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Article

A Refined Approach for Angle of Attack Estimation and Dynamic Force Hysteresis in H-Type Darrieus Wind Turbines

by
Jan Michna
and
Krzysztof Rogowski
*
Institute of Aeronautics and Applied Mechanics, Warsaw University of Technology, Nowowiejska 24, 00-665 Warsaw, Poland
*
Author to whom correspondence should be addressed.
Energies 2024, 17(24), 6264; https://doi.org/10.3390/en17246264
Submission received: 19 November 2024 / Revised: 9 December 2024 / Accepted: 10 December 2024 / Published: 12 December 2024
(This article belongs to the Special Issue Wind Turbine and Wind Farm Flows)
Browse Figures
Figure 1
<p>Diagram of an H-type vertical axis wind turbine structure with dimensions.</p> "> Figure 2
<p>Domain schema with dimensions (<b>upper</b>) and mesh layout with boundary conditions for the CFD simulation (<b>bottom</b>). The diagram shows the defined boundary conditions: velocity inlet on the left, pressure outlet on the right, symmetry on the top and bottom, wall on the airfoil surface, and interface region connecting different mesh zones. Enlarged sections illustrate the mesh refinement around the airfoil and interface areas.</p> "> Figure 3
<p>Variation of the mesh sensitivity test results: The plot shows the normal force coefficient C<sub>N</sub> (red) and the tangential force coefficient C<sub>T</sub> (blue) as a function of the number of mesh cells. C<sub>N</sub> is plotted on the left axis and C<sub>T</sub> on the right axis. The results indicate that both coefficients stabilize as the mesh density increases from 180,000 to 340,000 cells.</p> "> Figure 4
<p>Variation of the tangential force coefficient C<sub>T</sub> as a function of the number of rotor revolutions. Black dots represent values averaged over individual rotor revolutions, while the red dashed line shows the average C<sub>T</sub> over the last ten rotor revolutions. The figure illustrates the consistency of the aerodynamic loads over the examined revolutions, with minimal variation observed between individual revolutions and the overall average.</p> "> Figure 5
<p>Comparison of the tangential force coefficient C<sub>T</sub> as a function of azimuth angle <span class="html-italic">θ</span>. The red solid line represents the tangential load component calculated for the last rotor revolution, while the blue dashed line shows the same component, averaged over ten revolutions, both plotted as a function of the azimuth angle.</p> "> Figure 6
<p>Comparison of the V<sub>x</sub>/V<sub>0</sub> velocity component in the rotor wake for various downstream positions x/D, calculated using the SAS approach and validated against experimental data [<a href="#B46-energies-17-06264" class="html-bibr">46</a>].</p> "> Figure 7
<p>Comparison of the <span class="html-italic">V<sub>y</sub>/V<sub>0</sub></span> velocity component in the rotor wake for various downstream positions x/D, calculated using the SAS approach and validated against experimental data [<a href="#B46-energies-17-06264" class="html-bibr">46</a>].</p> "> Figure 8
<p>Comparison of simulated and experimental force coefficients [<a href="#B53-energies-17-06264" class="html-bibr">53</a>] as a function of azimuthal angle, <span class="html-italic">θ</span>. (<b>a</b>) Normal force coefficient C<sub>N</sub>, shows a general agreement between the SAS simulation and experimental data. (<b>b</b>) Tangential force coefficient, C.</p> "> Figure 9
<p>Illustration of the angle of attack (<span class="html-italic">α</span>) and relative velocity (<span class="html-italic">V<sub>rel</sub></span>) for a blade in a vertical-axis wind turbine. The pitch angle (<span class="html-italic">β</span> = 10°) and the azimuthal position angle (<span class="html-italic">θ</span> = 48°) are indicated.</p> "> Figure 10
<p>Velocity distribution around the airfoil at an azimuthal angle θ = 48°. The circular sampling line with evenly distributed points surrounds the airfoil, with arrows representing the local flow velocities at each point.</p> "> Figure 11
<p>Sensitivity analysis of the angle of attack (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>−</mo> <mover accent="true"> <mrow> <msup> <mrow> <mi>α</mi> </mrow> <mrow> <mfenced separators="|"> <mrow> <mn>2000</mn> </mrow> </mfenced> </mrow> </msup> </mrow> <mo>¯</mo> </mover> </mrow> </semantics></math>) based on the number of sampling points along the circular line surrounding the airfoil as a function of the azimuth angle θ. Standard deviation is chosen as an error measure.</p> "> Figure 12
<p>The mean relative velocity is represented by lines. Standard deviation is chosen as an error measure.</p> "> Figure 13
<p>Validation of angle of attack (<span class="html-italic">α</span>) and relative velocity (<span class="html-italic">V<sub>rel</sub></span>) results obtained using the SAS approach, compared with literature data from Melani et al. [<a href="#B45-energies-17-06264" class="html-bibr">45</a>] and Cacciali et al. [<a href="#B8-energies-17-06264" class="html-bibr">8</a>].</p> "> Figure 14
<p>The upper subfigures illustrate the drag coefficient C<sub>D</sub> (<b>a</b>) and lift coefficient C<sub>L</sub> (<b>b</b>) as functions of the azimuthal angle <span class="html-italic">θ</span>. The lower subfigures depict the same coefficients—drag (<b>c</b>) and lift (<b>d</b>)—analyzed for varying angles of attack α. The presented data include results obtained using the SAS approach, static results reported by Rogowski et al. [<a href="#B22-energies-17-06264" class="html-bibr">22</a>], and reference data from Melani et al. [<a href="#B45-energies-17-06264" class="html-bibr">45</a>].</p> "> Figure 15
<p>Effect of the number of blades on the normal and tangential force coefficients, as well as on the local angle of attack and relative velocity, as a function of azimuth <span class="html-italic">θ</span> at a tip-speed ratio of 4.5.</p> "> Figure 16
<p>The subfigures show the drag coefficient C<sub>D</sub> (<b>a</b>,<b>c</b>) and lift coefficient C<sub>L</sub> (<b>b</b>,<b>d</b>) for different azimuthal angles (<b>a</b>,<b>b</b>) and angles of attack (<b>c</b>,<b>d</b>). The presented data shows comparison of aerodynamic coefficients for 1-bladed, 2-bladed, 3-bladed, and 4-bladed configurations.</p> "> Figure 17
<p>Effect of the pitch angle on the normal and tangential force coefficients, as well as on the local angle of attack and relative velocity, as a function of azimuth <span class="html-italic">θ</span> for a 2-bladed rotor configuration at a tip-speed ratio of 4.5.</p> "> Figure 18
<p>The subfigures show the drag coefficient C<sub>D</sub> (<b>a</b>,<b>c</b>) and lift coefficient C<sub>L</sub> (<b>b</b>,<b>d</b>) for different azimuthal angles (<b>a</b>,<b>b</b>) and angles of attack (<b>c</b>,<b>d</b>). The data presented above shows aerodynamic coefficient comparison for varying blade pitch angles (<span class="html-italic">β</span> = −10°, 0°, 10°).</p> ">
Versions Notes

Abstract

:
This study investigates the aerodynamic performance and flow dynamics of an H-type Darrieus vertical axis wind turbine (VAWT) using combined numerical and experimental methods. The analysis examines the effects of operational parameters, such as rotor solidity and pitch angle, on aerodynamic loads and flow characteristics, using a 2-D URANS simulation with the Transition SST model to capture transient effects. Validation was conducted in a low-turbulence wind tunnel to observe the impact of variable flow conditions. The LineAverage method for determining the angle of attack demonstrated strong correlations between rotor configuration and load variations, particularly highlighting the influence of blade number and pitch angle on aerodynamic efficiency. This research supports optimization strategies for Darrieus VAWTs in urban environments, where turbulent, low-speed conditions challenge conventional wind turbine designs.

