Three-Dimensional Aeroelastic Investigation of a Novel Convex Bladed H-Darrieus Wind Turbine Based on a Two-Way Coupled Computational Fluid Dynamics and Finite Element Analysis Approach

Abstract

H-Darrieus vertical-axis wind turbines (VAWTs) capture wind regardless of its direction and operate effectively even in challenging and turbulent wind conditions. As a result, the blades operate under erratic and intricate aerodynamic loads, which cause them to bend. The performance of the H-Darrieus rotor will therefore be impacted by the blade’s deflection. This study aims at investigating the dynamic aerostructure influence on a novel convex-bladed H-Darrieus geometry. The results are compared to a straight-bladed baseline rotor. To do so, a two-way fluid–structure interaction (FSI)-coupled approach is performed to accurately address this issue. This approach allows for the simultaneous resolution of the fluid flow around the rotor and the mechanical structure responses inside the blades. The turbulent flows are resolved using the k-ω-SST model together with the URANS equations through computational fluid dynamics (CFD), while the structural deflections of the blades are assessed using finite element analysis (FEA). The results show that the performance of both H-Darrieus turbines decreases with increasing deformation. In addition, the study found that the carbon fiber composite (M1) material has the least deformation in the convex and straight blades, with values of 9.1 mm and 20.331 mm, respectively. The glass-fiber-reinforced epoxy composite (M3) material shows the most significant deflection across both types, with displacements of 32.50 mm and 73.78 mm for the straight blade and 19.02 mm and 43.03 mm for the convex blade. This study also reveals that the straight blade has a peak displacement of 73.785 mm when using the M3 material at TSR = 3, while the convex blade has a minimum displacement of 20.331 mm when using the M1 material, highlighting the varying performance characteristics of the materials. The maximum stress observed occurs in the straight blade, registering at 324.1 MPa with TSR = 3, which aligns closely with the peak displacement values, particularly for the aluminum alloy material (M2). In contrast, the convex blade made from the first material (M1) exhibits the lowest stress levels among the tested configurations.

1. Introduction

Wind power is one of the most promising renewable energy sources worldwide as a clean and sustainable alternative to conventional fossil fuels [1]. Using wind turbines, kinetic wind energy is converted to mechanical energy and then transformed into electrical form. While there are many different types of wind turbines, the vertical-axis wind turbine (VAWT) is currently attracting more attention based on its characteristics which enable it to operate either in offshore or urban settings, offering benefits including reduced noise and vibration [1,2,3,4,5].

However, one of the main challenges in the H-Darrieus VAWT design is to ensure that the blades withstand the various unexpected aerodynamic forces that it may encounter during operations [6,7]. The operational wind speed range and the more common blade profiles that enable effective wind energy capturing are taken into consideration when determining the best design parameters for this type of wind turbine.

Indeed, the deformation of the blade causes changes in the flow field around the turbine, which in turn affects the aerodynamic forces acting on the blade. For this purpose, from the earliest wind turbine design stage, it becomes necessary to carefully select the appropriate materials for the build-up of wind turbine parts. Currently, lightweight materials are frequently used in modern design, which inevitably leads to the blades’ distortion and breaking [8,9]. The use of this kind of material enhances the blades’ ability to adapt to changes in conditions, such as strong winds and storms, but it can also cause unwanted fluctuations that disrupt the flow and cause instability in the flow structure, especially behind the rotors. These factors make it essential to include material elasticity and deformation while investigating wind turbine performance.

In this context, recent works have been carried out to study the behavior of wind turbine rotors, taking flexibility into account, using various approaches. Krishna et al. [10] conducted a numerical investigation utilizing dual-channel fluid–structure interaction (FSI) analysis to predict the hydro-elastic response of a marine propeller. This approach enabled the projection of maximum stress and strain values generated during transitional phases, thereby enhancing prediction accuracy. In the same way, Krishna et al. [10] developed a two-way fluid–structure interaction (FSI) approach for modeling the transient response of a vertical axis tidal turbine rotor characterized by variable input. This study delved into mesh deformation techniques for accommodating substantial displacements, revealing that a two-way FSI approach improved design robustness compared to conventional one-way FSI. Implementing a practical two-way FSI with optimized processing resources necessitated innovative techniques. Results indicated that two-way FSI engendered elevated stresses and deformations, with fluid flow simulation using CFX offering superior precision compared to one-way simulations [11,12,13,14].

