar experimental and numerical study conducted by Korean researchers at KMOU increased the overall performance of CFT with a nozzle, improving the wave energy device efficiency by 5.49% [20]. The numerical study also proved that the augmentation channel shape is critical for operation, acting as a converging and diverging (vice versa) pathway for the fluid stream [14]. A more generalized method was adopted [15] for the turbine blade optimization process. An earlier study carried out by the same author proves “Banki–Michell Turbine” is a popular choice among researchers studying ocean renewable energy. The authors analyzed the above-mentioned studies and extended the literature survey further to meet the scope of the present study; a comprehensive, detailed list of previous research studies can be found in Appendix A. It is evident that most of the research work has been conducted for nearshore (regular) wave resources, which have lower wave heights and wave periods. The distinguishable disadvantage of regular nearshore waves is their significantly lower energy density compared to offshore waves. Appendix A contains the most recent global research (refer to Appendix A). Over 95% of the research provided in the table was biased toward regular wave conditions with significantly lower wave heights. The purpose was mainly to analyze the performance of a WEC device using numerical (CFD) or field-collected data (on-site wave data) under nearshore regions to analyze the behavior of a prescribed WEC system with key performance indicators (KPIs) such as structural, hydrodynamic or stability conditions. This is due to the short-term economic advantages of nearshore WEC projects. Nearshore wave energy analysis projects require a far smaller investment of time and monetary factors compared to offshore WEC projects. However, in terms of long-term energy profitability, the offshore WEC projects require an extended number of research studies to be carried out under different scenarios and compared to the nearshore WECs. This is a critical research gap, as fewer and fewer studies were conducted to explore offshore wave energy resource harnessing.

In addition to a suitable oceanographical location, the type of power take-off method used for WECs is important. It is difficult to assess the capability of CFT type as an offshore wave energy extracting device without a proper mathematical model based on numerical or experimental study results due to its technological novelty in offshore wave scenarios. By identifying this research gap between “CFT” and “offshore wave” resources, the present research focuses on developing a novel methodology for CFT to utilize as an offshore wave converter. It will be difficult to make accurate predictions, as the literature survey is limited to research findings for CFT working as an offshore wave energy converter. The authors present a comprehensive list summarizing previous studies’ merits, novelties and limitations, emphasizing the research questions the present research seeks to address. Moreover, none of the related listed research studies under different methodologies address methods of offshore wave resource harnessing. The present study is primarily based on the linear analysis and inviscid flow assumption method used to analyze the performance of the CFT behavior under offshore wave conditions.

Objectives of the Study

• To analyze the modified CFT design’s ability to harness offshore wave energy resources effectively with a higher energy density environment.
• To conduct research on the performance of an augmented channel cross-flow turbine with considerably higher significant wave heights and wave periods in offshore conditions.
• To design and optimize a geographical location data-specific cross-flow-type turbine and enhance its performance using Commercial CFD code.
• To evaluate the flow field: pressure and velocity variation of the turbine in a range of efficiency conditions.
• To study the effect of nozzle shape and preliminary turbine design on the performance of the turbine efficiency when operating under bidirectional conditions.

2. Materials and Methods

2.1. Offshore Wave Site Specification and Selection

The methodology initiated by the annual average capture width and power absorption of a turbine at a specific site is located. It involves extrapolating data from the available results and using power matrices to estimate the power absorption at different sea states. JONSWAP (Scheme 1) spectral analysis is also used to synthesize power matrices for irregular wave data. The study is based on a higher wave energy-dense site, and the design of the turbine unit is assessed based on the existing wave data extracted specifically for this case study, Ref.

Table 1. Wave specification data.

Wave Specification Data
Significant Wave Height (Hs)	3.0 m
Peak Wave Frequency	0.0177 Hz
Peak Wave Period	9.0 s
Peak Frequency	0.11 rad/s
Gamma Value	3.3

data. Figure 4a,b provides visual representations of the methodology and energy matrix for the specified model.

The initial estimation of the wave forces exerted on the turbine is generally measured as the capture width factor. Only a few data sources present the annual average capture width and wave resource. In some instances, it is necessary to extrapolate available results to estimate the Capture Width Ratio (CWR). The methodology is illustrated in Figure 4a. In some of the sources, power matrices are provided. In these cases, estimates of the annual average capture width at different locations are obtained. For this method, multiplying scatter diagram data with power matrices and summing the power contributions from each sea state generate the outcomes. In the work of Ref. A. Babarit et al. [22] site-specific data are extracted to maximize the capture factor for the estimated width/span of the turbine. The linear interpolation method was not used, as the bins of the power matrix chosen to match the given design criteria, the scatter diagram, were not required in this study.