1. Introduction

The 2021 special report by the International Energy Agency (Net Zero by 2050) [1] highlights the critical role of renewable energy sources in achieving global carbon neutrality by mid-century. Among these, wind energy is projected to supply a significant portion of future electricity demand. While offshore wind turbines have garnered attention for their capacity to harness high-quality winds, they face substantial challenges related to installation, maintenance, and scalability [2,3].
In contrast, vertical axis wind turbines (VAWTs), particularly the H-type Darrieus design, offer unique advantages for deployment in urban and low-wind-speed environments [4]. Their ability to operate efficiently without a yaw system makes them highly adaptable to the turbulent and unpredictable wind conditions typical of urban areas [5]. Moreover, their smaller physical footprint and lower noise emissions render them suitable for residential and distributed energy systems. However, these turbines face notable challenges, including limited self-starting capability and reduced aerodynamic efficiency at lower Reynolds numbers, which hinder their broader adoption [6,7].
This study aims to address these challenges by analyzing the aerodynamic characteristics of H-type Darrieus VAWTs across various geometrical configurations. A novel methodology is introduced, combining advanced numerical modeling with experimental validation, with a focus on improving turbulence modeling accuracy at low Reynolds numbers. The key contributions of this work include an in-depth analysis of the angle of attack and dynamic aerodynamic loads, which are critical for understanding the complex physical phenomena associated with the intricate flow around this type of turbine. These findings also play a vital role in efforts to improve engineering aerodynamic models for VAWTs, enhancing their performance and applicability in diverse operational conditions.
The airflow around the blades of a working wind turbine rotor is complex and highly unsteady. This complexity arises from the continuous shift in the tangential velocity direction of the blade, leading to constant changes in the angle of attack and relative velocity. Additionally, as the blades alternate between the upwind and downwind sections of the rotor, they interact with the rotor’s aerodynamic wake and operate in conditions of varying turbulence intensity [8,9]. Due to these complexities, computational fluid dynamics (CFD) methods utilizing the Reynolds-averaged Navier–Stokes (RANS) equations, paired with various turbulence models, have become essential tools in recent years for numerically analyzing aerodynamic characteristics [10,11,12].
Due to the necessity of employing dense computational meshes, simulation methods such as large eddy simulation (LES) and even 3-D unsteady Reynolds-averaged Navier–Stokes (URANS) are significantly constrained [13,14]. Additionally, RANS methods are often combined with turbulence models, typically employing one- or two-equation approaches such as the Spalart–Allmaras [15,16] model or the k-ω and k-ε families [12,17]. In recent years, transition models have gained increasing popularity, primarily due to the need for more accurate representations of transitional phenomena, such as laminar–turbulent bubbles and other forms of flow transition [18,19,20,21]. Although the accuracy of these models can be limited for certain flow conditions [9,22,23,24], and the application of URANS to 3-D problems remains restricted by computational resources and sometimes licensing costs [25,26], CFD methods remain invaluable for analyzing rotors with complex geometries. These methods also serve as powerful tools to gain insights into unsteady physical phenomena associated with Darrieus rotor flows [27,28,29]. Understanding these phenomena is essential for the aerodynamic optimization of the rotor [27] and can further contribute to the development of methods based on blade element theory, such as blade element momentum (BEM) [30], lifting-line methods [31], and actuator line methods [32].
When modifying blade element-based methods, it is essential to accurately estimate the local angle of attack, which changes with the azimuth during each rotor revolution. To leverage CFD results in improving such methods, it is necessary to extract the angle of attack from the velocity field around the blade. Unfortunately, this task is complicated by additional effects such as virtual camber [33] and dynamic stall [34]. Moreover, depending on the tip-speed ratio, the amplitude of the angle of attack variations can be huge.
Some of the popular methods for extracting the local angle of attack in horizontal-axis wind turbines are the inverse BEM method and the averaging technique. The first method uses the load distribution as input data but is limited to one-dimensional assumptions. The second method, averaging, provides good results in the middle section of the blade, though it requires many measurement points and is more challenging to apply under complex flow conditions [35]. In the approach utilizing a priori imposed forces, the angle of attack is calculated by predefined forces based on either experimental measurements or computed values. This method is widely recognized for providing reasonably reliable results, which can subsequently be used in BEM codes for further predictions. However, a primary limitation of this approach is its reliance on 1-D theory. Notably, the direction of the incoming wind is assumed not to be influenced by blade-flow interactions, leading to significant errors in cases of high deflection. Additionally, BEM models require tip correction, where accuracy is crucial for a reliable evaluation of airfoil data [36]. Another technique developed for horizontal-axis wind turbines (HAWTs) is the “averaging technique”, as applied by Hansen and Johansen [37] and further explored by Johansen et al. [38]. In this approach, the local angle of attack is calculated by reconstructing the velocity triangle on the blade: the peripheral speed is known in both magnitude and direction, while the relative speed is derived from CFD calculations by analyzing the flow in the rotor plane. However, as noted by Zhong Shen et al. [35], this method requires analyzing many points in the computational domain to capture local flow features due to its reliance on averaged data. Additionally, it is challenging to apply this approach under more general flow conditions, such as yawed flow.
Guntur and Sørensen [39] examined several methods of determining the angles of attack on wind turbine blades. One approach utilized computational fluid dynamics (CFD) to compute the local induced velocities at the rotor plane. By analyzing the velocity field from a full rotor 3-D CFD simulation, they could extract axial velocity profiles at different radial and azimuthal positions around the rotor. This data was processed by averaging velocities in specific annular sections or by obtaining pointwise values across the azimuth. Interpolating these velocities allowed them to estimate the induced axial flow at the rotor plane, providing a basis for determining the effective α at various blade sections and enhancing the accuracy of aerodynamic performance assessments [39,40]. Recent advances in input design and power spectrum optimization for aerodynamic analysis provide valuable methodologies that could enhance the precision of computational fluid dynamics (CFD) models, especially in capturing unsteady phenomena in rotor flows [41,42].
In Darrieus-type wind turbines, describing the flow field around the rotor is particularly challenging due to the variable angle of attack encountered during blade rotation, often exceeding the static stall angle. Balduzzi et al. [33] proposed an approach to estimate the angle of attack in such turbines based on an averaging technique and an inverse BEM method, which allowed for preliminary studies on flow curvature effects. While this approach yielded promising results, its application is limited to situations where the flow remains attached to the blades, and its accuracy is constrained by inherent limitations that require further analysis.
Saverin and Frank [43] introduced a method for improving the numerical prediction of the blade angle of attack (α). In their research, a method was employed to monitor the blade α within the simulation. The authors argued that using too few sampling points or a too-limited area around the blade leads to unreliable results. To obtain a credible estimation of the flow velocity components around the blade, a broader area or a greater number of sampling points is required, which allows for a more accurate capture of flow variations. Consequently, the researchers defined a finite control volume around the rotor blade, within which the velocity components of each finite volume element were averaged. Saverin and Frank emphasized that the control volume must not be too small. If the volume were too small, the averaged velocity components might not adequately reflect the actual flow experienced by the blade. It is crucial that the control volume encompasses a sufficient area around the blade to account for the flow’s impact on it. At the same time, the volume should not be excessively large, as regions outside the rotor’s direct influence might contribute to the averaged value, thereby distorting the results.
Bianchini et al. [36] proposed a new method for determining the angle of attack on an airfoil rotating around an axis orthogonal to the flow direction using CFD analysis. This approach incorporates the “virtual camber” effect, allowing the real airfoil to be treated as a virtual, straight-flow equivalent with a similar pressure distribution, which simplifies the assessment of aerodynamic performance. Pressure coefficient distributions for the virtual airfoil are normalized and compared with CFD results for the rotating airfoil, enabling accurate angle of attack estimation at various azimuthal positions. The method eliminates the need for user-selected variables, enhancing its robustness compared to previous BEM-based methods. Tests on a NACA0018 airfoil with different chord-to-radius ratios confirmed strong alignment with theoretical predictions, demonstrating the method’s effectiveness. This approach shows potential as a versatile tool for analyzing complex flow phenomena around vertical-axis wind turbines, particularly in dynamic stall effects and other unsteady flow conditions.
Delafin et al. [44] proposed a method for determining the angle of attack based on analyzing the position of the stagnation point on the blade, which allows for calculating an equivalent angle of attack depending on the azimuthal angle. The developed method uses the location of the stagnation point on the blade as an indicator of the angle of attack, enabling the consideration of unsteady phenomena, such as the wake of another blade or a tower. By comparing the stagnation point position at fixed angles of attack, it is possible to interpolate and determine the equivalent angle of attack for the turbine. Although the method does not account for the pitching motion of the blades, its local approach makes it well suited for aerodynamic analysis under variable flow conditions.
Melani et al. [45] mention three additional methods for estimating angles of attack on airfoils in cycloidal motion: the 3-Points method, the Trajectory method, and the Area Averaging Technique (AAT). The 3-Points method uses three sampling points on each side of the airfoil, allowing for angle of attack estimation that accounts for local velocity variations. The Trajectory method employs a single sampling point along the blade’s trajectory, simplifying calculations but offering less accuracy under dynamic conditions. The Area Averaging Technique (AAT), in turn, is based on analyzing the velocity field in the area around the turbine, enabling a more precise capture of the induced flow’s impact on the angle of attack. In this paper, the LineAverage method is used to determine the angle of attack for an airfoil rotating around an axis perpendicular to the flow direction. The choice of this method is due to its high accuracy, which has been validated in other studies, and its ability to account for circulation effects and flow distortions, which is particularly important in Darrieus turbine analysis. Additionally, the LineAverage method enables a more precise representation of unsteady aerodynamic phenomena, such as dynamic stall hysteresis cycles, making it a suitable tool for this type of analysis.

