# Performance Improvement of a Darrieus Tidal Turbine with Active Variable Pitch

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Turbine

^{2}/s)) varies between $0.6\times {10}^{6}$ and $1.2\times {10}^{6}$ at the tip speed ratio $\lambda =\omega R/{U}_{\infty}=3$ (with $\omega $ the turbine rotational speed). The characteristics of the SHIVA turbine are summarized in Table 1.

## 3. Model and Numerical Methods

#### 3.1. Computational Domain and Mesh

- an outer stator (square of side 60 D, not shown in Figure 3 for clarity);
- a rotating ring containing the 3 blades (rotor), located at the center of the outer stator; and
- an inner stator

#### 3.2. Boundary Conditions and Settings

#### 3.3. Numerical Methods

## 4. Validation

## 5. Measurement of the Angle of Attack

## 6. Fixed Pitch Cases

## 7. Variable Pitch Cases

#### 7.1. Aim, Strategy, and Definition of the Pitching Laws

#### 7.2. Results with Pitch Control

#### 7.2.1. Power Coefficient

#### 7.2.2. Thrust Coefficient

#### 7.2.3. Flow Field

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A. Details of the Pitching Laws

**Table A1.**Details of pitching law PL1 ($\theta $ is always in degrees (${}^{\circ}$) in the expressions).

${\theta}_{\mathrm{upstream}}^{\mathrm{start}}$ [${}^{\circ}$] | 25 |

Upstream law [${}^{\circ}$] | 6 - $\mathrm{atan}\left(\right)open="("\; close=")">\frac{sin\left(\theta \frac{\pi}{180}\right)}{\lambda +cos\left(\theta \frac{\pi}{180}\right)}$ |

${\theta}_{\mathrm{upstream}}^{\mathrm{end}}$ [${}^{\circ}$] | 167 |

${\theta}_{\mathrm{downstream}}^{\mathrm{start}}$ [${}^{\circ}$] | 193 |

Downstream law [${}^{\circ}$] | $0.1\times sin\left(\theta \frac{\pi}{180}\right)$ |

${\theta}_{\mathrm{downstream}}^{\mathrm{end}}$ [${}^{\circ}$] | 335 |

Int 1 [${}^{\circ}$] | $0+5.5854\times {10}^{-2}\theta +4.3288\times {10}^{-3}{\theta}^{2}-2.7370\times {10}^{-4}{\theta}^{3}$ |

Int 2 [${}^{\circ}$] | $-3.3687\times {10}^{-1}+4.6298\times {10}^{-1}\left(\right)open="("\; close=")">\theta -{\theta}_{\mathrm{upstream}}^{\mathrm{end}}+1.8578\times {10}^{-3}{\left(\right)}^{\theta}3$ |

Int 3 [${}^{\circ}$] | $0-9.7183\times {10}^{-2}\left(\right)open="("\; close=")">\theta -180-5.6463\times {10}^{-4}{\left(\right)}^{\theta}3$ |

Int 4 [${}^{\circ}$] | $-4.2262\times {10}^{-2}+1.5818\times {10}^{-3}\left(\right)open="("\; close=")">\theta -{\theta}_{\mathrm{downstream}}^{\mathrm{end}}+8.6488\times {10}^{-5}{\left(\right)}^{\theta}3$ |

**Table A2.**Details of pitching law PL2 ($\theta $ is always in degrees (${}^{\circ}$) in the expressions).

${\theta}_{\mathrm{upstream}}^{\mathrm{start}}$ [${}^{\circ}$] | 33 |

Upstream law [${}^{\circ}$] | 8 - $\mathrm{atan}\left(\right)open="("\; close=")">\frac{sin\left(\theta \frac{\pi}{180}\right)}{\lambda +cos\left(\theta \frac{\pi}{180}\right)}$ |

${\theta}_{\mathrm{upstream}}^{\mathrm{end}}$ [${}^{\circ}$] | 163 |

${\theta}_{\mathrm{downstream}}^{\mathrm{start}}$ [${}^{\circ}$] | 197 |

Downstream law [${}^{\circ}$] | $0.1\times sin\left(\theta \frac{\pi}{180}\right)$ |