Wendi Liu and Qing Xiao [15] studied the bending and twist deflection characteristics of a three-dimensional VAWT mounted with two flexible blades that permit passive spanwise deformation numerically under erratic external stresses. This study showed a strong relationship between the turbine’s tip-speed ratio and the bending and twist deflection peaks. Furthermore, they observed that, in comparison to less flexible or stiff blades, very flexible blades showed non-uniformly distributed structural stresses around the strut position, resulting in decreased energy extraction efficiency.

Using a flexible blade inside a water tunnel, Eldemerdash and Leweke [16] conducted an experiment aimed at understanding how flexibility influences the wake behind the blades and how hydrodynamic parameters affect blade deformations. Significant bending in forward motion and large-amplitude oscillations in reverse motion were found, according to their measurements of the flow field surrounding the blades and the deformations.

Fakhfekh et al. [17] explored the mean deflection progression of a flexible rotor blade employing a unique coupled model in which the Navier–Stokes equations and linear elasticity equations are resolved simultaneously. The authors validated the numerical model against experimental data, enabling the vorticity field to be compared with that of a rigid rotor. The study highlighted the non-monotonic relationship between blade deformation and rotation frequency, revealing the potential emergence of vortex ring states within a flexible rotor under certain conditions.

According to all the aforementioned works, using flexible straight blades extends the life of wind turbines and increases their blade lifespan in comparison to rigid blades. Furthermore, it has been demonstrated that flexible blades produce less stress concentrations than rigid blades, which lowers the risk of fatigue-related blade failure. These studies show that employing flexible blades in H-Darrieus-type VAWT rotors can have certain advantages.

Souaissa et al. [18,19] proposed novel convex- and concave-shaped blades where they numerically demonstrated their potential aerodynamic benefits in comparison to the classic straight blade at a large range of wind speed. They demonstrated that convex-shaped blades offered the best performance among the three evaluated configurations. However, in these works, the authors did not examine the impact of these configurations on the turbine’s aeroelastic performance. In fact, the choice of which materials may be used to manufacture the blades has a straightforward impact on their weight, stiffness, and durability.

Through aerodynamic analysis, earlier studies have shown that the convex-bladed H-Darrieus wind turbine outperforms the traditional straight-bladed one [18,19] since the H-Darrieus rotor’s performance is significantly impacted by the blade’s deformation during operation and when it subjected to high aerodynamic loads. This is a serious gap in the earlier works that has not been filled yet. Accordingly, this study aims at investigating the aeroelastic performance of the turbine’s convex-bladed rotor, proposed by one author of this work, and to compare it to the conventional straight one.

Studying the coupled effect of blade flexibility and surrounding airflow is necessary to fully understand how blade deformation affects the aerodynamic performance of H-Darrieus rotors during operation. To achieve this, a three-dimensional numerical model of the fluid–structure interaction was developed using ANSYS V 2019R1 software. In addition, other materials were examined to confirm how blade elasticity affected the H-Darrieus wind turbine’s performance.