2. Materials and Methods

2.1. Numerical Reference Model of the Rotor, Analyzed Rotor Configurations, and Operating Parameters

A simplified numerical model of the wind turbine rotor (Figure 1) was developed based on the experimental studies presented in [46]. The rotor model consists of only two blades, represented by symmetric NACA0018 airfoils. In the present study, the rotating tower and struts were omitted for simplicity. The rotor’s diameter is 0.5 m, with a chord length of 0.06 m. The blades are attached at a distance of 0.4c from the leading edge. The pitch angle is set to 0 degrees, meaning the blade’s chord is tangential to its path. The developed model is two-dimensional, meaning that blade edge vortex effects are not considered. However, it is worth noting that for the reference rotor model, the blade aspect ratio, defined as the ratio of blade length to chord length, is 16.67. As documented by studies such as [9,27], the contribution of 3-D effects to the overall rotor torque is minimal in the case of moderately loaded rotors. The solidity, σ, of the reference rotor is 0.24, defined as:
σ = B · c / R ,
where B is the number of blades, c is the chord length, and R is the rotor radius.
The above description refers to the reference rotor. In this study, several additional rotor configurations were also investigated. Two additional series of tests were conducted. In the first series, the number of rotor blades was varied, and rotors with 1, 2, 3, and 4 blades were analyzed. The blade chord remained constant. In the second series, the effect of fixed pitch angles was studied, considering three angles: −10°, 0°, and 10° (for positive values, the leading edge is pitched inwards).
The rotor described in [46] operated at a tip speed ratio (TSR) of 4.5, defined as the ratio of the blade’s tangential speed to the undisturbed flow velocity. The undisturbed flow velocity V 0 was 9.3 m/s, and the angular velocity ω was 83.7 rad/s. It was assumed that the turbulence intensity of the undisturbed flow was 1%, with a turbulence scale of 0.001 m. The impact of these parameters on the aerodynamic characteristics of the rotor was not investigated in this study.

2.2. Numerical Model Setup for the Vertical Axis Wind Turbine Simulation

Due to significant fluctuations in the aerodynamic loads on the rotor blades during a single rotation, the calculations must be performed in an unsteady mode. In this study, the unsteady Reynolds-averaged Navier–Stokes (URANS) equations were used to account for these transient effects. One implementation of this approach is the sliding mesh technique, which requires the development of an appropriate computational domain. This technique allows the independent mesh in the vicinity of the rotor path to rotate relative to the non-rotating mesh. An interface is defined between these areas to enable data exchange between the rotating and stationary meshes. It is critical that the distance between the surface of the blade profile and the interface is sufficient to avoid errors in the estimation of vorticity distributions. In this study, a circular region surrounding the rotor with a diameter slightly larger (1.5D) than the rotor itself was defined, with the radius of the rotating part of the mesh being 0.5 m. This radius is 50% larger than the turbine rotor to ensure adequate computational accuracy and stability. Figure 2 shows the rectangular computational domain within which the rotor is placed. The scale of both the rotor and the domain was maintained. This circular region can rotate with constant angular velocity ω during the simulation while the rectangular domain remains stationary. Both the rectangular and circular domains are meshed separately, and the data exchange between them is handled by an interface. Boundary conditions such as velocity inlet, pressure outlet, and symmetry were applied at the outer edges of the rectangular domain (Figure 2).
Due to the low Mach number, the flow was treated as incompressible. Air with a constant density of 1.225 kg/m3 and a viscosity of 1.7894 × 10−5 kg/(ms) was used as the working medium.
The numerical simulations were conducted using ANSYS Fluent 2023. The SIMPLE scheme was employed for pressure–velocity coupling. The spatial discretization of the equations was achieved using the least squares cell-based method for gradients, bounded central differencing for the momentum equation, and second-order upwind for all other equations. The residuals convergence criterion was set to 1 × 10−9 for all equations. At the beginning of the iterative process, constant initial conditions equal to the inlet values were applied.
The 2-D CFD simulations were conducted for 40 full rotor revolutions, with a time step size equivalent to an azimuth increment Δθ of 0.5 degrees. As demonstrated in our previous studies [9], this number of rotations is sufficient to obtain results independent of initial conditions for the considered tip-speed ratio of 4.5. In Rezaeiha et al. [47], the authors conducted several analyses for azimuth increments between 0.5 and 0.05 degrees, showing that within this range, the difference in the rotor power coefficient is negligible for a tip-speed ratio of 4.5. Moreover, simulating at least 25 full rotor revolutions ensures that the effect of initial conditions is negligible after 10 revolutions. The instantaneous blade load results presented in this work correspond to the final calculated rotation.
In this study, the Reynolds number was not explicitly analyzed. However, it can be estimated based on the chord length (c) of the blade using the following formula:
R e c = ω R c ρ / μ ,
where ω is the angular velocity of the rotor, R is the rotor radius, ρ is the air density, and μ is the dynamic viscosity of air. For the considered rotor, with an air density of 1.225 kg/m3 and viscosity of 1.7894 × 10−5 kg/(ms), the Reynolds number is approximately 160,000 [8]. While this value is an approximation, it provides a sufficiently accurate representation of the flow regime.