In real-world applications, data pertaining to the capture width or power absorption shall be extracted, provided only a minimal number of sea conditions are assumed. In this study, an estimation of the power matrix was carried out according to Ref. [23]. Initially, we assumed that the wave power absorption was achieved by sea states with a peak wave period of fewer than 5 s or higher than 13 s. This allowed the study to use the linear interpolation method to extract power absorption as a function of the peak wave period. The power matrix combined with the JONSWAP (Scheme 1) spectral analyses for irregular wave data synthetized the power matrix. The design of the turbine unit for assessment was based on a specifically chosen higher wave energy denser site, using the existing data of G. Iglesias et al. [23]. In Figure 4a, a specified site is examined. The site detail energy matrix is illustrated in Figure 4b for the specification model.

Spectral Analysis

The wave H_s and the T_(s) values were analyzed with the Joined North Sea Wave Analyzing Project or short-term “JONSWAP” spectrum. The author uses the site-specific data, utilizing the wave occurrence and the climate condition approximated study carried out by Erik Vanem, Norway [24]. This method involves a long-term series of sea states that are defined by the directional wave spectrum based on the calculation method adopted by Uk-Jae Lee et al. [25]. The spectrum data and analysis are described below. The results of the sea state data are illustrated in Table 1.

JONSWAP Spectrum: General not for fully developed sea states:

$$S\left(\omega \right)=\frac{\alpha g^2}{{\omega }^5}exp\left[-\frac{5}{4}{\left(\frac{{\omega }_p}{\omega }\right)}^4\right]{\gamma }^{exp\left[-\raisebox{1ex}{${\left(\omega -{\omega }_p\right)}^2$}\!\left/ \!\raisebox{-1ex}{$2{\sigma }^2{\omega }_p^2$}\right.\right]}$$

(1)

with fully developed conditions, the spectrum can be analyzed as follows:

$$\begin{array}{c}{S}_{JS}\left(f\right)=0.3125H_s^2{f}_p^4{f}^{-5}exp\left[-1.25{\left(\raisebox{1ex}{$f_p$}\!\left/ \!\raisebox{-1ex}{$f$}\right.\right)}^4\right] \mathrm{where} \alpha =exp\left[-\frac{{\left(f-f_p\right)}^2}{2{\sigma }^2{f}_p^2}\right]\\ S_{JS}\left(f\right)=C\left(\gamma \right)S_{PM}\left(f\right){\gamma }^{\alpha }\end{array}$$

(2)

Such that the calculation is performed using the following formation.

$$S_{JS}\left(f\right)=0.3125H_s^2{f}_p^4{f}^{-5}exp\left[-1.25{\left(\raisebox{1ex}{$f_p$}\!\left/ \!\raisebox{-1ex}{$f$}\right.\right)}^4\right]\left(1-0.287ln\gamma \right)\times {\gamma }^{exp\left\{-0.5\frac{{\left(\frac{f}{f_p}-1\right)}^2}{\sigma }\right\}}$$

(3)

The spectral density against the wave frequency had a narrow peak band, and the other dataset for that region, passing sea states five, six, seven, and eight, were analyzed. The results refer to Scheme 1.

2.2. Design Condition of Turbine

Figure 4b marks the highest energy bins with the highest occurrence of black circles. A significant wave height of 3.0 m and a corresponding wave period of 9.0 s were chosen for the simulation. Calculations were performed for a range of periods and adjacent wave heights of 2.0 m, 4.0 m, 5.0 m, and 6.0 m, with varying wave periods of 5.0 s, 7.0 s, 9.0 s, and 11.0 s. By adopting the preliminary design procedure, a summary of CFT design values was obtained. The relevant parameters are listed in the

Table 2. Design parameters of the cross-flow turbine.

Turbine Design Parameters
Designed Head (m)	3 m
Flow Rate Q (m3/s)	10 m3/s
Outer Diameter (D2)	2.0 m
Inner Diam. (D1)	1.36 m
Length/Width (L)	7.5 m
N (RPM)	35 RPM
Number of Blades (n)	18
Radial Rim Width (a)	0.34 m
Blade Curvature (p)	217.6 mm
Blade Span (t)	0.348 m
Nozzle Entry Arc (λ)	90~150 deg
Estimated Efficiency	75%
Blade Angle	16 deg
Entrance Angle	30 deg
Exit Angle	90 deg

and are considered for the base model of the analysis.

The CAD model for the turbine in the design dimensions was illustrated based on the derived parameters. Figure 5 illustrates the CAD model of the runner created for the test case. The nozzle shape was designed according to the shape optimization procedure involved. The relationship between the geometric functions allows for the smooth curve creation of the nozzle upper wall, as it becomes a convergence and divergence nozzle and vice versa. The nozzle is placed on the runner at the designed angle. The nozzle upper wall curvature is estimated to be tangent with the lower curve in Figure 6 [21] to satisfy the relationship between the ratios of λ/S₀ in the optimal range, as shown in Figure 6. However, the bottom of the nozzle curvature shape was developed by the author based on the cross-flow turbine nozzle design experience obtained during the case study.