2. Formulation of the Fluid–Structure Interaction

2.1. Mathematical Formulation

The equations governing the fluid–structure interaction problem are generally represented by the Navier–Stokes equations for fluid flow coupled to the elastic equations for solid motion, while taking into account the particular coupling conditions at interfaces and the boundary conditions of the problem. The model equations are [20,21]:

$${\rho }_f{\displaystyle \frac{\partial v}{\partial t}}+{\rho }_f\left(v\right)v=-\nabla p+\mu \nabla ·\left(\nabla v\right)+f_F$$

(1)

$$\nabla ·v=0$$

(2)

$${\rho }_s{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}-\nabla ·\sigma =f_s$$

(3)

where $v$ is the incompressible fluid velocity field, $u$ is the displacement of the solid blade, t is the time, $\mu$ is the dynamic viscosity, $f_F$ is the body force acting on the fluid (in this case, gravitational acceleration multiplied by ${\rho }_f$), and $f_s$ means the force per unit volume acting on the blade. ${\rho }_f$ and ${\rho }_s$ denote the fluid’s and solid’s density, respectively.

The stress field within the fluid, denoted as ${\sigma }_F$, is described by the following equation:

$${\sigma }_F=-pI+\mu \left(\nabla v+{\left(\nabla v\right)}^t\right)$$

(4)

Furthermore, Hooke’s law is used to identify the stress in the structure:

$${\sigma }_s=2{\mu }_s\epsilon -{\lambda }_{s}tr\left(\epsilon \right)I$$

(5)

$${\mu }_s={\displaystyle \frac{E}{2\left(1-\vartheta \right)}} \mathrm{and} {\lambda }_s={\displaystyle \frac{E}{\left(1+\vartheta \right)\left(1-2\vartheta \right)}}$$

(6)

where ${\sigma }_s$ are the stress fields, $\epsilon$ is the solid deformation, $E$ is Young’s modulus, $\vartheta$ is the Poisson coefficient, and $I$ is the identity tensor.

The coupling between the fluid and the structure occurs at the interface between the two domains. This coupling is described by two conditions, kinematic and dynamic. These conditions are modeled and given by the following equations [22]:

$$v\left(x,t\right)={\displaystyle \frac{\partial u}{\partial t}}$$

(7)

$$n·{\sigma }_s^{interface}=n·{\sigma }_F^{interface}$$

(8)

where n is the unit vector normal to the interface (blade).

In our study, to take turbulence into account, and to address the characteristics of turbulence in the fluid, the Navier–Stokes equations mentioned above are completed by the shear stress transport k-ω-SST turbulence model equations.

2.2. Fluid Structure Interaction Model Setup

The FSI (fluid–structure interaction) solution method is a numerical technique used to analyze the interaction between fluids and solid structures. It is a hybrid approach that combines the finite element method (FEM) for structural analysis with computational fluid dynamics (CFD) for fluid flow. This method is particularly useful in scenarios where both fluid and structural behavior need to be considered simultaneously, such as in aerodynamics, hydrodynamics, and mechanical engineering.

The flowchart in Figure 1 precisely describes the numerical simulation strategy used to analyze the complex interactions between a fluid and a structure, an essential method in fields such as aerodynamics, computational fluid dynamics (CFD), and structural dynamics. The simulation starts at a given point in time, where the fluid behavior is calculated for each interaction point on the rotating structure, capturing dynamic flow variations. Then, a coupling interaction is initiated between the CFD solver, responsible for the fluid analysis, and the structural solver, in charge of the structure’s response to the forces exerted. Steps include CFD analysis to evaluate fluid forces, mesh updating to integrate structural deformations, and the dynamic analysis of the structure to calculate deformations and transmit the new conditions to the fluid solver. This iterative process continues until the FSI solution converges, at which point variations between iterations become negligible, before moving on to the next time step. A specific interface enables the bi-directional coupling between the CFD and structural modules to be managed efficiently, guaranteeing smooth, consistent interaction throughout the simulation.

Within the two-way-based FSI and FEA framework, two fundamental solutions must be used to handle the closely coupled elasto-aerodynamic problem. The ANSYS Workbench environment’s system coupling module, as illustrated in Figure 1, connects the fluid dynamics solver, which employs the finite volume method (CFD), and the structural solver, which uses the finite element analysis (FEA), to manage the iterative process and data exchange (forces and displacements) between these two solvers. At each successive time step, the flow field is iterated to a new point on the turbine rotor by calling the CFD solver.