2.3. Advanced Turbulence Modeling Using Scale-Adaptive Simulation (SAS) for VAWT Analysis

While most computational fluid dynamics (CFD) simulations rely on Reynolds-averaged Navier–Stokes (RANS) turbulence models, certain flow scenarios exceed their capabilities. Scale-resolving simulation (SRS) models, which resolve part or all of the turbulence spectrum in specific regions, have emerged as a solution to address these limitations [48].
The use of SRS models is motivated by the need for greater detail and accuracy. For instance, the complex flow around a vertical axis wind turbine (VAWT) rotor involves dynamic interactions between unsteady airflow, rotor blades, and varying turbulence intensities—challenges that RANS struggles to capture. While RANS is effective for wall-bounded flows due to calibration with the law of the wall, its accuracy declines for free shear flows with larger turbulence scales [49].
Large eddy simulation (LES) can resolve these scales but remains computationally prohibitive, especially for industrial applications. Hybrid approaches, such as detached eddy simulation (DES) and scale-adaptive simulation (SAS), combine the strengths of RANS near walls with the scale-resolving capabilities of LES in free shear regions. SAS, in particular, is well-suited for capturing unsteady flow structures, making it an effective tool for simulating complex aerodynamic phenomena like those encountered in wind turbine rotors [50].
This chapter highlights the advantages of the SAS approach and demonstrates its application in analyzing intricate and dynamic flows.
The scale-adaptive simulation (SAS) model adjusts the turbulence length scale dynamically in response to local flow conditions. This enables it to capture a wider range of turbulence structures in unstable flow regions. Its methodology is based on von Karman’s length scale theory and Rotta’s theoretical developments from 1972. By including an additional source term in the governing equations, the model transitions naturally between RANS-like behavior in stable flows and LES-like performance in more complex, unsteady flow situations [51]. The SAS approach introduces a significant improvement over traditional RANS models by reformulating the scale-defining equation to include the second derivative of the velocity field, allowing for more accurate adaptation to varying flow conditions. This innovation builds on Rotta’s original transport equation for turbulence length scale and was further refined by Menter and Egorov [51] to overcome the limitations of earlier models. The SAS model introduces the von Karman length scale, a key term absent in standard RANS models, enabling it to dynamically adapt its length scale based on resolved flow structures. This adjustment, driven by the second velocity derivative, significantly improves the model’s ability to replicate experimental observations in flows with large separation zones or instabilities. Furthermore, the von Karman length scale can be integrated into different scale-defining equations, extending SAS functionality to various turbulence models, as demonstrated in the SAS-SST model. Importantly, the additional term in the ω -equation of the SAS-SST model has been carefully designed to maintain standard RANS behavior for wall-bounded flows while enabling a switch to SRS (scale-resolving simulation) mode in cases of unstable flow separations.
Detailed explanations of the SAS methodology and its derivation can be found in the referenced works, particularly Menter and Egorov [51], which provide comprehensive insights into the development and implementation of the model.
Despite the refinement of the computational mesh and the use of a fine azimuth increment, certain limitations persist in the numerical model. The 2-D simulations inherently neglect three-dimensional effects, such as tip vortices and spanwise flow. Additionally, the Reynolds-averaged Navier–Stokes (RANS) approach, while effective for steady and mildly unsteady flows, may struggle to fully capture transient phenomena, particularly in regions of high turbulence intensity and dynamic stall. Future studies could employ more advanced turbulence models, such as scale-resolving simulations (e.g., LES or DES), to address these challenges.

2.4. Mesh Sensitivity Test

An essential part of validating the developed numerical model is conducting a mesh sensitivity test. The purpose of this test is, of course, to demonstrate how changes in mesh density affect the final results. In this study, a two-bladed rotor model was analyzed using four different mesh densities. The nominal mesh (Figure 2) used had 1689 divisions along the edge of a single airfoil: 840 elements on the suction side, 840 elements on the pressure side, and 9 elements at the trailing edge. Volume statistics of the mesh reveal that the minimum cell volume is 1.05 × 10−10 m3, while the maximum cell volume is 9.5 × 10−2 m3. The generated mesh was refined in the rotor region to ensure accurate resolution of the blade wake and flow structures, while coarser cells were used further away from the rotor to reduce computational cost. The distribution of mesh elements was uniform. This mesh had already been employed in previous analyses of the clean airfoil’s performance [22,23]. For the present work, the mesh was subjected to similar sensitivity tests.
To analyze the sensitivity of the numerical solution to mesh density, four different meshes were used, each differing in the number of divisions along the airfoil’s edge. The number of mesh elements was either decreased or increased using a factor of the square root of two, which is a common practice in mesh sensitivity tests [52]. No other mesh parameters were examined in this study. As shown in Figure 3, the number of divisions used in the nominal mesh is sufficient. The size of the mesh used for further investigations is approximately 270,000 cells.

2.5. Aerodynamic Load Consistency Analysis Across Rotor Revolutions

Unlike the classical URANS approach, simulations using the SAS model capture more vortex structures in the flow [8]. As a result, the aerodynamic loads for each subsequent revolution are less repeatable than in typical URANS simulations, especially when a 2-bladed rotor operates, as in this case, at a tip speed ratio of 4.5, where moderate local angles of attack are expected. In Figure 4, the average tangential force coefficient is shown as a function of the number of revolutions from the 30th to the 40th. The black dots represent the average values for each individual rotor revolution, while the red dashed line indicates the average over the last ten revolutions. The vertical axis of the graph has been scaled to highlight the differences; however, as can be seen, the variations in force values between revolutions are minimal compared to the overall average. Moreover, as shown in Figure 5, the difference in the tangential force coefficient CT as a function of the azimuth between the last computed revolution and the average of all ten revolutions is also very small. Therefore, in the results section, only the aerodynamic load components for the last rotor revolution are compared.

3. Validation of the Reference Case: Velocities and Analysis of Loads

3.1. Velocity Profiles in the Rotor Wake

This section of the paper focuses on the validation of the 2-D SAS (scale-adaptive simulation) approach, addressing both the velocity field behind the rotor and the instantaneous sectional loads on the rotor blades.
Figure 6 compares the computed velocity profiles of the Vx component in the rotor wake for several positions downstream of the rotor, denoted by the coordinate x. The velocity profiles were collected using rakes, each two rotor diameters in length and positioned perpendicular to the rotor axis. Each rake consisted of 1000 checkpoints. The distances of the rakes, shown on the graph, are referenced to the rotor diameter D. For better visualization of the results, a color palette was used where the farther the rake is from the rotor, the colors transition from cool blue tones to warm orange for the most distant rake studied in this work.
The results presented in this figure highlight several important observations. First, the 2-D CFD results tend to underestimate compared to the experimental data. The computed velocities are slightly higher, particularly in the positive y/D range, which corresponds to the blade moving “upwind.” The second key observation is that the general trend of both the experimental and numerical results is similar: as the x-coordinate increases, the velocity Vx decreases more significantly. Lastly, there is a noticeable asymmetry in the experimental velocity profiles compared to the CFD results, with the experimental data showing greater asymmetry.
One of the likely reasons for the differences in velocity profiles is the absence of the rotating tower in our model, which was neglected. Another factor could be 3-D effects, which were also not accounted for. Additionally, it is possible that even with the SAS approach, the numerical dissipation is too high to predict the numerous structures in the aerodynamic wake accurately. In this study, the mesh density in the wake region was not analyzed. Another potential cause could be the aerodynamic characteristics of the airfoil. In this case, the airfoil moves through an environment with varying turbulence intensity, transitioning from high to low turbulence regions. As shown by Michna and Rogowski [23], turbulence intensity has a significant impact on the aerodynamic characteristics of clean airfoils.
The plot shows a noticeable deviation between the SAS simulation and the experimental data for the Vy velocity component in the rotor wake (Figure 7). The agreement between the CFD results and the experiment is better within the y/D range from −0.5 to 0.5. Although the differences between the SAS and experimental results appear significant, the absolute values of the Vy velocity component are relatively small. The likely causes for these discrepancies stem from certain assumptions made in the model, such as the 2-D approach, the absence of a rotating shaft, and the lack of struts. Nevertheless, the computed results preserve the correct trend, capturing not only the tilt but also the expected behavior as the x-coordinate increases.

3.2. Aerodynamic Loads Validation

In their study, Castelein et al. [53] measured the normal and tangential force coefficients for an H-type vertical axis wind turbine (VAWT) operating at various azimuthal positions using velocity fields obtained through Particle Image Velocimetry (PIV). By employing the Noca method [54], which relies on these PIV-derived velocity fields, they calculated the aerodynamic forces around the blade without direct force measurements. This indirect approach allowed them to observe force trends associated with different flow characteristics [53].
At a tip-speed ratio (TSR) of 4.5, the authors found that the normal force generally followed expected trends for VAWTs, while the tangential force showed minimal positive values, indicating limited power production between the leeward and upwind positions. The authors noted that a high deviation in tangential force was observed at the azimuthal position 270°, which they attributed to interactions with the wake from the turbine mast. Given the observed uncertainties, especially in the tangential force measurements, the authors recommended further research to reduce these uncertainties, suggesting an improvement in post-processing methods to enhance the accuracy of tangential force estimations [53].
Due to the high uncertainty in the tangential load component obtained in the experimental studies, only the normal blade load component was used for validation in this work. The loads presented in Figure 8 are expressed in the dimensionless form C N = F N / 0.5 · ρ · V 0 2 · R , where ρ is the air density, assumed to be 1.225 kg/m3 in the calculations, V0 is the undisturbed flow velocity, and R is the rotor radius. The results obtained using the SAS approach, shown in Figure 8, in the upstream part of the rotor, up to the maximum values of the normal force, are in reasonably good agreement with the experiment. In the remaining azimuthal range, larger differences are observed, most likely due to 3-D effects and secondary effects, such as struts or the rotating shaft, that are not taken into account in this investigation.