The turbine’s full span/width is 7.5 m, and the width or span is the primary factor in the energy capture factor for wave energy converters. The Capture Width Ratio (CWR) for a given wave energy device should be considered for a set width of the device. The CWR primarily measures the hydrodynamic efficiency for a given device in terms of energy but not in cost aspects, as demonstrated by the studies carried out by Ann Dallman et al. [22]. The device in this study aims to develop a novel CFT applicable to fixed-structure oscillating water column WECs. Referring to Figure 7, the turbine model examined in the present study reaches greater than 40% CWR.

The base model was designed for the optimization process using the steady-state method of the commercial CFD code Ansys CFX^® 17.2v. The numerical calculations were carried out for a turbine with a decreased length to reduce the computational cost and time. Transient (time-averaging) calculations were carried out with varying conditions to study the flow behavior inside the turbine runner and nozzle. Figure 8 illustrates the base model nozzle shape of the turbine, and Figure 9 illustrates the entire turbine with both nozzles fitted for bidirectional flow analysis. The upper nozzle entry arc angle will be altered in four cases, as 150 deg, 130 deg, 110 deg, and 90 deg nozzle entry arc angle variants. The most efficient nozzle shape will be considered for transient calculations as the geometrically optimized model.

Mesh Generation

The mesh for the full turbine setup was generated using Ansys ICEM CFD^® 17.2v software, with a fully structured hexahedral mesh composed of 2.99 million elements. Our simulation focused on a 0.5 m strip along the length of the turbine, which required mesh refinement during optimization to ensure an appropriate mesh size.

Figure 10 and Figure 11 illustrate the mesh generated around the turbine runner, which was modeled as a single runner section consisting of 18 symmetrical blades. Nodal point details were exaggerated to accurately capture the blade curvature shape. Figure 12 exaggerates the finer mesh elements and multiple prism layers to accurately capture boundary-layer effects around the runner blades and to enable the necessary fluid–blade surface interactions to produce accurate numerical solutions. In addition to blade meshing, extra-fine mesh regions with hexahedral elements were used around the nozzle arc that comes into contact with the runner; Ref to Figure 13 and Figure 14. The mesh nodal distribution was kept equal all around the volume mesh to ensure accuracy in this highly turbulent flow environment. Figure 15 and Figure 16 illustrate the resulting mesh around the nozzles.

The non-rotating sections of the turbine and nozzle parts contained a mesh element count of 0.514 million hexahedral elements. The meshing of the curved geometric locations of the nozzle was carried out with extra caution to ensure the accurate capture of boundary effects using prism layers. In Figure 17, the blade mesh is shown in green, while the rest of the regions are depicted in the default gray for clarity and contrast. Overall, the mesh was carefully generated to capture all necessary fluid dynamics phenomena and surface interactions. The final accuracy of the convergence will depend on the quality of the fluid domain, which will ensure accurate computational results.

2.3. Turbine Optimization Numerical Setup (ANSYS CFX v. 17.2)

The turbine optimization setup contained augmentation channel entrance passages. An equal 12.5 m long passage (see Figure 15a) was used to receive a smooth fluid flow towards the inlet/outlet of the nozzle and the runner sections. This was mainly carried out to ensure smooth fluid flow without excess turbulence and eddies. The same mesh model would be utilized in the time-averaging simulation but with changes to the inlet boundary conditions with CFX command language code to generate a wave profile. Similar to steady-state analysis, the exterior walls and turbine blades were modeled as no-slip wall boundary conditions. The domain interfaces were connected with the automatic mesh connection (GGI) method. The total flow time for the transient simulation (time averaging) model was set to 30 s, with a time step of 0.01 s for extended accuracy. The numerical modeling was performed and was validated by the study “Survivability of wave energy converters using CFD” (E.J. Ransley et al. [27]) using the numerical wave tank (NWT) CFD modeling method. All domains for steady-state and transient analyses were stationary except for the turbine domain. The simulation was performed based on the Reynolds Averaged Navier–Stokes (RANS) equation model provided by the SST (Shear–Stress–Transport) turbulence modeling scheme. Time discretization of equations was achieved with the implicit second-order Backward Euler scheme (see Figure 15b).

2.4. Governing Equations of CFD Code

CFD Code

In computational fluid dynamics (CFD), highly complex concepts are translated into computational algorithms and solved using numerical methods. The focus of this paper was to illustrate and achieve research targets using CFD codes of commercial software packages, and a brief overview of the governing equations was presented [27,28,29]. CFD codes are structured around numerical algorithms that can handle fluid flow problems (Figure 16) [26], and all codes contain three main elements: a pre-processor, a solver, and a post-processor. The solution to a flow problem (velocity, pressure, temperature, etc.) is defined at nodes inside each cell. The accuracy of CFD results depends on how well the solution converges to the final non-varying pattern, which is governed by the quality of the mesh or grid generated (number of nodes or cells) and how refined or coarse the mesh structure is. The governing equations of fluid dynamics are derived from the laws of physics in a simpler manner. A defined fluid element with six basic sides with δx, δy and δz is required for the derivation of conservation laws with a fluid element.