The interface between the structural module and the CFD module is facilitated through the system coupling command. Data exchange occurs at an external boundary in the CFD solver, with mesh displacement defined by the two-way coupling process. After using this command in the transient structural window of ANSYS, the interface boundary is created for each blade, i.e., Fluid–Structure Interface 1, Fluid–Structure Interface 2, and Fluid–Structure Interface 3 for blades 1, 2, and 3, respectively. Later, in CFD, each blade is correspondingly considered to match the fluid-solid interface. Due to these specifications, the CFD solver transfers forces to the structural module solver, and the CFD module transfers the total mesh displacement to CFD at the interface boundary surface [23].

Figure 1. Flowchart of FSI simulation strategy [24].

Figure 1. Flowchart of FSI simulation strategy [24].

2.3. Geometry

The fluid flow domain is split into two subdomains: a revolving cylindrical domain that models the rotor’s rotation at an angle of velocity, and a stationary rectangular domain that represents the fluid flow channel in the CFD model (Figure 2). The dimensions of the stationary domain for our investigation are 3 m (3D) width, 6 m (6D) height, and 16 m (16D) length, where D is the rotor diameter. The fluid domain’s inlet boundary is situated at a distance of 3D from the rotor, while its outlet boundary is situated at a distance of 13D from the rotor center (Figure 3). To ensure the solution convergence and stability, the domain dimensions and the rotor’s placement are carefully selected.

The solid region consists of three blades with an NACA4312 aerodynamic profile, as illustrated in

Table 1. Geometrical characteristics of the studied VAWT.

Parameter	Value
Number of blades N	3
Diameter D [m]	1
Height H [m]	1.065
Swept area [m2]	1.065
Solidity σ	0.32
Chord C (z) [m] straight blade	0.1665
Chord C(z) [m] convex blade	0.1z2 + 0.067
Airfoil	NACA4312

. In this study, two blade shapes were studied: convex blades and straight blades. For the simplification of calculations, only three blades were considered, and the connections between the arms and the rotor shaft were disregarded [19].

2.4. Boundary Conditions

To solve the abovementioned equation system, specific boundary conditions must be assigned to the computational domain in advance. These boundary conditions are applied to the walls of the computational domain. For instance, an inlet velocity of 7 m/s is assigned to the inlet wall, as illustrated in Figure 3. At the far outlet wall, a zero gauge pressure is considered. However, the four lateral walls of the cuboidal domain are assumed to be symmetric no-slip walls, meaning that the flow velocity perpendicular to these walls is zero. The tip-speed ratio, or TSR, is one of the primary VAWT characteristics. This parameter measures the ratio of the wind velocity to the turbine rotor’s periphery velocity. The TSR is defined as follows:

$$\mathrm{T}\mathrm{S}\mathrm{R}={\displaystyle \frac{R\omega }{U_{\mathrm{\infty }}}}$$

(9)

where ω is the angular velocity and R is the rotor radius; U_∞ here is the incoming wind speed. The results were processed to determine the coefficient of torque (C_T) and the coefficient of performance (C_P). C_T and C_P are given by Equations (10) and (11). The instantaneous torque was also observed.

$$C_T={\displaystyle \frac{T}{\frac{1}{2}\rho U_{\mathrm{\infty }}^{2}R\left[2RH\right]}}$$

(10)

The performance coefficient expressed by Equation (11) is deduced from Equation (10)

$$C_P={\displaystyle \frac{T\omega }{\frac{1}{2}\rho U_{\mathrm{\infty }}^{2}R\left[2RH\right]}}$$

(11)

The swept rotor area is (2R × H), where H is the rotor height.

