This article presents a comparison study of different aerodynamic models for an X-shaped vertical-axis wind turbine and offers insight into the 3D aerodynamics of this rotor at fixed pitch offsets. The study compares six different numerical models: a double-multiple streamtube (DMS) model, a 2D actuator cylinder (2DAC) model, an inviscid free vortex wake model (from CACTUS), a free vortex wake model with turbulent vorticity (from QBlade), a blade-resolved unsteady Reynolds-averaged Navier–Stokes (URANS) model, and a lattice Boltzmann method (from PowerFLOW). All models, except URANS and PowerFLOW use the same blade element characteristics other than the number of blade elements. This comparison covers the present rotor configuration for several tip-speed ratios and fixed blade pitch offsets without unsteady corrections, except for the URANS and PowerFLOW which cover a single case. The results show that DMS and 2DAC models are inaccurate – especially at highly loaded conditions, are unable to predict the downwind blade vortex interaction, and do not capture the vertical/axial induction this rotor exhibits. The vortex models are consistent with each other, and the differences when compared against the URANS and PowerFLOW mostly arise due to the unsteady and flow curvature effects. Furthermore, the influence of vertical induction is very prominent for this rotor, and this effect becomes more significant with fixed pitch offsets where the flow at the blade root is considerably altered.

Vertical-axis wind turbines (VAWTs) have attracted significant attention as a promising renewable energy source due to their wind direction independence and their low noise and vibration characteristics

A render of the X-Rotor turbine with geometrical dimensions from

The X-Rotor

The characterisation of the aerodynamics of the primary rotor is a critical challenge in the design of the X-Rotor turbine. An initial characterisation was completed by

Models considered for benchmarking.

Accurate prediction of VAWT aerodynamic performance is essential for effective design and optimisation, as it directly affects the power output and efficiency of the turbine. Different aerodynamic models for VAWTs have been researched and used, each with advantages and limitations.

With the unique configuration of the X-Rotor, there is a significant influence of induction in the vertical direction expected due to the coned blades of this geometry configuration. Therefore, to characterise the aerodynamics of the X-Rotor, it is necessary to understand the discrepancies between different aerodynamic models. In the aforementioned publications, the 3D aerodynamic effects of pitch offsets have been studied in detail but they do not cover that of VAWTs with coned blades, specifically for the X-Rotor.

Hence, the scope of this article is twofold: (1) present a comparative study of different aerodynamic models for the X-Rotor's primary rotor and (2) investigate the 3D aerodynamics of the X-Rotor associated with fixed blade pitch offsets. The results of this study will provide valuable insights into the development of accurate and efficient VAWT design tools and contribute to the advancement of renewable energy technologies.

The specific objectives of this paper are as follows:

understand the agreement between the models presented here, based on the power, thrust, and blade forces;

obtain the range of operating conditions over which this agreement holds (of particular interest is the pitch offset);

look at the effects of vertical induction to understand where 3D models are necessary for coned VAWT simulations.

The aerodynamic models used in this study are described here. A short summary of the models used along with their fidelity is presented in Table

The discretisation procedure of the X-Rotor in DMS from

The double-multiple streamtube model, developed by

The 2D actuator cylinder model, developed by

Vertical discretisation procedure of the X-Rotor inside the 2DAC. These are constrained to the radius of the rotor at the specific height of the vertical 2D segment.

The Code for Axial and Cross-flow TUrbine Simulation (CACTUS) developed by

Representation of

QBlade is a turbine design and simulation tool to perform aerodynamic studies to facilitate the design of wind turbines developed by

The X-Rotor is also simulated using a URANS model through OpenFOAM

PowerFLOW 6-2021-R2, developed by 3DS SIMULIA

The geometry for the X-Rotor blades and intermediate values can be found through linear interpolation.

The X-Rotor rotor geometry consists of a radius of 25 and 75

Top view of the upper blade tip with

Due to the coned blades, the local tip-speed ratio pertaining to the local blade elements varies along the span and height. Therefore, the tip-speed ratios considered for analysis represent the value at the blade tips. Each of the low-fidelity and mid-fidelity models simulated the aerodynamic performance and loads of the turbine at a tip-speed ratio range of

A homogeneous, constant, freestream velocity of

While the effects of flow curvature

Both high-fidelity models inherently include flow curvature, dynamic stall, and viscous terms, as the rotor in these models is blade resolved. Additionally, the PFLOW simulations include the secondary rotor in the lower blade, the tower, and the cross-beam that connects the two sets of blades together. As the secondary rotors experience high thrust, this is expected to decrease the tangential forces of the lower blade in the azimuth where the blade does not experience stall, compared to the other simulation cases. As the cross-beam is not as aerodynamically significant compared to the blades of the rotor, the aerodynamic effects from this surface are neglected. The lower blade loads are also expected to be affected by the wake of the tower. However, this is expected to only have a significant impact around an azimuth

Variation in rotor power coefficient (

The results for power and thrust of the X-Rotor are discussed in detail in Sects.

The variation of

The mid-fidelity models agree well with each other in pitch ranges of

Variation in rotor thrust coefficient (

The variation of

Overall, the vortex models seem to agree well with the high-fidelity models, suggesting the momentum models are not quite valid at these high tip-speed ratios.

As the URANS and PFLOW predicted the results for one case, the normal and tangential loads are compared with the momentum and the vortex models in Sect.

Non-dimensionalised normal forces at

Non-dimensionalised tangential forces at

The normal and tangential forces at

Non-dimensionalised normal forces from upper and lower blades for

Non-dimensionalised tangential forces from upper and lower blades for

By increasing

From the normal force data presented in Fig.