The conservative or divergence form of the system of equations governs the time-dependent three-dimensional fluid flow and heat transfer of a compressible Newtonian fluid:

$$\begin{gathered} Continuity \frac{\partial \rho }{\partial t}+div\left(\rho U\right)=0 \\ x-momentum; \frac{\partial \left(\rho u\right)}{\partial t}+div\left(\rho uU\right)=\frac{\partial p}{\partial x}+div\left(\mu .grad.u\right)+S_{Mx} \\ y-momentum; \frac{ \partial \left(\rho v\right)}{\partial t}+div\left(\rho vU\right)=\frac{\partial p}{\partial x}+div\left(\mu .grad.v\right)+S_{My} \\ z-momentum; \frac{\partial \left(\rho w\right)}{\partial t}+div\left(\rho wU\right)=\frac{\partial p}{\partial x}+div\left(\mu .grad.w\right)+S_{Mz} \\ Energy; \frac{\partial \left(\rho i\right)}{\partial t}+div\left(\rho iU\right)=-p.div.U+div\left(k.grad.T\right)+\varnothing +S_i \end{gathered}$$

(4)

These Navier–Stokes (NS) equations are computationally coded using computer programming languages to create a comprehensive software code. In CFD codes adopted for the simulation of wave energy converters, the turbulence modeling solver algorithm remains unchanged, as shown in Refs. [26,27,28,29]. The overall flow process of the research presented in this study is depicted in Scheme 2: Research Methodology, which is the formal path followed in this study.

3. Results and Discussion

3.1. Mesh (Computational Grid) Independence Test

The suggested CFD model was validated using a similar study carried out by Ref. [20] and numerical wave tank (NWT) model calculations from Refs. [8,19,22,23,24,25,26,28,30,31]. The model validation results showed a difference of 0.2%, and the results agreed with the numerical model used in the present study. The grid independence test of a CFD model is vital, as the performance of the system often relies on the mesh element count. The same hexahedral grid was used during the testing, and the nodal count was increased at each stage by increments of 20% to rely on a reliable mesh convergence parametric test. The results are shown in Figure 17.

The results at the initial stage, with mesh elements in the range of 1.5 million, showed higher efficiency due to the coarse mesh. As the mesh became finer, the efficiency plot at the turbine tended to decrease at a certain point, and increasing the number of elements further did not affect the turbine efficiency. The final two stages, with 3.5 million to 4.2 million elements, did not affect the performance. The efficiency difference for the latter two mesh cases was negligible. The y+ value of 3.14 and the turbulence model of k-Omega SST proved reliable results in the y+ value of 3.14. Initially, in the coarse mesh, the y+ value ranged from 143 to 78 and provided an inaccurate y+ value for k-Omega modeling. The efficiency steadily rested at the margin of 54.33% for the base design. This is because in Finite Volume Analysis (FVA), as the mesh becomes finer, the accuracy of the solution of the governing equations tends to provide more reliable results. The rest of the analysis used a mesh element count of 3.56 million models selected from a much finer model with 4.2 million elements due to the negligible amount of performance variation, as the mesh with 4.2 million elements requires much higher computational power and longer duration.

3.2. Base Turbine Performance

The preliminary design of the turbine was tested at a constant head of 3 m and a designed speed of 35 RPM using the CFD technique. The present study caries on steady state method and Transient method. And both boundary conditions for each simulation stage is presented in

Table 3. Numerical simulation setup parameters.

Simulation Type	Steady State	Transient (Time Averaging)
Mesh Type	Hexahedral	Hexahedral
Number of Nodes	3.56 million	3.56 million
Time step	Automatic	30 s
Turbulence model	Shea Stress Transport (SST)	Shea Stress Transport (SST)
Fluid Phase	Single Phase (Water)	Single Phase (Water)
Mesh Motion	Stationery	Moving
Inlet boundary	Wave inlet	Wave inlet
Outlet boundary	Opening	Opening
Turbine Rotational Speed	20, 30, 35, 40, and 45 rpm	25 and 35 rpm
Rotor–Stator Interface	Frozen Rotor	Transient Rotor Stator

. Base turbine performance refer to the steady state calculation results. The maximum performance of the turbine yielded an efficiency value of 54.33%. This efficiency was reached with a considerably lower head, as CFT designs tend to provide low efficiencies but at much lower head values. Some CFT designs operate at much lower efficiencies, ranging between 28% and 35%, with bidirectional flow operation. In the application of wave energy extraction, the efficiency of double-sided flow path designs achieves lower values. The base model was initially simulated with two different CFD codes: Ansys CFX and Star CCM+. The mesh was created by ICEM CFD, and Star CCM+ allowed the import of mesh types of similar platforms. The results from both platforms yielded very similar results (

Table 4. Base Simulation Results (Steady State) for the CFT.