Finally, for each simulation, a rotational velocity will be prescribed to the rotating zone to cover the entire range of the tip-speed ratio (TSR) that the wind turbine might encounter. In the structural domain, several rotational velocities (ω = 14, 28, 42, and 56 rad/s) assigned to the rotor and gravity acceleration (g = 9.8 m/s) are applied as boundary conditions to the structure component. Additionally, as depicted in Figure 4, a fixed support maintains the mid-blades of the rotor. The blade surfaces are handled as fluid–structure coupling interfaces to enable data flow between CFD and FE modules. Please refer to Figure 2 for a thorough understanding of the rotor’s geometry and aerodynamic characteristics.

2.5. Blade Materials

Using blades made of various flexible materials, such as aluminum alloy, epoxy reinforced with fiberglass, and epoxy reinforced with carbon fiber, preliminary analyses were carried out.

Table 2. Material properties of the blades.

	Density (kg/m3)	Young Module (GPa)	Poisson Coefficient
Carbon fiber composite (M1)	1565	54.65	0.306
Aluminum allow (M2)	2680	73	0.33
Glass-fiber-reinforced epoxy composite (M3)	1600	30	0.3

lists these materials’ primary properties.

Note that the Darrieus VAWT rotor was subjected to a complete 3D coupled simulation without considering the shaft and blade arms.

2.6. Meshing Strategy and Solution Mesh Grid Independency

The coupling system was developed in ANSYS Workbench to facilitate the two-way connection between the fluid dynamics and structural analysis systems. The selection of a precise and resilient digital model is crucial for fluid–structure interaction (FSI) simulations. In this context, the coupling between pressure and velocity in the fluid component was accomplished using the Coupled Schema. This model is especially well suited to FSI simulations because it ensures good convergence and increasing precision in the results while effectively capturing complicated dynamic interactions between the fluid and the structure.

2.6.1. Meshing Strategy

Since a dynamic 3D meshing domain was adopted to resolve the FSI interactions, the use of a structured mesh was not possible. Therefore, as depicted in Figure 5, an unstructured tetrahedral mesh was employed in the fluid domain (stationary zone) and particularly refined around the blade and in the rotating domain. To accurately capture the interactions between the fluid flow and the blades, a fine mesh was used around them, resulting in a mesh size of 0.005 m near the coupling interfaces. A dynamic meshing approach was applied to the area near the flexible blade, where fluid–structure interaction occurs, using remeshing, layering, and smoothing methods. The element size in the rotating zone was 0.07 m, while in the stationary zone, it was 0.21 m. The structure domain was discretized by a structured tetrahedral mesh with 170,709 nodes and 32,100 elements. In the blade, the element size was 0.001 m to accurately capture deformation. In summary, the mesh for the calculation domain was carefully selected using different mesh types to achieve a trade-off between accuracy and computation time. Mesh refinements were made around the blades to ensure an accurate representation of their geometry, and a dynamic meshing approach was used to capture fluid–structure interaction.

2.6.2. Grid Independence Solution Study

To determine the grid independency, the H-Darrieus rotor’s performance was examined for several unstructured meshes at TSR = 3, at the optimal operating condition. Four meshes of the fluid part (Mesh 1, Mesh 2, Mesh 3, and Mesh 4), as reported in

Table 3. Grid independence analysis (fluid part mesh).

	Cell Number	Cp	Cp-Ref	Error %
Mesh 1	2,145,210	0.288	0.3	4
Mesh 2	2,690,881	0.290	0.3	3.4
Mesh 3	3,254,541	0.293	0.3	2.3
Mesh 4	3,797,835	0.295	0.3	1.6

, with progressively refinement levels, were generated and compared to the H-Darrieus benchmark performance to evaluate the mesh independence solution. A particular cell density was established near the blades for every mesh. Upon computation, Table 3 displays the expected performance and error differences. Comparing all mesh errors to the baseline, it was evident that they were less than 5%. Therefore, based on computational cost considerations, Mesh 1 which is depicted in Figure 6, was chosen for the rest of this study.