4. Reference Case: Local Angle of Attack, Relative Velocity, and Dynamic Lift and Drag Coefficients Assessment

4.1. Methodology for Determining Local Angle of Attack and Relative Velocity

The previous section discussed both the methodology used in this paper and the accuracy of the approach in determining the velocity field. A reliable description of the velocity field is essential for estimating the local angle of attack and relative velocity, which are critical parameters for blade element theory-based methods, such as the actuator line approach. While comparing velocities or loads obtained from different methods is relatively straightforward, extracting the angle of attack, especially for vertical axis wind turbines (VAWTs), poses a greater challenge.
One of the primary contributions of this paper is the analysis of the angle of attack and relative velocity for a blade profile in a Darrieus-type VAWT. The aim of this study is not to compare various existing methods for determining this parameter but rather to adopt one of the approaches proposed by Melani et al. [45] for calculating the angle of attack. Before delving into the method used to determine the local angle of attack, we will briefly introduce the basic definitions of velocity and angle of attack.
The angle of attack is defined as the angle between the chord line of the blade profile and the direction of the relative velocity (Figure 9). For the reference rotor, the chord is tangential to the path (pitch angle β is equal to 0°). The relative velocity vector, Vrel, results from the vector sum of the blade’s tangential velocity V B T = ω R (taken with a negative sign) and the airflow velocity V acting on the blade. The airflow velocity at the turbine rotor is lower than the undisturbed wind speed far upstream as the rotor decelerates the air passing through it. In the one-dimensional stream-tube theory used in the classical blade element momentum (BEM) method, only one component of the flow velocity in the undisturbed flow direction, Vx, is considered. In this study, however, the airflow velocity component Vy, perpendicular to the undisturbed flow direction, is also taken into account, as shown in Figure 9.
Based on the figure above and considering the pitch angle β, with its sign convention illustrated in this figure, the angle of attack can be calculated using the normal and tangential components, Vn and Vt, of the relative velocity vector according to the following mathematical formula:
α = tan − 1 ⁡ − V n / V t + β
where
V t = V x cos ⁡ θ + V y sin ⁡ θ + ω R
V n = − V x sin ⁡ θ + V y cos ⁡ θ
where ω is the angular velocity of the rotor, and R is the rotor radius.
In this study, only one method for extracting the local angle of attack was used: the so-called Line Average method, which was introduced by Jost et al. [55] and successfully implemented by Melani et al. [47]. The goal of Jost et al. [55] was to achieve greater accuracy compared to previously existing approaches in accounting for the effects of shed and trailing vorticity on the measured angle of attack. According to Melani et al. [45], a circular sampling line with a radius of r = 1c was defined around the airfoil profile, centered at the aerodynamic center located at 1/4 of the chord length from the leading edge (Figure 10). Along this line, a series of points were evenly distributed. The authors of this method tested different sampling resolutions, ranging from six to eighty points. The contribution of this study is to test this method further for a larger range of control point counts. Figure 11 presents a sensitivity analysis of the angle of attack based on the number of sampling points used. In these analyses, a maximum of 2000 points was considered. For each tested number of points, the process of picking control points on the circular line was performed 100 times with a random shift but with the same even distribution. The mean and standard deviation of difference relative to the maximum points case were calculated. As shown in Figure 11, the highest sensitivity was observed for the lowest number of control points (five points) and for azimuth angles between 80° and 180°. The distribution of velocity components Vx and Vy as a function of the angle of attack for various azimuth angles reveals a local peak in these components within a narrow range of azimuths, resulting from the presence of the wake behind the profile. For instance, at an azimuth of 0°, the average component is 7.94 m/s, while its extreme value reaches −8.81 m/s. Moreover, the area of the line where this peak occurs constitutes 1.63% of the total. Thus, evenly but sparsely distributed control points may lead to significant errors in estimating the flow velocity. The desired flow velocity V is the integral average of the flow velocity field along a closed line around the airfoil. In conclusion, the figures above demonstrate that 200 sampling points provide sufficient accuracy for determining the true angle of attack. This resolution ensures that the local flow characteristics are captured effectively, minimizing errors and enhancing the reliability of the angle of attack estimation.
In the previous paragraph, we discussed the methodology for estimating the local angle of attack. This angle is essential for determining the local flow velocity at the rotor, specifically on the rotor blade. Based on this velocity and the blade’s tangential velocity, the relative airflow velocity vector can be estimated. Using the previously determined tangential and normal components of this velocity, we can express it as follows:
V → r e l = V → n + V → t
The sensitivity analysis for relative velocity was also performed in this case (Figure 12), using the mean and the standard deviation approach as previously applied for the angle of attack. As shown in the figure above, 200 sampling points are sufficient to achieve an accurate estimation of the relative velocity. This resolution minimizes fluctuations and ensures a reliable measurement across the azimuth range, demonstrating that higher sampling densities provide better stability and precision in capturing the true relative velocity profile.

4.2. Angle of Attack and Relative Velocity Analysis

Figure 13 presents the results of the local angle of attack and relative velocity obtained using the scale-adaptive simulation (SAS) approach with the Transition SST model and the angle of attack estimation method described in the previous subsection of this paper. In this study, the local angle of attack was calculated based on the velocity field around the blade, derived from 2-D CFD simulations with a resolution of Δ θ = 5 ° . The α ( θ ) and V r e l ( θ ) curves shown in this figure are thus developed from 200 points evenly distributed around the blade trajectory. Due to fluctuations in the normal component of the aerodynamic load, our local angle of attack predictions do not capture all the details of angle variations at Δ θ resolutions smaller than 5°. Despite these limitations, Figure 13 shows that the maximum local angle of attack in the upwind part of the rotor reaches 9.65°, which closely matches the independent result obtained by Melani et al. [45].
Our analysis does not account for the rotating rotor shaft in the downwind region, which explains the absence of an additional peak in α ( θ ) around an azimuth of 270°. Nevertheless, it is noteworthy that the local angle of attack in this part of the rotor also aligns well with the results from [45]. The local angle of attack in the downwind part remains relatively constant over most azimuths, ranging between −5.8° and −5.3°. For the second independent dataset from Cacciali et al. [8], angles of attack starting from an azimuth of approximately 92° are larger than those presented in this study and those reported by Melani et al. [45]. This discrepancy arises from a different sampling approach for the velocity used to calculate the angle of attack. Additionally, Cacciali et al. [8] incorporated extra submodels in their ALM approach along with the Smagorinsky–Lilly model. In Cacciali et al.’s study [8], the maximum local angle of attack reaches around 11°. Further differences are due to the use of external experimental airfoil data by the cited researchers. As demonstrated in [22], the Transition SST approach used in this paper and the k-ω SST model applied in Melani et al. [45] yield significant differences in lift coefficient characteristics (CL) for angles below the critical angle, which is also documented in recent independent experimental research by Rogowski et al. [56].
The relative velocity derived from the calculated local velocity field around the blade is shown in the lower part of Figure 13. Despite the greater discrepancies in angle of attack estimates compared to Cacciali et al. [8], the differences in relative velocity values are minimal. The maximum local relative velocity calculated using the SAS approach is 50.36 m/s, occurring at an azimuth of 15°. The minimum value of this velocity is still observed in the upwind part of the rotor, at an azimuth of 175°, and is approximately 35 m/s.