Criterion	Numerical Value at BEP
Head	3.00 m of Head
RPM	35 RPM
Efficiency	54.33%
Y+ Value	3.14156
Mass Flow Rate	2025.56 kg/s
Extracted Power	33.366 kW (0.5 m Width)
Real Scale Model Power	500.49 kW
CFD Code	Efficiency
Ansys CFX	54.33%
Star CCM+	52.38%

), with a difference of approximately 1% margin.

3.2.1. Turbine Performance Results and Analyses for Steady State

The numerical modeling was scaled down to a 1:15 scale of the length of the CFT front profile diameter, which did not change. Initial stage simulation carried out to extract the base turbine performance under steady state conditions This reduction in the length of the turbine allowed faster and more accurate numerical calculations with a finer grid. A 0.5 m wide strip of runner produced 33.366 kW, and the real scale model, which was 7.5 m wide, produced 500 kW of power, summery of the results are tabulated in Table 4. The steady-state results of the turbine were used to numerically evaluate its performance. The efficiency versus the turbine RPM was plotted in Figure 18, and the turbine power variation versus RPM was plotted in Figure 19.

In the single-directional flow analysis, the CFT behaved similarly to the rest of the CFTs. At both lower and higher RPM values which were the inverse of the turbine’s best operating point, the efficiency dropped considerably. The numerical calculations proved that the turbine operates at the BEP, as the turbine design point was correct and agreed with the preliminary design procedure. Once again, the turbine efficiency was plotted for a varying and vast range of rotational speeds ranging from 15 RPM to 55 RPM; the spread of efficiency values can be used to analyze the effective point of the operation. The lowest efficiency was recorded at 15 RPM, which was 18.3%. The turbine operated at off-operation points, resulting in considerably lower power production.

In Figure 19, the turbine produces 33.366 kW of peak power at 35 RPM, which is the maximum power extraction point for this turbine. The BEP lies in this point, demonstrating the accuracy of the results. Even though the CFT tends to operate with a flat efficiency line for a larger flow rate range, the bidirectional application-oriented turbines tend to deviate from this argument. The lowest power rating, 13.4 kW, was produced at 15 RPM due to the very low rotation rate and the turbine’s lack of ability to cope with the flow. The reverse of this case was observed at 55 RPM. These far points are considered the turbine’s off-operational points. The turbine was also tested by changing the head variant with different RPM values (Figure 20); the graph tends to agree with the cumulative results, as the turbine design condition was held constant with 3 m of head and a 35 RPM value. The turbine achieved optimal performance at the 35 RPM designed value. When the turbine RPM was at a constant fixed rate, as the head varied with time, the behavior of the turbine can be predicted by the graph in Figure 20. The accumulated results show that the efficacy increases in a head range from 4 m to 5 m at the full spectrum of the rotational speeds.

3.2.2. Geometric Modification Results: Nozzle Entry Arc Angle

The nozzle entry arc angle (δ) was optimized to analyze its effect on turbine performance. The angle was varied from 90° to 150° in increments of 20°. The best performance was observed for the base case with a 150° entry arc angle, achieving a peak efficiency of 54.33% at 35 RPM. The lowest performance was obtained for the 90° nozzle shape model. A larger entry angle exposes more blades to incoming flow and imparts more energy onto the blades, as shown in Figure 21. This increases power production while keeping the inlet conditions constant. The 150° entry arc angle model, referred to as the base model, is used for time-averaging (oscillated) calculations to integrate with wave-energy applications. Figure 22a–c illustrate the flow fields of velocity vectors, velocity streamlines, and pressure contour variations for all the nozzle shapes analyzed. The optimized results were compared with the base model using the 150 Arc entry model, which had the highest efficiency and performance. The analysis is based on CFD post-processing.

3.3. Flow Field Analysis of the Turbine

The CFD post-processing results are analyzed with flow field variables. The flow path through the turbine runner and the blades affects the power extraction from the fluid to the blades. Low-head turbines behave more like impulse types. The momentum transfer from the fluid to the blades (see Figure 23) is shown in the velocity vector field of the turbine and through the nozzles.

As flow passes from the channel into the nozzle, it converges with higher efficiency and is smoothly streamlined through the nozzle arc interfacing runner. This optimized nozzle shape allows the flow to strike at the designed flow angle to the runner blades. Figure 23 and Figure 24 demonstrate that the nozzle converges the flow with a high impact on the runner blades. The velocity is decreased, and the flow pressure field is increased where the nozzle arc strikes the blades. These phenomena impart the maximum amount of pressure energy of the flow towards the blades as kinetic energy of striking.

CFTs behave similarly to impulse turbines, and depending on the degree of reaction, they will start to exhibit reaction-type behavior in some regions of the runner. In the circular section A in Figure 25, the flow reaches the highest velocity at the striking blade point, and at point B, the flow achieves a higher velocity at the exit from the second stage of the runner. This phenomenon occurs because even at the second stage of the flow pass, the flow field transfers much of the available kinetic energy to the second row of blade sets in the runner. This occurs because of the complex flow behavior resulting from the nozzle shape, as explained in the illustration of the flow path direction in Figure 26.