Figure 7 displays variations in the rotor’s total instantaneous torque coefficient vs. times at TSR = 3 for the four meshes listed in Table 3. The resulting torque coefficient makes it evident that the turbine experienced a positive torque during full rotation. We could observe a slight difference at the higher torque coefficient, yet the torque for all four meshes converged at the lower torque peak.

Table 4. Grid independence analysis (structural part mesh).

	Element Size (mm)	Maximum Displacement of the Structure (mm)	Relative Error (%)
Mesh 1	3	29.22	-
Mesh 2	4	29.82	2
Mesh 3	5	30	6
Mesh 4	6	30.9	3

reports the maximum displacements measured at the blade ends for the four meshes of the structural part (blades). The results showed that the discrepancy was less than 6%.

2.6.3. Time Step Sensitivity

Since the time step has a direct impact on the accuracy of the simulation results, it was crucial to analyze its impact in this study. When a time step is too tiny, computation time is greatly increased without appreciable accuracy gain, transient phenomena are undervalued, and detail is lost. Thus, several time steps were tested: 0.01, 0.05, 0.075, 0.005 and 0.0025 s. By comparing the moment coefficients corresponding to each time step, as reported in

Table 5. Influence of time step.

Δt	Cp	Cp-Ref	Error (%)
0.01	0.27	0.3	10
0.05	0.285	0.3	5
0.075	0.3	0.3	0
0.005	0.309	0.3	3
0.0025	0.309	0.3	3

, it was possible to make a trade-off between accuracy and computational cost. In this case, after a thorough analysis of the results, it was inferred that the 0.005 s time step was the most appropriate for this simulation.

3. FSI Model Validation

To assess the accuracy of the proposed 3D-FSI numerical model, the calculated results were compared to available experimental data. The experimental model for validation was a vertical-axis tidal current turbine tested in a water tunnel (Khalid et al. [23]). The turbine rotor was made up of four steel blades that were each 5.5 m in span and 4 m in diameter. The main features of the baseline are described in

Table 6. Main features of the experiment baseline.

Parameter	Dimension
Turbine diameter	4 m
Number of blades	4
Blade span	5.5 m
Blade profile	NACA 0018
Velocity	3.5 m/s
Material	Steel
Fluid	Water

. The turbine’s axis was positioned near the midline of the tunnel. The outlet was 21 m away from the center, while the inlet and bilateral sides were 7 m. The incoming water flow had a constant velocity of 3.5 m/s. The numerical model was conducted over a TSR range of 1 to 4 to identify the performance characteristic curve as a function of the turbine’s TSR.

Figure 8 compares the H-Darrieus’ aeroelastic performance vs. TSR, which accounts for the influence of blade structure deformation, denoted by the FSI curve in Figure 8, with the H-Darrieus’ aerodynamic performance, where only the flow around the rotor is resolved, which does not account for the effect of structural deformation, represented by the CFD curve. A comparison between these two H-Darrieus performance responses as a function of TSR and an existing experimental baseline result carried out by Khalid et al. [23] is shown by the EXP curve in the same figure.

Figure 8 makes it evident that the FSI model accurately reproduced the turbine’s aeroelastic behavior and the optimum operating TSR at which the turbine operated at its peak efficiency. Using the 3D-FSI numerical model, a high agreement between the experimental and numerical results was obtained.

4. Results and Discussions

4.1. Maximum Average Displacement and Stress

This section discusses how different materials affect the performance of both straight and convex Darrieus-type VAWT as well as the deflection of the blades at specific TSRs (TSR = 2 and TSR = 3). Figure 9 compares the maximum displacements of the blade tip at TSR = 2 and TSR = 3 for the three materials as reported in Table 2. As can be observed, the convex blades show the lowest maximum displacement, whatever material and TSR are considered. The results clearly show that the M1 material deformed the least in the blades, with 9.1 mm and 20.331 for the convex blade at TSR = 2 and TSR = 3, respectively. For the straight blade at TSR = 2 and TSR = 3, the values were 13.9 mm and 31.6 mm, respectively. On the other hand, the M3 material exhibited the highest deflection at the convex and straight blades. The straight really deflected at TSR = 2 and TSR = 3, with equivalent deflections of 32.50 mm and 73.78 mm. At TSR = 2 and TSR = 3, respectively, the convex blade showed 19.02 mm and 43.03 mm less than the straight one. Additionally, the results show that when the third material (M3) was utilized, the straight blade achieved the maximum displacement of 73.785 mm at TSR = 3. However, when the initial material (M1) was used, the convex blade achieved the lowest displacement with 20.331 mm.

