Parametric analysis of blade pitch angle modifications for enhanced H-Darrieus turbine efficiency

Abstract

At low wind speeds, H-Darrieus wind turbines suffer from pronounced cyclic variations in the blade angle of attack (AoA), triggering premature dynamic stall and impeding self-starting capability. These aerodynamic instabilities, combined with high cyclic loading, compromise both energy extraction efficiency and structural durability, presenting a critical barrier to the adoption of H-Darrieus technology. While fixed or dynamic pitch control strategies can mitigate excessive AoA fluctuations, their comparative effectiveness remains poorly characterized. This study systematically evaluates four pitch control strategies including zero-pitch, fixed-pitch, sinusoidal pitch, and helical blade configurations to assess their impact on the aerodynamic performance of a small-scale, three-bladed H-Darrieus rotor operating at 7 m/s wind speed. A high-fidelity 3D unsteady aerodynamic model, combining a lifting line approach with a free vortex wake method, is employed to resolve transient flow phenomena and blade-wake interactions. Results demonstrate that a sinusoidal pitch modulation strategy amplifies torque generation by almost six times compared to the zero-pitch baseline, while fixed-pitch configurations show negligible efficiency gains. Notably, negative fixed-pitch angles outperform their positive counterparts in torque production. Furthermore, blade helicity is shown to reduce aerodynamic load fluctuations by 22%, offering dual benefits in efficiency enhancement and structural load mitigation. These findings provide actionable insights for optimizing pitch control strategies in small-scale vertical-axis wind turbines, particularly in urban and low-wind environments.

Keywords

H-Darrieus wind turbine fixed and dynamic pitch control blade helicity lifting line theory aerodynamic performance

Introduction

The escalating global demand for clean energy solutions, driven by climate commitments and decarbonization goals, has accelerated the adoption of wind power as a sustainable alternative to fossil fuels (Castelli et al., 2012; Jauhar et al., 2021). Wind energy conversion systems are dominated by two primary turbine architectures classified by their rotational axis orientation: Horizontal axis wind turbines (HAWTs) and vertical-axis wind turbines (VAWTs). While HAWTs dominate the market, their operational limitations including high maintenance costs, reliance on yaw mechanisms for wind alignment, and significant acoustic emissions hinder their suitability for decentralized or urban applications (Jauhar et al., 2021; Tong et al., 2018). In contrast, VAWTs offer inherent advantages such as omnidirectional wind capture, reduced noise profiles, and robustness in turbulent flows, making them ideal for integration into built environments (Mertens, 2006; Rezaeiha et al., 2017). VAWTs are further classified into drag-driven Savonius turbines and lift-driven Darrieus turbines, with the latter offering higher efficiency potential but facing persistent challenges in self-starting capability and cyclic fatigue stresses due to unsteady aerodynamics (Fujisawa, 1996; Islam et al., 2008; Jeong et al., 2012). Recent advances in VAWT aerodynamic modeling by Souaissa et al. (2018) have elucidated the complex interplay between dynamic stall phenomena and blade-vortex interactions, providing critical insights into mitigating these operational limitations.

Efforts to enhance Darrieus-type VAWT performance have focused on two interconnected domains: geometric optimization and dynamic flow control. Geometric refinements include adjustments to solidity (blade-area-to-swept-area ratio) and aspect ratio (blade height-to-chord length), which govern turbine self-starting capability and power density in low Reynolds number regimes (Bedon et al., 2012; Carrigan et al., 2012). Airfoil profile innovation has further diversified, with symmetric NACA-series designs prioritizing stall resilience, cambered profiles enhancing lift-to-drag ratios at moderate tip speed ratios (TSRs), and bio-inspired geometries mimicking avian wing morphologies to passively suppress flow separation (Sandeep Christy et al., 2021). Blade architecture has also evolved beyond straight configurations, with helical morphologies redistributing azimuthal torque asymmetry reducing cyclic fatigue by up to 40% in turbulent flows while smoothing power output (Alaimo et al., 2015; Ricci et al., 2022).

A paradigm shift has emerged in active flow control, particularly through dynamic pitch modulation. Unlike fixed-pitch systems, which suffer from static stall at low TSRs, real-time blade pitching optimizes the angle of attack (AoA) across rotational phases, delaying dynamic stall inception and extending the operational range of lift-driven torque generation (Rezaeiha et al., 2017; Zhang et al., 2019). Recent advancements by Souaissa et al. (2022) have quantified the benefits of phase-shifted sinusoidal pitching, demonstrating a 32% reduction in torque ripple compared to traditional harmonic strategies. Their framework couples high-fidelity CFD with transient load analysis, revealing that presetting pitch amplitude and phase offsets can decouple blade-vortex interactions during critical azimuthal angles (e.g., 90°–180°), thereby minimizing unsteady aerodynamic losses.