4.3. Dynamic Lift and Drag Coefficients Analysis

Figure 14 presents a comparison of dynamic lift (CL) and drag (CD) coefficients with static data based on different turbulence models: Transition SST (γ-Reθ) and k-ω SST [23]. The same coefficients are shown in two ways: as a function of azimuth (top plots) and as a function of angle of attack (bottom plots). In each plot, the red open circles represent results for the azimuth range from 0 to 180 degrees (SAS), the red filled circles represent results for the range from 180 to 360 degrees (SAS), and the black open circles indicate reference data from Melani et al. [45]. Presenting the results in this way allows for an understanding of the differences in aerodynamic characteristics at various stages of the azimuthal cycle and provides an assessment of the SAS-based results compared to established literature data. The results obtained using the scale-adaptive simulation (SAS) approach exhibit significantly more chaotic behavior, particularly in the upwind part of the rotor, starting at an azimuth of approximately 50–60 degrees. This effect coincides with frequent changes in the normal force, as shown in Figure 8. Since these fluctuations occur more frequently than the sampling points for the angle of attack, we opted to display only the points on these plots to capture the local values of the coefficients better.
It is notable that the slope of the static CL curve corresponds closely to that of the dynamic curve despite the fact that the latter does not pass through zero angle of attack. As noted by Bianchini et al. [36], this is due to the “virtual camber” effect, which can influence results for the dynamic angle of attack. The k-ω SST model, used in the study by Melani et al. [45], appears to yield better results for more turbulent regions of the flow, whereas the Transition SST model seems more suitable for regions with lower turbulence intensity, such as the downwind part of the rotor. Presenting both dynamic and static results provides valuable insight into how different turbulence models impact the accuracy of aerodynamic calculations in various phases of the turbine’s operating cycle.
Rezaeiha et al. [57], who also considered the reference rotor presented in this study, obtained fairly similar hysteresis loops for both lift and drag forces. In their research, a slightly different method of sampling the velocity components was used to determine the local angle of attack. Specifically, they applied velocity sampling at monitoring points on a circular path with a fixed distance of 0.2d upstream of the turbine. Additionally, these researchers determined the aerodynamic force components using the geometric angle of attack. The differences, particularly in the drag coefficient, are highly dependent on both sides of the rotor. This suggests that our use of the SAS approach does not significantly impact aerodynamic force predictions; rather, the primary influence comes from the turbulence model.

5. Impact of Rotor Configuration on Aerodynamic Force Components and Local Angle of Attack

5.1. Number of Blade Effects

Figure 15 shows the coefficients of the aerodynamic blade load components, tangential and normal. The results presented in these figures correspond to a rotor operating at a tip speed ratio of 4.5. The colors, ranging from red to blue, represent four rotor configurations based on the number of blades: red for the single-bladed rotor and blue for the four-bladed rotor. The results displayed in these graphs suggest a significant influence of the blade number on turbine performance. The primary consequence of a reduced blade count is notably higher loads, especially tangential loads, in the downwind part of the rotor. For the two- and three-bladed rotors, the tangential force is zero or even negative, indicating a low or negative contribution of this component to energy production. In the case of the normal components, the differences in load are less pronounced. However, for both the tangential and normal components, the largest discrepancies are observed for the single-bladed configuration.
Figure 15, bottom-left and bottom-right plots, compare the local angle of attack, α, and relative velocity, V r e l , respectively, for a rotor operating at a tip speed ratio of 4.5, as a function of the number of blades. The most significant differences in local angle of attack are observed across the entire rotor, while for relative velocity, the differences are mainly present in the downwind part of the rotor. The large variations in the angle of attack correlate with significant differences in tangential force, as shown in this figure. The maximum local angle of attack is observed at an azimuth angle of approximately 100–105 degrees and decreases linearly from 10.16 degrees for a single-bladed rotor to 6.34 degrees for a four-bladed rotor. As the number of blades decreases, the shape of the angle of attack curve becomes more sinusoidal and approaches the geometric angle of attack, as the angle of attack in the downwind region of the rotor increases. Increasing the number of blades causes the local angles of attack in the downwind part to decrease, and their distribution becomes almost flat, similar to the tangential force characteristics.
A reduction in the number of blades also causes the maximum local angle of attack to approach or even slightly exceed the critical static angle of attack. This explains the local peaks in the angle of attack, as well as the characteristic wavy pattern of the tangential and normal force distributions in the azimuth range of approximately 70 to 210 degrees, which results from the numerous vortices generated on the airfoil.
Figure 15 also shows that at zero azimuth, the local angle of attack is non-zero and slightly increases with the number of blades, from −1.67 degrees for the single-bladed rotor to −2.38 degrees for the four-bladed rotor.
Despite the significant influence of rotor solidity on the local angle of attack in both the upstream and downstream regions, notable differences in relative velocity are only observed in the downstream part of the rotor, starting around an azimuth angle of 155 degrees, still within the wake region. A higher number of blades causes a greater deflection of the relative velocity curve. Interestingly, all curves intersect at the same point, approximately 265 degrees. In the azimuth range from 155 to 265 degrees, the relative velocity decreases as the number of blades increases, while beyond this characteristic azimuth, the trend is reversed.
Figure 16 presents a comparison of aerodynamic coefficients for different rotor configurations with one to four blades. The plots of drag coefficient CD and lift coefficient CL as a function of azimuth angle θ (Figure 16a,b) illustrate how the number of blades influences the distribution of aerodynamic forces during the last rotation of the rotor. The same coefficients are also shown as a function of the angle of attack α (Figure 16c,d), allowing the observation of hysteresis loops. It is evident that as the number of blades increases, and consequently the solidity increases (Equation (1)), the range of angles of attack decreases. This, in turn, leads to a reduction in tangential force, as discussed in detail in the previous paragraph. Additionally, the aerodynamic force characteristics acting on the rotor blade elements exhibit fewer oscillations with an increasing number of blades, indicating more stable aerodynamic properties.

5.2. Pitch Angle Effects

Figure 17, top-left and top-right plots, compare the load characteristics of a two-bladed rotor with three different pitch angles: −10, 0, and 10 degrees. As shown in the first figure, both large positive and negative pitch angles result in a significant deterioration of the rotor’s aerodynamic performance. However, current research trends in Darrieus wind turbines are focused on evaluating the aerodynamic wake for different rotor configurations and the potential for active and passive control of this wake. Therefore, in this comparison, the more critical load component is the normal force. The figure illustrates the expected trend in estimated loads and highlights the significant impact of dynamic effects, which generate vortex structures manifesting as oscillations in the curves.
Figure 17’s bottom-left and bottom-right plots illustrate the effect of pitch angles on the local angle of attack and relative velocity, respectively. As observed in both figures, regarding the shape of the curves, the pitch angle does not significantly influence the characteristics of the relative velocity compared to the number of rotor blades. For the angle of attack, however, the pitch angle generally causes a shift in the curve by about 10 degrees. The most notable differences are seen with a pitch angle of 10 degrees. The angle of attack curve (Figure 17) displays several sharp peaks. This curve could be smoother, and the number of peaks could increase if the angle of attack were sampled with a much smaller azimuthal increment. Additionally, the case with a 10-degree pitch angle represents the rotor under the heaviest load compared to the 0-degree and −10-degree cases. However, this does not apply to the tangential load component, where the average tangential force coefficients are −0.24, 0.08, and −0.14 for pitch angles of 10 degrees, 0 degrees, and −10 degrees, respectively. For the normal load, the average force coefficients are 2.2, 0.5, and −1.16 for the same pitch angles, respectively. The maximum magnitudes of the normal force coefficients are 5.45, 3.05, and 3.85, respectively.
As demonstrated in their work by Melani et al. [45], the static critical angle of attack for the NACA0018 airfoil, as predicted by the four-equation transition turbulence model, is approximately 12 degrees. As shown in Figure 17, this angle is reached at an azimuth of about 30 degrees. In the case of a 10-degree pitch angle, the absence of oscillations in the normal force component is observed from around 28–30 degrees azimuth. Significant oscillations in both aerodynamic load components and a sudden loss of tangential force begin to appear around an azimuth of approximately 55 degrees. Therefore, it can be inferred that in the azimuth range of approximately 30 to 55 degrees, dynamic effects occur that do not cause a loss of lift. In the case of the downstream part of the rotor, the obtained angle of attack values for a pitch angle of −10 degrees may indicate that the rotor is operating within the range of maximum dynamic lift coefficients.
As demonstrated in [9], due to significant 3-D effects in this case resulting from large tip losses near the blade ends, the results obtained using the 2-D CFD approach should be interpreted more qualitatively than quantitatively. For a more accurate quantitative comparison, a rotor with a much higher blade aspect ratio would need to be constructed than the reference rotor used in this study. However, in the same study [9], we showed that for a pitch angle of −10 degrees, the impact of these tip losses is not as significant. The oscillations in both the angle of attack and aerodynamic loads visible in the downstream part of the rotor indicate high turbulence intensity and numerous vortices.
Figure 18 illustrates the effect of blade pitch angle on the aerodynamic coefficients of the rotor. In plot (a), the drag coefficient CD is shown as a function of azimuth angle θ. For non-zero pitch angles, the drag coefficient is generally higher, which can be attributed to the exceeding of critical static angles of attack in both cases. It is clearly visible that for β = 10°, a sharp increase in drag occurs in the windward region of the rotor, whereas for the negative pitch angle (β = −10°), a similar increase is observed in the leeward region. In plot (d), a noticeable shift in the CL characteristic can be seen, depending on the pitch angle. The highest values of the lift coefficient CL are observed for β = 10° in the windward section of the rotor, reaching values close to 2. This analysis clearly shows that increasing the pitch angle intensifies the lift force while significantly affecting the distribution of aerodynamic forces as a function of rotor azimuth.