With referring to Figure 26, this high-velocity flow path starts at the back end of the nozzle from the lower side, and it has little directional or angular movement when traveling from the first stage to the second stage. Flows have minimal energy waste when making a minor upper directional movement and leave the second blade stage by striking the blades, releasing kinetic energy. When a fluid flow path experiences abrupt flow path changes, more energy is wasted, and this clearly reduces the available energy of the flow path, affecting its ability to carry out useful work. This is another reason for the higher efficiency of the turbine compared to similar CFT model types. The flow field also has many regions of relative impotence. The complex flow behavior in rotodynamic machines is difficult to analyze due to the high turbulence field. The flow path in the non-rotating region in the middle of the shaft has many regions of flow separations and flow vortices. Figure 27 refers to the above flow separation regions that exhibit blade losses in real operation conditions more than in the CFD simulations.

These flow separation regions are heavily influenced by the pressure variation inside the runner and in the non-rotating region in which a high-velocity jet of flow passes through. The adjacent regions counter-rotate with the rotational direction of the runner, causing low-velocity particles to separate from the mainstream. The flow separation is visible in Figure 27 highlighted at points C (black circle) and D (black box), and in real field conditions, the rate of change of flow separation leads to vortex formation. Point E represents the relative eddy phenomena in the blades, which is essentially the blade circulation. This phenomenon can be captured by the CFD method, primarily due to the clarity of the generated mesh and nodal distribution. Additionally, the implementation of the prism layer near the blade surfaces is crucial for capturing the fluid flow phenomena that affect performance. It is important to analyze the aforementioned effects to effectively optimize turbine design.

Eddy formations can be clearly observed near the blades and between two adjacent blades. Figure 28 illustrates the 2nd stage of blades where the eddy formation is greater, the circular section F; main eddies in between the two successive blade passages, and circular section G; the eddy structure very close to the trailing edge of a 2nd stage runner blade. Figure 29 presents the 3D illustration of the complex flow filed in between two successive blade passages and the relative eddy formation on the exit of the flow from the turbine. These formations occur due to the relative eddy phenomena caused by the frictionless fluid passing through the runner blades without rotation. Since the runner rotates at a positive angular velocity, the fluid must have a rotation relative to the impeller of the opposite direction, forming relative eddies (Figure 30). The size of these eddies may vary with an increase in the blade gap; thus, it is crucial to minimize blade losses in turbine design as they are considered a form of slip loss.

Another important flow field phenomenon is blade circulation, which is clearly visible and can be captured by the CFD code. The pressure difference on the two sides of the blades causes this slip formation. As shown in and Figure 31 (simulation) two yellow arrows on either side of the blade illustrates this phenomena, and Figure 32 (theoretical) the sides of the blades have nearly different pressure gradients, and hence circulation occurs at the blade. The pressure in the fluid on one side of the blade is greater than the other, causing the velocity near the backside of the blade to be greater than that near the forward side. This velocity difference on the two sides of the blade creates blade circulation associated with lift. However, the non-uniform velocity distribution is responsible for the mean direction of flow leaving the runner, which deviates from the ideal flow situation. This effect reduces the tangential velocity component (velocity of whirl), which is therefore known as slip. It is crucial to analyze this phenomenon using CFD in turbine optimization.

Two fundamental reasons explain the actual energy transfer achieved by a hydraulic machine, which is reduced compared to the prediction made using Euler’s Equation. Firstly, the velocities in the blade passages and at the impeller outlet are non-uniform due to the presence of the blades, and the real flow is three-dimensional (3D). This diminishes the velocity of the whirl component, theoretically reducing Euler’s head in the design. This effect is not due to friction and does not represent a loss but follows from the ideal flow analysis of pressure and velocity distributions. Secondly, in a real turbine runner, energy is lost due to friction, separation, and wakes associated with the development of boundary layers. The flow field also has little or no flow in the upper section of the turbine runner due to the middle water jet passing with very minimal flow directional change. The upper region shows recirculation flow due to vortex formation. This also affects the bottom half of the turbine runner and the blades. Additionally, the fluid stream is not significantly distorted in the bottom portion of the turbine, compared to the much more distorted flow in the upper part of the blades and nozzle. Figure 33 shows these two regions separately.

The flow in the bottom half of the runner section is organized and streamlined, with the water jet dividing the runner into two halves. This is greatly enhanced by the fact that (Figure 34) bottom of the blades entering the 1st stage runner blading region has less formation of relative eddies. This influences the fluid particles to adhere to each other, causing pressure distribution in two regions. As a result, there is less formation of eddies and more transformation of pressure energy to the blades, partially in the specified region.

The flow through the upper nozzle (outlet nozzle) has greater flow separation and backflow regions, as shown in Figure 35. CFD post-processing indicates that the upper nozzle with a black border marking signifies the reversal of the flow field. This phenomenon is due to the decrease in the speed of the fluid in a particular region, and the boundary layer thickness increases significantly. The adverse pressure gradient has increased such that the flow is forced back against the actual flow direction. This results in the backflow area detaching the laminar flow itself from the contour at this point.