${\theta}_{\mathrm{downstream}}^{\mathrm{end}}$ [${}^{\circ}$] | 327 |

Int 1 [${}^{\circ}$] | $0+5.7635\times {10}^{-2}\theta +3.3873\times {10}^{-3}{\theta}^{2}-1.5767\times {10}^{-4}{\theta}^{3}$ |

Int 2 [${}^{\circ}$] | $-1.4151\times {10}^{-1}+4.3507\times {10}^{-1}\left(\right)open="("\; close=")">\theta -{\theta}_{\mathrm{upstream}}^{\mathrm{end}}+1.0901\times {10}^{-3}{\left(\right)}^{\theta}3$ |

Int 3 [${}^{\circ}$] | $0-1.0340\times {10}^{-1}\left(\right)open="("\; close=")">\theta -180-3.5165\times {10}^{-4}{\left(\right)}^{\theta}3$ |

Int 4 [${}^{\circ}$] | $-5.4464\times {10}^{-2}+1.4638\times {10}^{-3}\left(\right)open="("\; close=")">\theta -{\theta}_{\mathrm{downstream}}^{\mathrm{end}}+5.1237\times {10}^{-5}{\left(\right)}^{\theta}3$ |

**Table A3.**Details of pitching law PL3 ($\theta $ is always in degrees (${}^{\circ}$) in the expressions).

${\theta}_{\mathrm{upstream}}^{\mathrm{start}}$ [${}^{\circ}$] | 42 |

Upstream law [${}^{\circ}$] | 10 - $\mathrm{atan}\left(\frac{sin\left(\theta \frac{\pi}{180}\right)}{\lambda +cos\left(\theta \frac{\pi}{180}\right)}\right)\times \frac{180}{\Pi}$ |

${\theta}_{\mathrm{upstream}}^{\mathrm{end}}$ [${}^{\circ}$] | 158 |

${\theta}_{\mathrm{downstream}}^{\mathrm{start}}$ [${}^{\circ}$] | 202 |

Downstream law [${}^{\circ}$] | $0.1\times sin\left(\theta \frac{\pi}{180}\right)$ |

${\theta}_{\mathrm{downstream}}^{\mathrm{end}}$ [${}^{\circ}$] | 318 |

Int 1 [${}^{\circ}$] | $0+5.4294\times {10}^{-2}\theta +2.5025\times {10}^{-3}{\theta}^{2}-9.2188\times {10}^{-5}{\theta}^{3}$ |

Int 2 [${}^{\circ}$] | $-2.4412\times {10}^{-1}+4.0153\times {10}^{-1}\left(\right)open="("\; close=")">\theta -{\theta}_{\mathrm{upstream}}^{\mathrm{end}}+5.9175\times {10}^{-4}{\left(\right)}^{\theta}3$ |

Int 3 [${}^{\circ}$] | $0-9.2933\times {10}^{-2}\left(\right)open="("\; close=")">\theta -180-1.8832\times {10}^{-4}{\left(\right)}^{\theta}3$ |

Int 4 [${}^{\circ}$] | $-6.6913\times {10}^{-2}+1.2970\times {10}^{-3}\left(\right)open="("\; close=")">\theta -{\theta}_{\mathrm{downstream}}^{\mathrm{end}}+2.9708\times {10}^{-5}{\left(\right)}^{\theta}3$ |

Pitching Law | Range of Validity |
---|---|

Int 1 | $\theta \in [{0}^{\circ},{\theta}_{\mathrm{upstream}}^{\mathrm{start}}]$ |

Upstream law | $\theta \in [{\theta}_{\mathrm{upstream}}^{\mathrm{start}},{\theta}_{\mathrm{upstream}}^{\mathrm{end}}]$ |

Int 2 | $\theta \in [{\theta}_{\mathrm{upstream}}^{\mathrm{end}},{180}^{\circ}]$ |

Int 3 | $\theta \in [{180}^{\circ},{\theta}_{\mathrm{downstream}}^{\mathrm{start}}]$ |