The maximum stress that the straight and convex blades can produce when operating is shown in Figure 10. The highest stress was seen for the straight blade at 324.1 MPa and TSR = 3, which is similar to the maximum displacement values for the second material (M2). Furthermore, the convex blade for the first material (M1) produced the least amount of stress.

Considering the two investigated blade configurations, Figure 11 shows the total displacement variation as a function of the tip-speed ratio (TSR) for three different materials. It is evident that the deformation of a blade in response to aerodynamic pressures rises with blade flexibility. The forces applied to the blades rise in tandem with TSR, causing the blades to bend and shift. In fact, the displacement for material M1, as shown in Figure 12, varied from 1 mm to 25 mm for the convex design and from 3 mm to 42 mm for the straight blade arrangement.

Similarly, for material M2, values ranged from 4 mm to 56 mm for the straight design and from 3 mm to 32 mm for the convex form, with displacement variations displayed in Figure 11. With respect to material M3, displacement varied between 5 and 110 mm in a straight configuration and between 4 and 40 mm in a convex shape. These observations demonstrate how the blade’s flexibility has a major impact on how it responds to aerodynamic forces and how this reaction changes as the TSR is changed.

The displacement contour plots with two selected materials, M1 and M3, at TSR = 2 and TSR = 3 is shown in Figure 12 for both straight- and convex-bladed rotors. It is evident that when the TSR rises, there is an increase in deformation for both convex and straight blades. The minimal displacement is obtained at the mid-blade, the point where the shaft and trusts join. The findings showed that the convex blade shape significantly lessens the blade’s deflection.

The stress contour plots in the rotor’s straight and convex blades for the two distinct materials, M1 and M3, are shown in Figure 13. For both blade configurations, it is evident that stress rises as TSR rises. It is interesting to note that the stress in a convex blade is about half that of a straight blade, regardless of the material and TSR. The maximum stress was located close to the strut joints. In addition, the minimum stress was generated by the convex blade with material (M1).

The development of instantaneous blade deformation in time for the three materials is shown in Figure 14a for both straight and convex blades. The deformation experienced a transient phase before starting to stabilize around an average value, after a few revolutions. The deformation’s amplitude oscillated with a period that matched the blade’s rotational frequency. It can be observed that the starting time of stabilization differed from one material to another. Indeed, as can be seen from Figure 14a,b, material M1 for both straight and convex blades achieved a rapid stabilization of deformation. In comparison to other materials, material M3 did, however, show a certain delay in stabilizing the deformation.

Figure 15 depicts the evolution of the instantaneous maximum blade stress over time for both convex and straight blades made of the three materials. After a few revolutions, the stress goes through a transitory period before beginning to stabilize around an average value. Similarly, it is worth noting that the amplitude of the stress oscillates at a period equal to the rotating frequency of the blade. It is evident that different materials have different stabilizing onset times. In fact, material M1 for both straight and convex blades achieved quick stress stabilization, as shown in Figure 15a,b. However, material M2 did exhibit the maximum stress when compared to the other materials.

Figure 15 shows that for material M2, the convex configuration stabilized at 120 MPa, while the straight configuration stabilized at 200 MPa. Furthermore, during stabilization, the straight configuration of material M3 had a stress fluctuation of 110 MPa, whereas the convex shape had a stress variation of 67 MPa. It is noteworthy that for all three materials, the convex configuration presented lesser stress since it reached stabilization before the straight configuration.