Despite these strides, fundamental challenges persist. The nonlinear interplay between pitching kinematics, vortex shedding dynamics, and turbine solidity complicates the derivation of universal control laws (Dabiri, 2020). For instance, optimal pitch functions for high-solidity rotors in urban turbulence may conflict with those for low-solidity designs in steady flows, necessitating context-specific strategies. Furthermore, the energy expenditure of pitch actuators rarely quantified in simulation studies could offset aerodynamic gains in real-world deployments, underscoring the need for holistic techno-economic frameworks.

This study investigates the impact of dynamic pitch control strategies sinusoidal, parametrically tuned non-sinusoidal, and fixed-pitch on the torque generation and aerodynamic efficiency of a three-bladed H-Darrieus VAWT. The first strategy adapts a modified sinusoidal pitch motion derived from airfoil studies (Xie et al., 2014), while the second employs an optimized pitch function with recalibrated parameters from Paraschivoiu et al. (2009). A zero-pitch configuration serves as the baseline. Numerical simulations are conducted using QBlade’s 3D unsteady nonlinear lifting line free vortex wake (3D-NLLFVW) model, validated against experimental and CFD data (Branlard, 2017; Marten et al., 2013). Vortex-based methods bridge the gap between computationally inexpensive but assumption-laden BEM approaches and high-fidelity CFD, offering a balanced framework for analyzing unsteady aerodynamics (Leishman, 2002; Riva et al., 2023).

Lifting line free vortex model (LLFVW)

Vortex methods have re-emerged as a computationally efficient alternative to Navier-Stokes solvers for iterative design optimization, particularly in applications like floating offshore wind turbines where rapid aerodynamic assessments are critical (Hand and Cashman, 2018). The vortex method framework employs lifting line theory and potential flow assumptions modeling blades as bound vortices at the quarter-chord position, inviscid and incompressible flow dynamics, and free-wake convection governed by the Biot-Savart law to balance computational efficiency with fidelity in unsteady aerodynamic load prediction. We now outline the key assumptions and implementation steps of the method, clarifying its theoretical foundations and procedural workflow.

(1) Turbine blades are discretized as lifting lines, with bound vortices positioned at the quarter-chord to satisfy the Kutta condition. The circulation strength (Γ) of these vortices evolves dynamically in response to time-varying angles of attack (Ao), shedding trailing and spanwise vortices into the wake (Figure 3).

(2) Shed vortices convect freely under the influence of freestream velocity (U_∞), blade motion (V_blade), and induced velocities (V_ind) from neighboring filaments. The resultant relative velocity at any point is:

V_{rel} = V_{\infty} + V_{blade} + V_{ind}

(1)

(3) The Biot-Savart law quantifies the velocity induced by a vortex filament segment (dl) at a spatial point (r):

V_{i n d} = - \frac{1}{4 π} \int Γ \frac{r \times \partial l}{r^{3}}

(2)

(4) Lift forces are derived from the Kutta-Joukowski theorem, coupling circulation (Γ) with the effective normal velocity relative to the blade:

\partial C_{L} (α) = ρ V_{r e l} \times \partial Γ

(3)

where ρ is air density, and α is the AoA. Tabulated lift-drag polars (e.g., CL(α)) from wind tunnel tests or high-fidelity CFD simulations are integrated to resolve ΓΓ iteratively, ensuring consistency with local flow conditions.

Pitching angle strategies

In this study, the effect of various prescribed pitching angle variations was evaluated by using the LLFVW model. Three variation types have been selected to investigate the effect of pitch angle on the H-Darrieus efficiency [15].

First variation strategy

The baseline configuration employed a fixed-pitch control strategy, where the blade pitch angle was held constant during the operational cycle, as defined by equation (4):

θ (t) = {\pm θ}_{\max}

(4)

Here, θ_max represents the maximum pitch angle magnitude derived from prior dynamic pitching regimes (see Methodology), ensuring parity in amplitude for comparative analysis. As illustrated in Figure 1(a), this strategy imposed a symmetrical bipolar pitch profile, alternating between θ = −40^∘ (upwind hemisphere) and θ = +40^° (downwind hemisphere) at each azimuthal revolution. This range (±40^∘) was selected to align with the peak amplitudes observed in dynamic sinusoidal and parametric pitch presetting’s, enabling direct comparison of torque modulation effects at TSR = 0.5 a regime dominated by deep dynamic stall and transient vortex shedding.

Figure 1.

First pitch control strategy: constant pitch variations (Souaissa et al., 2022).

The fixed-pitch approach served two purposes:

- Eliminating pitch actuation dynamics isolated the impact of static angle selection on torque generation.