3.4. Transient Calculations Results and Bidirectional Flow Analysis

Transient calculations are required to analyze the bidirectional performance of the turbine [27,28,29] when integrated as a WEC turbine. The bidirectional flow is created by an Ansys CFX CEL expression attached to the inlet boundary conditions. The wave period is 9.0 s, and the wave height is 3.0 m. Numerical simulations are conducted for a 45 RPM case to compare the results. The wave-energy calculation is based on the equations below. The available water power (P_WP) is given by the following Equation (7), where C_p is the phase velocity, C_g is the group velocity, g is the acceleration due to gravity, ρ is the water density, and P_Wave is the wave energy flux or wave power.

$$c_p=\sqrt{\frac{g\lambda }{2\pi }\tan \mathrm{h}\left(\frac{2\pi h}{\lambda }\right)}; \mathrm{Wave} \mathrm{Celerity}$$

(5)

$$c_g=\frac{1}{2}{C}_p\left\{1+\frac{4\pi h}{\lambda }\left(\frac{1}{sinh\left(\raisebox{1ex}{$4\pi h$}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.\right)}\right)\right\}; \mathrm{Group} \mathrm{Velocity}$$

(6)

$$P_{Wave}=\frac{1}{8}\rho g{H}^2{c}_g; \mathrm{Wave} \mathrm{Power}$$

(7)

$$P_{turbine}=\tau \times \mathsf{\omega }; \mathrm{Turbine} \mathrm{Power}$$

(8)

$$\eta =\frac{P_{turbine}}{P_{Wave}\times W_G}; \mathrm{Efficiency}$$

(9)

$$\begin{gathered} Q=\frac{V}{T}=\frac{A_{cs}\times \left(2\Delta Y\right)}{T} \\ {P}_{WP }=\rho gQ\Delta H \end{gathered}$$

(10)

$$P_{Avail}=P_{wave}\times W_G$$

(11)

$$C_f=\raisebox{1ex}{$P_{WP}$}\!\left/ \!\raisebox{-1ex}{$P_{Avail}$}\right.$$

(12)

For the given period T, there are two oscillations in the rear wave maker; that is, the water level rises to a maximum and falls to a minimum, displacing twice the volume. Thus, ΔY is multiplied by a factor of “two” in Equation (12). A_CS is the rear chamber cross-sectional area, which was 1.15 m². Primary energy conversion is then obtained.

The Bidirectional Flow Results (Transient Simulation)

In the performance analysis, a simulation run time of 6 s was counted with 0.5 s periods. In each time period, the wave power and water power imparted by the wave motion were calculated, and the efficiency graph for the cyclic period was obtained (see Figure 36).

In these findings, the cyclic efficiency of turbine power for a fixed turbine speed increases as the wave period increases. The maximum cyclic efficiency for 25 RPM is 40.59%, and the minimum is 9.08%. Similarly, for 35 RPM (see Figure 37), the maximum efficiency is 56.83%, and the lowest is 11.55%. The fluctuation of the turbine torque (see Figure 38) with time shows similar behavior with the bilateral wave flows. The maximum average cycle efficiency was achieved with 35 RPM at 36.52%. The turbine extracts more energy from the incoming and outgoing flow through the augmentation channel effect. The results indicate that higher power is achievable from incoming waves with longer wavelengths. However, the efficiency increases with increasing rotational speed. It reaches a maximum and decreases from the best efficiency point onwards. In this study, there were 18 blades. The variables were wave period and turbine speed. Under the actual conditions, there is a point scale in which the turbine reaches the highest efficiency. The present study results were compared with Simonetti et al. (2017) [29], and a similar result pattern was obtained. The optimization method and results agreed with the results of [29] and [20].

The flow is constant at a given wave period. When the turbine RPM is greater, water passage across the turbine blade is unable to impart kinetic energy effectively because the time frame in which two successive blades come in contact with the fluid is minute. When the turbine RPM reaches lower values, the water passage rushes through the blade stages and imparts less kinetic energy. It is critical to obtain the speed at which the turbine produces maximum power with peak efficiency under given wave conditions. The peak in efficiency indicates that the interaction between the turbine and flow is maximized at the optimum rotational speed. The turbine flow field was analyzed in 0.5 s intervals, and the velocity vector changes varied significantly for each frame. Figure 39 illustrates the turbine’s internal flow behavior under the bidirectional influence.

The flow in Figure 39a–g initially flows from the front passage to the downstream passage (from the bottom nozzle to the upheld nozzle via the runner). In the time point of 4 s and 4.5 s, Figure 39g,h the flow almost comes to a complete stop, and then the pressure field changes direction. From Figure 39i, with a time of 5.0 s, the flow completely stops and begins to reverse. And from the velocity vector in Figure 39j–l (time 5.5 s to 6.5 s) shows the flow is completely reversed its flow direction, it flows in the opposite direction. While the turbine runner does not change the direction of rotation regardless of the bilateral flow.