4.2. Performances

The performance curves of the straight- and convex-bladed rotors, for the three materials M1, M2, and M3, are compared in Figure 16a,b. Although the convex shape continuously had a higher Cp than the straight configuration over the whole range of TSR, it can be observed that material M1 had a maximum performance of 0.28 at TSR = 3, whereas material M2 had a maximum performance of 0.27 at the same TSR and material M3 had a maximum performance of 0.22 at the same TSR.

The findings show that material stiffness has a significant impact on both the bending of wind turbine blades and the VAWT’s performance. The elasto-aerodynamic performance of convex-shaped blades is evident, and this is explained by the more effective distribution of aerodynamic loads throughout their surface. Compared to a straight design, this specific shape lowers von Mises stresses and total displacements by reducing local stresses.

4.3. Wake Structure

The contour plots of the x-component of the flow velocity for both straight- and convex-bladed rotors are shown in Figure 17. It is evident that the blade’s bending induced two large wakes behind the mid-blade. The convex blade caused the formation of a stretched wake structure behind it. That is why minimal displacement was obtained in the middle of the blade, at the junction point between the axis and the arms. This result clearly shows that the convex blade tended to reduce the deflection of the blade. For this, a low deformation and a low stress were obtained at this location as explained previously. Nevertheless, the tips of the straight blades generated two much finer wakes that also minimized the deformation of the tips. On the other hand, much larger wakes were observed attached to the ends of the convex blades. The structure of the wake and vortices necessarily had a direct impact on the deflection and endurance of the blade. The restricted structure of the wake in a convex blade generated low stresses and subsequently increased the endurance of the blade, which was approximately half that of a straight blade, regardless of the material and TSR.

Accordingly, the differences in wake and vortex structure induced by the differences in blade shapes may significantly impact the overall performance and efficiency of the wind turbine rotor. Consequently, understanding these aerodynamic effects is crucial for optimizing rotor design and operation in various applications.

5. Conclusions

A fluid–structure interaction (FSI) model was developed in this work to examine the performance of Darrieus-type vertical-axis wind turbine blades. The realistic handling of the interactions between the fluid (the surrounding air) and the structures (the wind turbine blades) was made possible by the FSI model. A two-way coupled FSI strategy was adopted, enabling the exchange of data between the two aforementioned domains.

Straight and novel convex blade designs are the two primary types of blades that are the subject of this study. In this investigation, three materials with various levels of stiffness were also taken into account to assess how the blades’ flexibility affects the wind turbine’s overall performance.

The results show at TSR = 2 and TSR = 3 that the M1 material exhibited the least deformation in the blades, recording values of 9.1 mm and 20.331 mm for the convex blade, respectively. In contrast, the straight blade demonstrated greater deflection, with measurements of 13.9 mm and 31.6 mm. Conversely, the M3 material showed the most significant deflection across both blade types, with the straight blade experiencing displacements of 32.50 mm and 73.78 mm, while the convex blade recorded deflections of 19.02 mm and 43.03 mm, which were notably lower than those of the straight blade.

Furthermore, the findings reveal that the straight blade reached its peak displacement of 73.785 mm when the M3 material was employed at TSR = 3. In contrast, the convex blade, when utilizing the initial M1 material, achieved the minimum displacement of 20.331 mm. This comparison underscores the varying performance characteristics of the materials used, highlighting the superior deflection properties of M3 in the straight blade configuration, while M1 demonstrated optimal stability in the convex blade design.

The maximum stress recorded was 324.1 MPa for the straight blade at a TSR of 3, which corresponded closely to the peak displacement values observed for the second material (M2). Additionally, the convex blade made from the first material (M1) exhibited the lowest stress levels.

This study demonstrates the crucial importance of blade flexibility and configuration on the performance of a vertical-axis wind turbine. Additionally, these findings highlight the intricate relationships between fluid and structure and offer insights for improving wind turbine design to increase energy efficiency.