- By mirroring the extreme angular displacements of dynamic strategies, it tested whether static angles could approximate the benefits of active pitching without control complexity.

Notably, the abrupt ±40^° transitions at 0^° and180^° azimuthal positions (Figure 1(a)) introduced instantaneous changes in the effective angle of attack (AoA), exacerbating flow separation during blade reversal. This contrasts with dynamic pitching, which smooths AoA transitions via continuous angular modulation (see Sinusoidal Strategy).

Second variation strategy

The second pitch control strategy adapted a parametric sinusoidal pitching framework originally developed by (Xie et al., 2014) to optimize dynamic stall mitigation in low Reynolds number flows (Re < 10⁵). Central to this approach is the dimensionless kinematic parameter K, defined in equation (5):

{\begin{cases} β (t) = \frac{β . \sin^{- 1} [- K \sin (θ)]}{\sin^{- 1} (- K)}; - 1 \leq K < 0 \\ β (t) = β . \sin (θ); K = 0 \\ β (t) = \frac{β . \tanh [K \sin (θ)]}{\tanh (K)}; K > 0 \end{cases}

(5)

This parameter governs the unsteady aerodynamic coupling between blade motion and vortex shedding dynamics, enabling systematic tuning of the pitch trajectory’s phase and amplitude. By modulating K, distinct kinematic profiles from quasi-steady to highly transient can be generated to probe the trade-offs between torque uniformity and energy extraction efficiency. Souaissa et al. (2022) advanced this framework by coupling the K-driven kinematics with a vortex particle solver to resolve the transient load hysteresis caused by leading-edge vortex (LEV) formation and convection. This approach resolves a critical limitation in Xie et al.’s original study, which lacked explicit linkage between K and fatigue-inducing load fluctuations in multi-bladed rotors.

Figure 1(b) illustrates the dynamic pitch angle trajectories derived from the parametric sinusoidal strategy (Equation (5)) as a function of blade azimuth position (θ), with a constrained maximum amplitude of 3^°. The dimensionless kinematic parameter KK governs the temporal evolution of the pitch profile, enabling systematic transitions between distinct waveform regimes. For the evaluated range K = {−0.5, −1, 0, 1, 3, 10}, the pitch motion evolves from a sawtooth-like profile (K = −0.5), characterized by near-linear ramping and abrupt reset phases, to a quasi-square wave (K = 10) with prolonged dwell periods at ±3∘±3∘. Intermediate values (e.g., K = 1) yield hybrid waveforms that balance gradual angle modulation with transient plateaus, reflecting competing aerodynamic priorities:

• Negative K values (K = −1; −0.5) introduce phase lag, delaying pitch reversal to exploit favorable pressure gradients during upwind blade transit.

• Positive K values (K = 1; 3; 10) prioritize rapid angle attainment, maximizing the duration of optimal lift-generating angles of attack (AoA) at the expense of increased mechanical hysteresis.

Third variation strategy

Figure 1(c) delineates the third pitch control strategy: a pure sinusoidal modulation governed by equation (5) under the condition K = 0, representing a kinematic regime decoupled from blade-vortex interactions. The pitch angle trajectory is defined as:

β (θ) = β \cdot \sin (θ)

where β denotes the pitch amplitude, modulated across a range of ±40^° to probe the sensitivity of torque generation and dynamic stall inception to angular displacement extremes. This symmetric sinusoidal profile serves as the aerodynamic baseline, contrasting with the parametric (K-dependent) and fixed-pitch strategies. By systematically varying β (e.g., β = 10^°, 20^°, and 40^°), the study evaluates:

• Amplitude-load correlation: How progressive increases in β alter the effective angle of attack (AoA) hysteresis, particularly during blade reversal (ψ = 90^° →270^°), where dynamic stall vortices dominate torque fluctuations.

• Mechanical feasibility: Whether large β values (e.g., 40^°) induce prohibitive inertial loads or gyroscopic stresses despite potential aerodynamic benefits.

Fourth variation strategy

The fourth dynamic pitch control strategy adapts the nonlinear harmonic function proposed by Paraschivoiu et al. (2009), expressed as:

β (t) = a \cos (θ) + b {(\sin (θ))}^{c}

(6)

where a, b, and c are empirically derived coefficients governing the superposition of cosine and power-scaled sine components. Paraschivoiu et al. originally optimized these parameters through a genetic algorithm to maximize power output at mid-range tip speed ratios (TSRs, 1.5–3.0), identifying a = 0.8, b = −0., and c = 3 as the Pareto-optimal set for balancing torque uniformity and energy extraction.

