Wake Expansion and the Finite Blade Functions for Horizontal-Axis Wind Turbines

Wood, David

doi:10.3390/en14227653

Cited by 6 publications

(3 citation statements)

References 23 publications

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“…It should be noted, that in this section, the wind turbine wake is treated as a helix with constant radius while the wake expands behind a real wind turbine. However, Wood 26 demonstrates that the approximation of the Kawada‐Hardin equation for the axial induced velocity presented, here, remains accurate when compared to a Biot‐Savart evaluation of an expanding wake.…”

Section: Sweep Correction Modelmentioning

confidence: 81%

An efficient blade sweep correction model for blade element momentum theory

2022

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This article proposes an efficient correction model that enables the extension of the blade element momentum method (BEM) for swept blades. Standard BEM algorithms, assuming a straight blade in the rotor plane, cannot account for the changes in the induction system introduced by blade sweep. The proposed extension corrects the axial induction regarding two aspects: the azimuthal displacement of the trailed vorticity system and the induction of the curved bound vortex on itself. The extended algorithm requires little additional processing work and maintains BEM's streamtube independent approach. The proposed correction model is applied to simulations of swept blade geometries based on the IEA 15 MW reference wind turbine. Results show good agreement with lifting line simulations that inherently can account for the swept blade geometry. Blade sweep couples bending and torsion deformations by curving the blade axis in the inplane direction. As such, it can be used to passively alleviate loads and, thus, aeroelastically tailor wind turbine blades. The implementation of aeroelastic tailoring techniques, and the aeroelastic analysis in general, becomes increasingly significant with the size of wind turbine rotors continually rising. Due to its low computing complexity, BEM remains a crucial tool in the aerodynamic and aeroelastic analysis of wind turbine rotors. Thus, the proposed correction model contributes to a fast and accurate evaluation of swept blade designs.

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Section: Sweep Correction Modelmentioning

confidence: 81%

An efficient blade sweep correction model for blade element momentum theory

2022

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“…Simple and accurate approximations to the KH solutions are described in and were used to analyse the effects of low λ by Wood et al (2016) and Vaz & Wood (2016), and blade sweep by Fritz et al (2022). Wood (2021) showed that the expansion of the trailing vortices in the wake of a real turbine is easily accounted for in the approximations to the KH equations. The present analysis implies that the remaining restriction of the KH equations to constant p is not significant.…”

Section: The Constancy Of Vortex Pitchmentioning

confidence: 99%

Optimal performance of actuator disc models for horizontal-axis turbines

Wood

Hammam²

2022

Front. Energy Res.

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This study analyzes actuator disc (AD) models of horizontal-axis turbines to determine optimal performance, defined as the maximum power extracted at any tip speed ratio. We use the calculus of variations to maximize rotor torque relative to the thrust without making any assumptions about the rotor loading. The torque was obtained from the angular momentum equation and the thrust from the Kutta-Joukowsky equation which depends on the circumferential velocity and tip speed ratio. The optimality requirement is that the pitch of the vorticity exiting the rotor must be constant across the wake and equal to the ratio of torque to thrust. This result generalizes the classical finding of Betz and Goldstein that optimal lightly-loaded ADs have constant pitch. Optimizing the torque in the far-wake, well downstream of the rotor, leads to the same requirement of constant pitch. This implies that the pitch of an optimal rotor is constant everywhere in the wake at all tip speed ratios. We show that it is not possible for the pitch to reach its optimal value because of the vorticity distribution in the wake, and propose modifications to the pitch at the rotor and in the far-wake. The axial and circumferential velocities in the far-wake, which are easily determined, were used to find those at the rotor from the “disc loading equation” for the angular momentum which is also the normalized bound circulation at the rotor. For the simplest case of a lightly-loaded rotor at zero tip speed ratio, the induced circumferential velocity is linear in radius and the axial component is quadratic, As the tip speed ratio increases, the optimal power and thrust asymptote to the familiar Betz-Joukowsky values, and the induced axial velocity and rotor bound circulation become constant. At low tip speed ratios, the optimal wakes are constrained by the need to avoid breakdown of the flow at high swirl, and the conventional thrust equation, involving the axial velocity only, is inaccurate. As found in previous studies, the power coefficient increases monotonically with tip speed ratio, but the thrust coefficient reaches a maximum value slightly above the Betz-Joukowsky limit at a tip speed ratio of two, before decreasing towards the limit.

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“…Nearly all BEMT codes for wind turbines use Prandtl's tip loss factor, F P to approximate F u and F w . Some of the inaccuracies of using F P are documented in Wood et al (2016) and Wood (2021), while reviewed the numerical methods that produce accurate estimates for F u and F w . Wood et al (2016) showed that F u , F w → F p and F p → 1 as λ increases, suggesting that azimuthal variations become less important in the high-thrust region.…”

Section: Introductionmentioning

confidence: 99%

A revision of blade element/momentum theory for wind turbines in their high-thrust region

Wood,

Golmirzaee

2023

Front. Energy Res.

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Modern horizontal-axis wind turbines produce maximum power at an optimal tip speed ratio, λopt, of around 7. This is also the approximate start of the high-thrust region, which extends to runaway at λR ≈ 2λopt where no power is produced and the thrust is maximized. The runaway thrust coefficient often exceeds unity. It is well known that the conventional axial momentum equation must be modified whenever the thrust coefficient approaches unity, but most past modifications have no sound physical basis. Our main revision is to include the “wake vorticity” term in the axial momentum balance. This term is related to blade element drag and acts to decouple the thrust from the induced axial velocity when it becomes large near the edge of the rotor as the runaway is approached. The wake vorticity term dominates the axial momentum equation in these conditions and leads to estimates of power and thrust that are consistent with the limited amount of high-quality experimental data in the high-thrust region.

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Wake Expansion and the Finite Blade Functions for Horizontal-Axis Wind Turbines

Cited by 6 publications

References 23 publications

An efficient blade sweep correction model for blade element momentum theory

An efficient blade sweep correction model for blade element momentum theory

Optimal performance of actuator disc models for horizontal-axis turbines

A revision of blade element/momentum theory for wind turbines in their high-thrust region

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