6. Conclusions

This paper analyzes the aerodynamics of a reference VAWT rotor using the well-established Transition SST turbulence model combined with the scale-adaptive simulation (SAS) approach to enhance the accuracy of velocity field representation in the rotor area. Here are some key conclusions drawn from our study:
  • The study demonstrates that the Line Average method is an effective approach for accurately determining the local angle of attack, as it considers the effects of shed and trailing vorticity around the airfoil. By averaging velocity components along a circular line centered at the aerodynamic center, this method minimizes errors from wake effects, particularly when using a sufficient number of sampling points. Our analysis shows that 200 sampling points offer a reliable resolution, capturing essential flow characteristics while minimizing sensitivity to the azimuth angle, especially in regions prone to high variability. This method was validated for different rotor configurations, and the trend in resulting values appears consistent and accurate across configurations, supporting the robustness of this approach.
  • Furthermore, the findings of this work hold significant potential for advancing engineering methodologies tailored to the aerodynamic analysis of vertical axis wind turbines (VAWTs). Accurate determination of the local angle of attack is a crucial step in aerodynamic modeling, as it enables precise estimation of aerodynamic loads acting on the blades. These loads directly influence the design, performance optimization, and operational stability of wind turbines. The methodology developed and validated in this study could be instrumental in improving predictive capabilities for aerodynamic performance and dynamic loading in future studies. We believe the results presented here provide a strong foundation for the continued development of engineering models aimed at enhancing the design and efficiency of wind turbines with vertical axes.
  • The comparison of turbulence models highlights their varying performance in capturing aerodynamic and dynamic characteristics within the rotor region. While the SAS approach with the Transition SST model allows for more precise velocity field mapping and captures oscillations in normal forces, the overall aerodynamic loads remain similar to those obtained using the standard Transition SST model or even the classical URANS approach reported by Rezaeiha et al. [57]. Additionally, the k-ω SST model yields results comparable to the Transition SST model for aerodynamic load components, local angle of attack, and relative velocity. However, it differs in capturing the dynamic characteristics of drag and lift coefficients as functions of the local angle of attack. Furthermore, the Transition SST model shows better performance in the upwind region of the rotor due to transitional phenomena, while the k-ω SST model is more effective in handling the higher turbulence levels in the downwind region.
  • The analysis demonstrates that the number of blades has a significant impact on the aerodynamic performance of the rotor. Fewer blades lead to notably higher aerodynamic loads, especially tangential loads, in the downwind region, with single-bladed configurations often showing minimal or even negative contributions to energy production. Variations in the local angle of attack and relative velocity are also strongly influenced by blade count, with fewer blades causing larger fluctuations in the angle of attack, which approaches a sinusoidal pattern and even exceeds the critical static angle at times. Higher blade counts stabilize the angle of attack and in the downwind region, resulting in a flatter distribution of loads. This highlights the crucial role of blade count in shaping the aerodynamic forces and flow stability around the rotor.
  • The analysis showed that large positive and negative pitch angles significantly impact the rotor’s aerodynamic performance, especially by increasing loads on the normal force component. Dynamic effects, including vortex formations, cause oscillations, especially pronounced at a 10-degree pitch angle. While pitch angle has minimal effect on relative velocity, it shifts the angle of attack curve, with the most considerable differences observed for a 10-degree pitch. Notably, high turbulence intensity and vortex shedding are evident in the downstream region, especially for a −10-degree pitch angle. Varying the blade pitch angle alters the aerodynamic force characteristics, with positive pitch angles (β > 0°) enhancing lift generation but also increasing drag, particularly in the windward region. Negative pitch angles (β < 0°) reduce drag in the upwind section but lead to higher drag in the downwind region, highlighting the trade-off between aerodynamic efficiency and force distribution. Overall, these results indicate that extreme pitch angles lead to performance fluctuations, which are further amplified by 3-D effects and tip losses, suggesting the need for higher blade aspect ratios for quantitative accuracy.
The next step of this work will be employing the Line Average method in the pitching airfoil model. This model can be a simplified model of VAWT’s blade during rotation. Determining the angle of attack during that airfoil pitch motion is crucial to investigating dynamic phenomena influencing the airfoil aerodynamic loads. That will be the next important step in analyzing the modeling of the aerodynamic performance of vertical axis wind turbines methods.