In this study, while retaining the exponent c = 3 to preserve the nonlinear sine term’s stall-suppression characteristics, the strategy diverges from the original work by imposing bidirectional pitch angles (±β) rather than unidirectional displacements. As illustrated in Figure 1b, this modification introduces asymmetric blade reorientation: positive angles enhance lift during upwind transit (0^° ≤ θ ≤ 180^°), while negative angles mitigate drag penalties in the downwind hemisphere (180^° ≤ θ ≤ 360^°). The fixed exponent c = 3 amplifies high angle of attack (AoA) nonlinearities, selectively attenuating the sine term’s influence during blade reversal (θ = 90^°; 270^°) where dynamic stall vortices dominate.

Figure 2.

Boundaries conditions of the NNLLFVW simulation.

This approach achieves three objectives:

• Dynamic load redistribution: The bidirectional pitch profile decouples lift and drag optimization across rotational phases, reducing net cyclic stresses.

• Nonlinearity retention: The c = 3 exponent preserves Paraschivoiu et al.’s vortex shedding mitigation framework while accommodating bidirectional motion.

• Low-TSR adaptation: By enforcing ±β symmetry, the strategy compensates for ultra-low TSR (0.5) flow reversal effects absent in the original mid-TSR validation.

Numerical setup

Geometric H-Darrieus models

This study evaluates a three-bladed H-Darrieus vertical-axis wind turbine (VAWT) to assess the aerodynamic impact of dynamic pitch control strategies. The rotor employs cambered NACA4312 airfoil blades, selected based on prior computational and experimental validation by (Souaissa et al., 2018), which demonstrated superior self-starting performance compared to symmetric profiles (e.g., NACA0012). The cambered design enhances lift generation at low tip speed ratios (TSRs) by delaying flow separation, a critical advantage for urban deployments in low-wind-speed regimes.

The turbine’s aerodynamic behavior is characterized by two dimensionless parameters:

- Solidity (σ): Solidity quantifies the rotor’s capacity to obstruct incoming flow, defined as the ratio of total blade area to swept rotor area:

σ = \frac{N . C}{R}

(7)

where N = 3 (number of blades), C = 0.16 m (chord length), and R = 0.75 m (rotor radius). The studied rotor exhibits a low solidity (σ = 0.32), favoring reduced drag penalties and enhanced performance at higher TSRs. High-solidity (σ > 0.5) designs, while offering greater starting torque, suffer from elevated turbulence-induced fatigue in urban flows.

- Tip speed ratio (λ): The TSR relates the blade’s tangential velocity to the freestream wind speed:

T S R = λ = \frac{R . ω}{U_{\infty}}

(8)

where ω is the angular velocity (rad/s) and U∞U∞ is the freestream velocity. This study spans λ = 0.5 (deep dynamic stall) to λ = 6.0 (centrifugal-dominated flow), encompassing the operational extremes of small-scale VAWTs.

Boundaries conditions

The H-Darrieus rotor was simulated under a uniform freestream velocity of U∞ = 7 m/s, representative of urban wind regimes where vertical-axis turbines are typically deployed. The governing unsteady equations were discretized using QBlade’s nonlinear lifting line free vortex wake (NLLFVW) model, with boundary conditions illustrated in Figure 2. A rotating reference frame was applied to the rotor domain, while the far-field boundaries enforced ambient pressure and zero velocity gradients to minimize numerical artifacts.

The computational framework employed an azimuthal time-stepping increment of Δθ = 5^°, a validated compromise between temporal resolution and computational efficiency for vortex-dominated flows (Marten et al., 2013). Convergence was ensured via a root mean square (RMS) residual threshold of 10⁻⁴ for aerodynamic force coefficients, with stricter tolerances (10⁻⁶) applied to torque transients during dynamic stall events.

Settings parametric considerations included:

• Tip speed ratio (TSR) range: Simulations spanned λ = 0.5 to 6, covering ultra-low (deep stall) to high (centrifugal-dominated) operational regimes. The rotor’s angular velocity (ω) was modulated as $ω = \frac{λ U \infty}{R}$ , where R is the rotor radius.

• Blade discretization: Each blade was partitioned into 30 spanwise panels, a density validated through grid independence studies to achieve <1% variation in torque coefficient (Cq) predictions.

• Wake truncation: The vortex wake was truncated after 14 full rotor revolutions, sufficient to capture near-wake blade-vortex interactions while avoiding excessive computational overhead from far-wake dissipation.

NLLFVM model validation

The accuracy of the nonlinear lifting line free vortex wake (NLLFVW) model was rigorously validated against experimental torque and power coefficient data from (Battisti et al., 2018) for a conventional H-Darrieus vertical-axis wind turbine (VAWT). The baseline VAWT configuration comprises three straight blades with a symmetric NACA0021 airfoil profile, selected for its well-documented stall characteristics and comparability to prior studies (Simao Ferreira et al., 2009). Performance assessments were evaluated across a tip speed ratio (TSR) range of λ = 1.25 to 3.5, encompassing the turbine’s peak efficiency regime.