4. Conclusions

The present research study focused on the design and performance evaluation of a cross-flow turbine designed for a wave energy harnessing application. The study also examined the impact and benefits of wave energy on the global energy crisis and identified potential sites with higher wave energy. The study was aimed at a specific offshore location based on available wave data, with a chosen wave height and period selected as design input data for wave energy converters (WECs). The cross-flow turbine was designed based on a 3.0 m significant wave height and a 9.0 s wave period. The turbine design used a preliminary calculation procedure originally adopted by Banki–Mitchell as a hydro-turbine, but in this study, the turbine design was improved to suit offshore wave conditions. Flow-guiding augmentation channels were designed to improve turbine efficiency. The turbine had a 2.0 m outer diameter, a 1.36 m internal diameter, 18 blades and a rotational speed of 35 rev/min.

The operational performance of the cross-flow turbine was analyzed numerically using the ANSYS CFX 17.2v CFD code. Initially, the CAD model and mesh were generated, and a mesh sensitivity study was conducted in steady-state conditions to reduce the computational effort. The base turbine was analyzed with four different nozzle shapes and with varying nozzle entry angles of 90°, 110°, 130°, and 150°. The simulations were carried out for different RPM values from 15 RPM to 55 RPM, as offshore larger turbines’ rotational speeds are much lower compared to traditional turbines. The head was also changed from 2.0 m to 6.0 m. The 150° nozzle outperformed all other nozzle shapes. The highest efficiency for the simulated 0.5 m turbine strip was 54.3%, with a peak power output of 33.4 kW at the 3.0 m head. The simulation results proved that the correct preliminary design procedure was adopted. Computational flow analysis was carried out for the turbine runner and nozzle. The flow path had fewer directional changes when passing from the first stage to the second stage through the non-rotating passage, imparting maximum energy flow to the turbine runner blades. The flow separation, vortex formation, blade circulation and relative eddy formation were observed. However, the viscous flow phenomena visible during CFD post-processing were not considered during performance calculations. Slip and cavitation effects were also not considered in the present model but were identified during CFD post-processing. These factors are minor effects as far as offshore wave turbines are concerned, and they are more often considered in hydro-turbine design analysis.

With the modifications carried out in steady-state simulations, the real behavior of the system was analyzed using “transient simulations” to study the bidirectional performance of the turbine. CFD was used for the 3.0 m equivalent head and 9.0 s wave periods. The wave formation was carried out using ANSYS CFX 17.2v Command Expression Language (CEL). Transient simulations or calculations are much more similar to the actual behavior of the turbine under real wave flow conditions. In fact, two base cases were analyzed at 25 RPM and 35 RPM. The highest cyclic efficiency was achieved at 35 RPM with a performance of 36.5%.

The novel cross-flow turbine provides a comparative difference in performance with similar case studies, but none of the related case studies pertained to offshore wave conditions. This is a major drawback. The authors identified this as a major research gap since few numerical bidirectional turbines were researched for offshore wave conditions. The proposed design and CFD model can be improved. Limitations of the present study include the inability to quantify the performance of the turbine for various sea states with regular and irregular wave patterns. It is necessary to carry out simulations that consider different wave conditions. Another limitation is the lack of experimental results that validate the model’s performance with the CFD results.

In future research, it is strongly recommended and proposed to conduct a scaled-down experimental producer for this CFD model-based study to investigate real-world test cases. Further research is needed to fully understand the long-term performance and durability of CFTs in offshore wave power applications, as well as their potential impacts on marine ecosystems. It will be wise to identify the scale-down effects and performance differences based on the CFD results to conduct an accurate experimental setup. The authors propose a novel numerical (CFD-based) method to extract offshore wave energy resources utilizing a CFT, and the orifice geometry is improved to increase efficiency. The research can be expanded for different further studies.

The authors propose another future study to combine the scaled-down experimental work with oscillating water column (OWC) wave-energy devices. The present research study can be tailored using a power take-off (PTO) method for hydraulic PTO OWC research. OWC technology is used during bilateral wave generation. In the real operational condition of the OWC structure, the air compressing chamber and the section with the water column create a pressure drop across these planes, affecting the oscillating amplitude of the system. This repetitive cyclic process becomes the key parameter in the generation of power, which is the PTO (power take-off) device. However, in practical situations, it is difficult to construct, install and experiment with the turbine behavior in a scaled-down experimental setup because of the relatively complex blade shape and the limited number of resources at the lab scale. The “Orifice Plate” based pressure difference method is proposed as the ideal substitute sub-device in place of the actual turbine, investigating similar performance characteristics such as the pressure drop and mass flow rate under similar operational conditions. This ideology can be implemented to use for validation of the actual turbine setup. These values are useful during one-to-one experimental studies, helping to identify the level of pressure intensity to expect in real oceanic offshore conditions.