Percentage difference of non-dimensional normal forces at fixed pitch offsets relative to zero pitch offset of vortex models. Light and dark blue indicate

The tangential forces from Fig.

To enhance this difference seen in the lower blade, Fig.

Vertical induction normalised by freestream velocity (

The X-Rotor consists of coned blades in its primary rotor geometry; therefore, a component of the normal forces from the blade acts in the axial or vertical direction. As observed in Fig.

Vertical induction normalised by freestream (

As the tip-speed ratio is different along the span, the vertical induction also varies along the span. To obtain an overview of the spanwise variation of vertical induction at these pitch cases, Fig.

Overall, the force field of the rotor is subject to change with pitch offsets as the vorticity system changes

A numerical comparison of different aerodynamic models was conducted to understand the aerodynamic characteristics and the performance of an X-shaped VAWT for a range of tip-speed ratios and pitch offsets. This study contributed the following: (1) a comparative study of different aerodynamic models for the X-Rotor's primary rotor and (2) the significance of 3D aerodynamics for the X-Rotor associated with fixed blade pitch offsets.

The models presented were the double-multiple streamtube (DMS), 2D actuator cylinder (2DAC), QBlade lifting line (QBlade), CACTUS lifting line (CACTUS), PowerFLOW (PFLOW), and unsteady Reynolds-averaged Navier–Stokes (URANS) CFD model.

This study showed that the DMS model performed almost on par with the other models in calculating rotor performance parameters when there was no pitch offset but predicted significantly different results once blade pitch was introduced, as the validity of the streamtubes enclosing the downwind actuator began to fail. This worsened at high tip-speed ratios where the DMS models predicted power coefficients at the Betz limit with pitch offsets.

The 2DAC model offered consistent data at small pitch offsets (between

The thrust predicted by the momentum models was different from that of the vortex models. Moreover, these models were unable to capture the vertical induction effects and the BVI that occurred in the downwind cycle of the rotor. Therefore, the 2D momentum models were mostly inaccurate in predicting the thrust, power, or blade loads for the X-Rotor due to the influence of 3D aerodynamics.

The QBlade and the CACTUS open-source simulations offered great consistency with each other, including capturing the 3D aerodynamics effectively. The QBlade model showed huge turbulent viscosity spikes in its results occurring throughout the normal and tangential forces, which was a result of the vortex core and turbulent viscosity model used in the solver. This behaviour also translated to the blade-integrated forces where the QBlade slightly overpredicted the results compared to CACTUS. However, both of their results remained consistent, suggesting their reliability over low-fidelity models for this specific rotor geometry.

URANS and PFLOW results were compared with the vortex and momentum models for power, thrust, and blade forces for one test case. Primarily, the URANS and PFLOW agreed very well with the thrust of the vortex models, although they predicted slightly less power. This was because URANS had small domain sizes, which caused enough blockage effects to affect the results, and the PFLOW modelled the tower as well as the secondary rotors, which also caused slight power loss. Secondly, the downwind half of the turbine showed significantly lower force magnitudes, which was due to a combination of flow separation at the blades and the inherent flow curvature effects present in the CFD solvers.

The distribution of vertical induction over the span and azimuth was studied to understand the 3D aerodynamics of the X-Rotor and also with pitch offsets. The vertical induction was dominated primarily by the tip vortices (due to spanwise lift variation) in cases with no pitch offset. Interestingly, with pitch offsets the root vortex showed greater influence on the vertical induction for the X-Rotor despite operating at a much lower local tip-speed ratio. Due to the large vertical induction fluctuations through the azimuthal cycle of VAWTs at fixed pitch offsets, it concluded that the 2D models lose their validity in these conditions and for coned VAWTs.

Parametric definition of the upper/top blade.

Parametric definition of the lower/bottom blade.

The number of cells is 72 million.

The mesh background is created using blockMesh. The mesh dimensions are described in Table

The mesh has been created using snappyHexMesh, where three refinement regions have been used:

annular disk in the upper blades; axis (0, 0, 70 m) to (0, 0, 100 m); radius 90 m with inner radius 60 m;

annular disk in the lower blades; axis (0, 0,

cylinder axis (0, 0,

With regard to meshing levels, the rotor surface is meshed with level 8, which leads to a minimum element size of 0.96

The rotation of the turbine is imposed in the domain using the arbitrary mesh interface (AMI) capability available in OpenFOAM. The boundary condition of the domain is listed in Table

The turbulence model used in the simulations is

Simulation domain description.

Boundary conditions.

The finest cell size is ^{®} Gold 6148 (500-core Skylake) 2.4

Grid and time-step sizes.

Secondary rotor specifications.

The simulations conducted are of the primary rotor coupled with the secondary rotor and the tower. The dimensions and the operating condition of the secondary rotor are mentioned in Table

Thrust of the secondary rotor over the azimuthal cycle of the primary rotor non-dimensionalised by the inflow velocity

The generated and processed data sets used in this paper are available at

AGA and LM did performed main research work; performed and analysed the numerical results from the momentum and vortex models; and wrote the paper, with AGA writing the most. YW developed and provided the numerical results for PowerFLOW. DB, AC, and OP performed the URANS simulations and presented the results. Through meetings and feedback, AGA, LM, YW, and CF contributed to interpreting and discussing the results. CF offered general guidance and also reviewed the paper. The paper was revised and improved by all authors.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The authors acknowledge the use of computational resources of the DelftBlue supercomputer (

This research has been supported by the funding received from the European Union's Horizon 2020 research and innovation programme under grant agreement no. 101007135 as part of the XROTOR project (

This paper was edited by Mingming Zhang and reviewed by two anonymous referees.