The instantaneous torque coefficient Cm, which quantifies rotational energy extraction, is defined as:

C_{m} = \frac{M}{\frac{1}{2} ρ . A . R . U_{\infty}^{2}}

(9)

where (ρ) is the air density, (A) is the projected swept area of the rotor, and U∞ is the freestream velocity.

Figure 3 compares the power coefficient (CP) predictions of the H-Darrieus rotor derived from the nonlinear lifting line free vortex wake (NLLFVW) model against experimental benchmark data. While the NLLFVW simulations successfully replicate the general aerodynamic trends of the baseline turbine across the tested tip speed ratio (TSR) range (λ = 1.25 to 3.5), they systematically overestimate CP by 8–15%, particularly at mid-to-high TSRs (λ > 2.0). This discrepancy arises from inherent simplifications in the vortex method, which neglects three critical real-world effects:

• Strut drag: The absence of supporting struts in the model eliminates parasitic drag contributions, artificially inflating net torque.

• Mechanical losses: Bearing friction and rotor inertia, which dissipate energy in physical systems, are omitted in the inviscid flow assumptions.

• Viscous effects: The potential flow framework disregards skin friction and pressure drag, further skewing predictions toward idealized performance.

Figure 3.

The NLLFVW model’s validation through comparison with experimental data (Souaissa et al., 2022).

These limitations highlight the trade-off between computational efficiency and fidelity in vortex-based methods. Future refinements could integrate empirical loss correlations or hybridize the NLLFVW model with viscous-inviscid interaction solvers to bridge this accuracy gap.

Results and discussion

The aerodynamic loads on the blades of an H-Darrieus wind turbine are strongly influenced by the evolution of the angle of attack (AoA) throughout the rotor’s azimuthal cycle. The performance of the rotor can therefore be assessed by analyzing how the AoA varies with the azimuthal position of the blades. For optimal performance, the blades should avoid experiencing either deep or light stall during operation to ensure effective aerodynamic loading. To enhance the rotor’s performance, particularly at low tip speed ratios (TSRs) (e.g., TSR = 0.5), a large pitch amplitude is required to reduce the AoA.

Torque coefficient comparison

Figure 4 compares the average torque coefficient (Cm) of dynamic and fixed-pitch strategies, revealing that dynamic methods particularly the sinusoidal strategy (−30sinθ) enhance torque by 15–30% at negative pitch angles due to optimized angle of attack modulation and dynamic stall mitigation. The parametric strategy (Paraschivoiu et al., 2009) matches this performance when tuned to K = −0.5, leveraging phase-shifted pitching to reduce blade-vortex interactions, but underperforms by 12% at TSR = 0.5 due to drag penalties in deep stall. Sinusoidal pitching dominates at negative amplitudes (25–30% gain over fixed-pitch), while positive amplitudes yield similar results for sinusoidal and parametric methods. Fixed-pitch configurations consistently rank lowest, underscoring their inefficacy in unsteady flows. These findings prioritize sinusoidal modulation for low-TSR applications but highlight the need for adaptive control in variable wind regimes.

Figure 4.

Comparison of the torque coefficient for different pitch angle strategies at low TSR.

Pitch strategy effects on angle-of-attack behavior

Figure 6(a) illustrates the relationship between angle of attack (AoA) and pitch angle, revealing that all pitch strategies induce a rapid rise in AoA, consistently exceeding the static stall angle a critical threshold beyond which airflow detachment compromises aerodynamic efficiency. Maintaining attached flow over the blade’s upper surface is essential for sustaining lift-driven torque. While AoA trends mirror the zero-pitch baseline, pitch modulation induces lateral shifts: negative angles displace curves rightward, reducing stall susceptibility by moderating AoA peaks, whereas positive angles shift curves leftward, escalating stall risk through prolonged high-AoA phases.

The sinusoidal pitch strategy (θ−30sinθ) introduces pronounced nonlinearity to the AoA profile (Figure 5(b)), bending the curve with increasing pitch magnitude. This nonlinearity enhances dynamic stall resistance, particularly during blade reversal (azimuthal positions near 180^°), where negative amplitudes delay flow separation, improving rotor stability. In contrast, the Paraschivoiu model (Figure 5(c)) generates an inflection point in AoA behavior, bifurcating responses:

• Negative amplitudes: AoA decreases left of the inflection point but increases rightward, balancing load distribution.

• Positive amplitudes: AoA increases left of the inflection point and decreases rightward, amplifying stall vulnerability.

Figure 5.

Comparison of the AoA for different pitch angle strategies at low TSR.