Author Contributions

Conceptualization, K.R. and J.M.; methodology, K.R. and J.M.; software, K.R. and J.M.; validation, K.R. and J.M.; formal analysis, K.R. and J.M.; investigation, K.R. and J.M.; resources, K.R. and J.M.; data curation, K.R. and J.M.; writing—original draft preparation, K.R. and J.M.; writing—review and editing, K.R. and J.M.; visualization, K.R. and J.M.; supervision, K.R.; project administration, K.R.; funding acquisition, K.R. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the POB Energy program at Warsaw University of Technology, under the Excellence Initiative: Research University (IDUB) (Grant No. 1820/355/Z01/POB7/2021). Computational resources were provided by the Interdisciplinary Centre for Mathematical and Computational Modelling at the University of Warsaw (ICM UW) through computational allocations no. G93-1588 and G94-1718.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Diagram of an H-type vertical axis wind turbine structure with dimensions.
Figure 1. Diagram of an H-type vertical axis wind turbine structure with dimensions.
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Figure 2. Domain schema with dimensions (upper) and mesh layout with boundary conditions for the CFD simulation (bottom). The diagram shows the defined boundary conditions: velocity inlet on the left, pressure outlet on the right, symmetry on the top and bottom, wall on the airfoil surface, and interface region connecting different mesh zones. Enlarged sections illustrate the mesh refinement around the airfoil and interface areas.
Figure 2. Domain schema with dimensions (upper) and mesh layout with boundary conditions for the CFD simulation (bottom). The diagram shows the defined boundary conditions: velocity inlet on the left, pressure outlet on the right, symmetry on the top and bottom, wall on the airfoil surface, and interface region connecting different mesh zones. Enlarged sections illustrate the mesh refinement around the airfoil and interface areas.
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Figure 3. Variation of the mesh sensitivity test results: The plot shows the normal force coefficient CN (red) and the tangential force coefficient CT (blue) as a function of the number of mesh cells. CN is plotted on the left axis and CT on the right axis. The results indicate that both coefficients stabilize as the mesh density increases from 180,000 to 340,000 cells.
Figure 3. Variation of the mesh sensitivity test results: The plot shows the normal force coefficient CN (red) and the tangential force coefficient CT (blue) as a function of the number of mesh cells. CN is plotted on the left axis and CT on the right axis. The results indicate that both coefficients stabilize as the mesh density increases from 180,000 to 340,000 cells.
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Figure 4. Variation of the tangential force coefficient CT as a function of the number of rotor revolutions. Black dots represent values averaged over individual rotor revolutions, while the red dashed line shows the average CT over the last ten rotor revolutions. The figure illustrates the consistency of the aerodynamic loads over the examined revolutions, with minimal variation observed between individual revolutions and the overall average.
Figure 4. Variation of the tangential force coefficient CT as a function of the number of rotor revolutions. Black dots represent values averaged over individual rotor revolutions, while the red dashed line shows the average CT over the last ten rotor revolutions. The figure illustrates the consistency of the aerodynamic loads over the examined revolutions, with minimal variation observed between individual revolutions and the overall average.
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Figure 5. Comparison of the tangential force coefficient CT as a function of azimuth angle θ. The red solid line represents the tangential load component calculated for the last rotor revolution, while the blue dashed line shows the same component, averaged over ten revolutions, both plotted as a function of the azimuth angle.
Figure 5. Comparison of the tangential force coefficient CT as a function of azimuth angle θ. The red solid line represents the tangential load component calculated for the last rotor revolution, while the blue dashed line shows the same component, averaged over ten revolutions, both plotted as a function of the azimuth angle.
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Figure 6. Comparison of the Vx/V0 velocity component in the rotor wake for various downstream positions x/D, calculated using the SAS approach and validated against experimental data [46].
Figure 6. Comparison of the Vx/V0 velocity component in the rotor wake for various downstream positions x/D, calculated using the SAS approach and validated against experimental data [46].
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Figure 7. Comparison of the Vy/V0 velocity component in the rotor wake for various downstream positions x/D, calculated using the SAS approach and validated against experimental data [46].
Figure 7. Comparison of the Vy/V0 velocity component in the rotor wake for various downstream positions x/D, calculated using the SAS approach and validated against experimental data [46].
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Figure 8. Comparison of simulated and experimental force coefficients [53] as a function of azimuthal angle, θ. (a) Normal force coefficient CN, shows a general agreement between the SAS simulation and experimental data. (b) Tangential force coefficient, C.
Figure 8. Comparison of simulated and experimental force coefficients [53] as a function of azimuthal angle, θ. (a) Normal force coefficient CN, shows a general agreement between the SAS simulation and experimental data. (b) Tangential force coefficient, C.
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Figure 9. Illustration of the angle of attack (α) and relative velocity (Vrel) for a blade in a vertical-axis wind turbine. The pitch angle (β = 10°) and the azimuthal position angle (θ = 48°) are indicated.
Figure 9. Illustration of the angle of attack (α) and relative velocity (Vrel) for a blade in a vertical-axis wind turbine. The pitch angle (β = 10°) and the azimuthal position angle (θ = 48°) are indicated.
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Figure 10. Velocity distribution around the airfoil at an azimuthal angle θ = 48°. The circular sampling line with evenly distributed points surrounds the airfoil, with arrows representing the local flow velocities at each point.
Figure 10. Velocity distribution around the airfoil at an azimuthal angle θ = 48°. The circular sampling line with evenly distributed points surrounds the airfoil, with arrows representing the local flow velocities at each point.
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Figure 11. Sensitivity analysis of the angle of attack ( α − α 2000 ¯ ) based on the number of sampling points along the circular line surrounding the airfoil as a function of the azimuth angle θ. Standard deviation is chosen as an error measure.
Figure 11. Sensitivity analysis of the angle of attack ( α − α 2000 ¯ ) based on the number of sampling points along the circular line surrounding the airfoil as a function of the azimuth angle θ. Standard deviation is chosen as an error measure.
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Figure 12. The mean relative velocity is represented by lines. Standard deviation is chosen as an error measure.
Figure 12. The mean relative velocity is represented by lines. Standard deviation is chosen as an error measure.
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Figure 13. Validation of angle of attack (α) and relative velocity (Vrel) results obtained using the SAS approach, compared with literature data from Melani et al. [45] and Cacciali et al. [8].
Figure 13. Validation of angle of attack (α) and relative velocity (Vrel) results obtained using the SAS approach, compared with literature data from Melani et al. [45] and Cacciali et al. [8].
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Figure 14. The upper subfigures illustrate the drag coefficient CD (a) and lift coefficient CL (b) as functions of the azimuthal angle θ. The lower subfigures depict the same coefficients—drag (c) and lift (d)—analyzed for varying angles of attack α. The presented data include results obtained using the SAS approach, static results reported by Rogowski et al. [22], and reference data from Melani et al. [45].
Figure 14. The upper subfigures illustrate the drag coefficient CD (a) and lift coefficient CL (b) as functions of the azimuthal angle θ. The lower subfigures depict the same coefficients—drag (c) and lift (d)—analyzed for varying angles of attack α. The presented data include results obtained using the SAS approach, static results reported by Rogowski et al. [22], and reference data from Melani et al. [45].
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Figure 15. Effect of the number of blades on the normal and tangential force coefficients, as well as on the local angle of attack and relative velocity, as a function of azimuth θ at a tip-speed ratio of 4.5.
Figure 15. Effect of the number of blades on the normal and tangential force coefficients, as well as on the local angle of attack and relative velocity, as a function of azimuth θ at a tip-speed ratio of 4.5.
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Figure 16. The subfigures show the drag coefficient CD (a,c) and lift coefficient CL (b,d) for different azimuthal angles (a,b) and angles of attack (c,d). The presented data shows comparison of aerodynamic coefficients for 1-bladed, 2-bladed, 3-bladed, and 4-bladed configurations.
Figure 16. The subfigures show the drag coefficient CD (a,c) and lift coefficient CL (b,d) for different azimuthal angles (a,b) and angles of attack (c,d). The presented data shows comparison of aerodynamic coefficients for 1-bladed, 2-bladed, 3-bladed, and 4-bladed configurations.
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Figure 17. Effect of the pitch angle on the normal and tangential force coefficients, as well as on the local angle of attack and relative velocity, as a function of azimuth θ for a 2-bladed rotor configuration at a tip-speed ratio of 4.5.
Figure 17. Effect of the pitch angle on the normal and tangential force coefficients, as well as on the local angle of attack and relative velocity, as a function of azimuth θ for a 2-bladed rotor configuration at a tip-speed ratio of 4.5.
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Figure 18. The subfigures show the drag coefficient CD (a,c) and lift coefficient CL (b,d) for different azimuthal angles (a,b) and angles of attack (c,d). The data presented above shows aerodynamic coefficient comparison for varying blade pitch angles (β = −10°, 0°, 10°).
Figure 18. The subfigures show the drag coefficient CD (a,c) and lift coefficient CL (b,d) for different azimuthal angles (a,b) and angles of attack (c,d). The data presented above shows aerodynamic coefficient comparison for varying blade pitch angles (β = −10°, 0°, 10°).
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Michna, J.; Rogowski, K. A Refined Approach for Angle of Attack Estimation and Dynamic Force Hysteresis in H-Type Darrieus Wind Turbines. Energies 2024, 17, 6264. https://doi.org/10.3390/en17246264

AMA Style

Michna J, Rogowski K. A Refined Approach for Angle of Attack Estimation and Dynamic Force Hysteresis in H-Type Darrieus Wind Turbines. Energies. 2024; 17(24):6264. https://doi.org/10.3390/en17246264

Chicago/Turabian Style

Michna, Jan, and Krzysztof Rogowski. 2024. "A Refined Approach for Angle of Attack Estimation and Dynamic Force Hysteresis in H-Type Darrieus Wind Turbines" Energies 17, no. 24: 6264. https://doi.org/10.3390/en17246264

APA Style

Michna, J., & Rogowski, K. (2024). A Refined Approach for Angle of Attack Estimation and Dynamic Force Hysteresis in H-Type Darrieus Wind Turbines. Energies, 17(24), 6264. https://doi.org/10.3390/en17246264

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