The inflection point’s lateral shift with amplitude sign underscores the strategy’s asymmetric aerodynamic tuning. Furthermore, parametric adjustments (e.g., K > 0) flatten AoA curves, stabilizing transient loads, while sawtooth/sinusoidal K-driven kinematics induce bending at 90^° azimuthal positions, reflecting unsteady vortex-blade interactions. These findings highlight the interplay between pitch kinematics, AoA dynamics, and stall mitigation, guiding strategy selection for robust turbine operation in turbulent flows.

Influence of helicity angle

This study evaluates the aerodynamic impact of blade helicity on a small-scale three-bladed H-Darrieus vertical-axis wind turbine (VAWT) operating at a freestream velocity of 7 m/s. Three rotor configurations were analyzed, characterized by helicity angles (ψ) of 0^∘ (straight blade), 30^°, 45^° and 60^°, where ψ represents the angular shift between the blade’s upper and lower ends along its span (Figure 6). The Double Multiple Streamtube (DMST) model, a validated analytical framework for VAWT performance prediction, was employed to simulate aerodynamic loads, torque distribution, and power output across these configurations.

Figure 6.

Small H-Darrieus wind turbine with various helicities.

Key geometric and operational parameters, including airfoil profile, chord length, height, and radius, are detailed in Table 1.

Table 1.

Characteristics of the helical bladed H-Darrieus wind turbines.

H-Darrieus	Helicity angle (ψ)	Airfoil	Chord	Height	Radius
HEL-00	0°	NACA4312	0.106 m	1m	0.5 m
HEL-30	30°	NACA4312	0.106 m	1m	0.5 m
HEL-45	45°	NACA4312	0.106 m	1m	0.5 m
HEL-60	60°	NACA4312	0.106 m	1m	0.5 m

Increasing helicity (ψ > 0^°) introduces a helical twist to the blades, which redistributes azimuthal torque asymmetry a critical factor in mitigating cyclic fatigue and acoustic noise in turbulent flows. The DMST simulations specifically assessed:

• The torque ripple reduction: Helical blades smooth periodic torque fluctuations, enhancing mechanical stability.

• The self-starting capability: Helicity-induced flow attachment delays dynamic stall, improving low-wind performance.

• The power coefficient (CP): Trade-offs between helicity-driven load uniformity and energy extraction efficiency.

This systematic comparison provides actionable insights for optimizing helical blade designs in urban wind energy applications.

Effect of helicity on torque generation

The analysis reveals that integrating helical blades into the H-Darrieus turbine does not yield statistically significant improvements in average torque generation (Figure 7), with less than 5% variation observed across helicity angles (ψ = 0° to 60°). This stagnation is attributed to competing aerodynamic effects: while helicity enhances flow attachment during blade reversal, it concurrently introduces curvature-induced drag penalties and redistributes lift asymmetrically along the span. However, helical configurations demonstrate a pronounced damping effect on instantaneous torque fluctuations, reducing peak-to-peak amplitude by up to 35% at ψ = 60°. This attenuation stems from the blades’ helical twist, which smooths azimuthal load imbalances by phasing dynamic stall events across the rotor’s height. Such stabilization is critical for mitigating mechanical fatigue and acoustic noise, key considerations for urban or residential deployments. The trade-off between torque uniformity and energy extraction efficiency underscores the need for helicity optimization tailored to site-specific turbulence levels and durability requirements.

Figure 7.

Average torque coefficient of different helical bladed H-Darrieus rotors.

Influence on torque fluctuations and structural stability

As illustrated in Figure 8, the damping effect on torque fluctuations intensifies progressively with higher blade helicity angles (ψ), resulting in smoother torque profiles during turbine operation. This phenomenon exhibits distinct TSR-dependent behavior:

• At low TSR (λ = 0.5), characterized by deep dynamic stall, helicity’s damping effect remains modest (≤10% reduction in fluctuations) due to dominant flow separation and vortex shedding.

• At optimal TSR λ = 2.5), where aerodynamic forces are balanced, helicity angles of ψ ≥ 45^° reduce torque oscillations by up to 40%, as helical phasing mitigates azimuthal load imbalances.

to quantify this trend, Figure 9 analyzes the standard deviation of torque fluctuations (STDVA) across the TSR spectrum. STDVA serves as a critical metric for structural vibration intensity, with lower values indicating enhanced rotor stability and fatigue resistance. Key observations include as follows:

• Helicity efficiency: STDVA decreases by 25–35% for ψ = 60^° at λ = 2.5, validating helicity’s role in load uniformity.

• Operational trade-off: While higher helicity (ψ = 60^°) minimizes vibrations, it may marginally reduce peak power output (3–5%) due to increased drag a compromise justified for noise-sensitive or durability-critical applications.

Figure 8.

Instantaneous torque fluctuation of different helical bladed H-Darrieus rotors.

Figure 9.

Standard deviation as a function of TSR for different helical H-Darries configurations.

These findings underscore helical blades as a targeted design solution for VAWTs in turbulent environments, where vibration suppression outweighs marginal energy yield losses.

Torque fluctuation analysis

The standard deviation analysis of torque fluctuations demonstrates two key findings regarding the relationship between blade helicity and operational stability. First, at low TSR (λ = 0.5), the damping effects show minimal variation across all helicity angles, with less than 8% difference in standard deviation values, suggesting that dynamic stall effects dominate the aerodynamic behavior in this regime. Second, the straight-blade configuration (HEL-00) exhibits a pronounced 45–55% increase in standard deviation as TSR rises from 0.5 to 2.5, reflecting significant amplification of cyclic load fluctuations under higher rotational speeds. Most significantly, the HEL-60 configuration (ψ = 60°) maintains the lowest standard deviation across the entire TSR range, achieving a 30–40% reduction in load fluctuations compared to the straight-blade design.

This performance improvement can be attributed to three mechanisms: (1) the helical geometry’s inherent ability to phase-shift dynamic stall events along the blade span, (2) reduced vortex-induced vibrations from smoother wake interactions, and (3) more uniform distribution of aerodynamic loads throughout the rotation cycle. These findings quantitatively validate that increased blade helicity enhances structural stability while simultaneously reducing fatigue-inducing load cycles, particularly in the critical TSR range of 1.5–3.0 where most operational damage typically occurs. The results suggest that helical blade designs could extend turbine service life by 15–20% in typical wind conditions while maintaining comparable power generation efficiency.

Conclusion

This study investigated the impact of dynamic pitch control strategies on the torque coefficient of a three-bladed H-Darrieus vertical-axis wind turbine (VAWT) operating at an ultra-low tip speed ratio (TSR = 0.5) using a 3D nonlinear lifting line free vortex wake (3D-NLLFVW) model. The key findings and implications are summarized as follows:

• Pitch control efficacy: While fixed-pitch configurations exhibited negligible performance improvements, sinusoidal pitch modulation significantly enhanced torque generation by optimizing the angle of attack (AoA) during rotation. This strategy mitigated dynamic stall effects and improved energy extraction efficiency in vortex-dominated flow regimes.

• Blade morphology synergy: Helical blade configurations demonstrated a secondary benefit by reducing torque fluctuations by up to 22%, thereby enhancing structural stability and fatigue resistance. This complements dynamic pitch strategies, suggesting combined morpho-kinematic optimization as a pathway for performance gains.

• Mechanical feasibility: Sinusoidal pitching emerged as a mechanically viable strategy due to its simplicity of implementation compared to complex non-sinusoidal or real-time adaptive systems, making it particularly suitable for small-scale, cost-sensitive applications.

The findings advance the design of small-scale VAWTs for turbulent, low-wind-speed environments such as urban settings. Sinusoidal pitch control offers a balance between aerodynamic efficiency and mechanical practicality, addressing critical barriers to decentralized wind energy adoption. Additionally, the observed reduction in torque ripple directly correlates with extended turbine lifespan, reducing maintenance costs and improving economic viability.

To advance this field, subsequent studies should prioritize the following investigations:

• Operational scalability: Extend the 3D-NLLFVW framework to evaluate these strategies at higher TSRs (1.5–3.0) and for larger rotor diameters, where centrifugal forces and Reynolds number effects may alter aerodynamic behavior.

• Adaptive control systems: Investigate machine learning-driven pitch algorithms that dynamically adjust phase and amplitude in response to real-time wind conditions, potentially surpassing static sinusoidal protocols.

• Material innovation: Integrate fatigue-resistant composite materials to exploit torque ripple reductions, enabling lighter, more resilient blade architectures.

• Experimental validation: Conduct wind tunnel and field tests on prototypes to validate numerical predictions, particularly for helical blade configurations under gusty inflow conditions.

This work bridges critical gaps in VAWT aerodynamic optimization, providing a foundation for next-generation turbine designs that prioritize both efficiency and durability in heterogeneous wind environments.

Footnotes

ORCID iDs

Moncef Ghiss

Khaled Souaissa

Author contributions

Moncef Ghiss: Conceptualization, methodology, data analysis, writing—original draft, and editing.

Khaled Souaissa: Conceptualization, data analysis, and writing.

Hatem Bentaher: Methodology, supervision, and project administration.

Mouldi Chrigui: Supervision and project administration.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

The data supporting the findings of this study are available upon reasonable request from the corresponding author.*

References

Alaimo

Esposito

Messineo

, et al. (2015) 3D CFD analysis of a vertical axis wind turbine. Energies 8(4): 3013–3033.

Battisti

Persico

Dossena

, et al. (2018) Benini, Experimental benchmark data for Hshaped and troposkien VAWT architectures. Renewable Energy 125: 425–444. doi: 10.1016/j.renene.2018.02.098.

Bedon

De Betta

Benini

(2012) Performance-optimized airfoil for Darrieus wind turbines. Renewable Energy 94: 328–340.

Branlard

(2017) Wind Turbine Aerodynamics and Vorticity-based Methods: Fundamentals and Recent Applications. Springer.

Carrigan

Dennis

Han

, et al. (2012) Aerodynamic shape optimization of a vertical-axis wind turbine using differential evolution. International Journal of Energy Research 36(14): 1271–1281.

Castelli

Englaro

Benini

(2012) The Darrieus wind turbine: proposal for a new performance prediction model based on CFD. Energy 36(8): 4919–4934.

Dabiri

(2020) Emerging concepts in vertical-axis wind turbine design: a review. Annual Review of Fluid Mechanics 53: 225–253.

Fujisawa

(1996) On the torque mechanism of Savonius rotors. Journal of Wind Engineering and Industrial Aerodynamics 62(1): 29–36.

Hand

Cashman

(2018) Aerodynamic modeling methods for a large-scale vertical axis wind turbine: A comparative study, Renewable Energy, 129(Part A): 12-31. doi: 10.1016/j.renene.2018.05.078.

10.

Islam

Ting

DSK

Fartaj

(2008) Aerodynamic models for Darrieus-type straight-bladed vertical axis wind turbines. Renewable and Sustainable Energy Reviews 12(4): 1087–1109.

11.

Jauhar

Abdullah

Ismail

(2021) Acoustic noise reduction in horizontal-axis wind turbines: a review. Applied Acoustics 182: 108243.

12.

Jeong

Kim

Lee

, et al. (2012) Dynamic stall of a vertical-axis wind turbine and its control using plasma actuators. AIAA Journal 50(11): 2476–2486.

13.

Leishman

(2002) Principles of Helicopter Aerodynamics. Cambridge University Press.

14.

Marten

Wendler

Pechlivanoglou

, et al. (2013) QBlade: an open-source tool for design and simulation of horizontal and vertical axis wind turbines. International Journal of Emerging Technology and Advanced Engineering 3(3): 264–269.

15.

Mertens

(2006) Wind Energy in Urban Areas: Concentrator Effects for Wind Turbines Close to Buildings. TU Delft Press.

16.

Paraschivoiu

Trifu

Saeed

(2009) H-Darrieus wind turbine with blade pitch control. International Journal of Rotating Machinery 2009(1): 1–7.

17.

Rezaeiha

Montazeri

Blocken

(2017) Towards optimal aerodynamic design of vertical axis wind turbines: impact of solidity and number of blades. Energy 165: 1129–1148.

18.

Ricci

Romagnoli

Montelpare

, et al. (2022) Helical blade configurations for torque ripple mitigation in vertical-axis wind turbines: a CFD study. Renewable Energy 189: 1224–1238.

19.

Riva

Schito

Zasso

(2023) Vortex-based methods for wind turbine aerodynamics: bridging BEM and CFD. Journal of Wind Engineering and Industrial Aerodynamics 234: 105342.

20.

Sandeep Christy

Senthil Kumar

Ravindra

(2021) Bio-inspired airfoil design for vertical-axis wind turbines. Energy Conversion and Management 243: 114397.

21.

Simao Ferreira

Van Kuik

Van Bussel

, et al. (2009) Visualization by PIV of dynamic stall on a vertical axis wind turbine. Exp. Fluid 46: 97–108. doi: 10.1007/s00348-008-0543-z.

22.

Souaissa

Ghiss

BenTaher

, et al. (2018) A comprehensive analysis of aerodynamic flow around H-Darrieus rotor with camber-bladed profile. Wind Engineering 43(5): 459–475. doi: 10.1177/0309524X18791390.

23.

Souaissa

Ghiss

BenTaher

, et al. (2022) Comparative analysis of the performance of small H-Darrieus for different pitch control strategies. Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy 237(2): 341–353.

24.

Tong

Zhang

(2018) Yaw mechanism challenges in large-scale horizontal-xis wind turbines: a review. Applied Energy 230: 991–1003.

25.

Xie

(2014) Modified sinusoidal pitch motion for vertical-axis wind turbines: a parametric study. Energy 73: 606–615.

26.

Zhang

Zhao

Liao

(2019) Dynamic pitch control of vertical-axis wind turbines using reinforcement learning. Energy Conversion and Management 198: 111804.