Downstream law | $\theta \in [{\theta}_{\mathrm{downstream}}^{\mathrm{start}},{\theta}_{\mathrm{downstream}}^{\mathrm{end}}]$ |

Int 4 | $\theta \in [{\theta}_{\mathrm{downstream}}^{\mathrm{end}},{360}^{\circ}]$ |

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**Figure 2.**Definition of the blade pitch angle $\beta $. (${\overrightarrow{e}}_{ct},{\overrightarrow{e}}_{cn}$): coordinate frame attached to the blade (i.e., ${\overrightarrow{e}}_{ct}$ remains parallel to the blade’s chord at all time). (${\overrightarrow{e}}_{t},{\overrightarrow{e}}_{n}$): coordinate frame that follows the blade path with ${\overrightarrow{e}}_{t}$ parallel to the tangent to the trajectory of the quarter chord at all time. The pitch angle $\beta $ is the angle between ${\overrightarrow{e}}_{ct}$ and ${\overrightarrow{e}}_{t}$.

**Figure 7.**Power coefficient of the fixed pitch turbine as a function of the tip speed ratio ($\lambda $), ${\mathrm{U}}_{\infty}=2\mathrm{m}/\mathrm{s}$.

**Figure 10.**(

**a**) Blades’ angle of attack of the Darrieus turbine (from Equation (4)) and of the 3 variable-pitch cases; (

**b**) pitching laws studied, $\lambda =3$.

**Figure 12.**Torque coefficient (${C}_{Q}$) of one blade for the Darrieus and the pitching law cases, $\lambda =3$.

**Figure 13.**Thrust coefficient (${C}_{T}$) for the Darrieus and the pitching law cases, $\lambda =3$.

**Figure 14.**Isocontours of non-dimensioned streamwise velocity (${U}_{x}/{U}_{\infty}$) for the Darrieus (

**left**) and the PL2 (

**right**) cases, $\lambda =3$. Instantaneous flow field with blade 1 located at $\theta ={0}^{\circ}$.

**Figure 15.**Non-dimensioned flow speed (${U}_{x}/{U}_{\infty}$) at the center of the turbine for the Darrieus and the pitching law cases, $\lambda =3$.

**Figure 17.**Non-dimensioned instantaneous streamwise velocity (${U}_{x}/{U}_{\infty}$) plotted on a line crossing the center of the turbine ($y=0$ m) in the streamwise direction. Data for the Darrieus and the pitching law cases, $\lambda =3$.

The SHIVA Turbine | |
---|---|

Rotor diameter ($D=2R$) | 1.6 [m] |

Number of blades (N) | 3 |

Blade length (l) | 1 [m] |

Blades cross section | NACA 0018 |

Chord length (c) | 0.15 [m] |

Solidity ($\sigma =Nc/R$) | 0.563 |

Blade rotation axis location | $0.25\times c$ |

**Table 2.**Power coefficient: average ($\overline{CP}$) and ripple factor ($C{P}_{F}=C{P}_{max}-C{P}_{min}$) values for the fixed and variable pitch cases.

$\overline{\mathit{CP}}$ | ${\mathit{CP}}_{\mathit{F}}$ | |
---|---|---|

Fixed pitch | 0.360 (ref) | 0.60 (ref) |

PL1 | 0.503 ($+39.8\%$) | 0.20 ($-66.7\%$) |

PL2 | 0.507 ($+40.9\%$) | 0.28 ($-52.8\%$) |

PL3 | 0.499 ($+38.7\%$) | 0.38 ($-37.1\%$) |

**Table 3.**Thrust coefficient: average ($\overline{{C}_{T}}$) and ripple factor (${C}_{{T}_{F}}={C}_{Tmax}-{C}_{Tmin}$) values for the fixed and variable pitch cases.

$\overline{{\mathit{C}}_{\mathit{T}}}$ | ${\mathit{C}}_{{\mathit{T}}_{\mathit{F}}}$ | |
---|---|---|

Darrieus | 0.982 (ref) | 0.77 (ref) |

PL1 | 0.919 ($-6.5\%$) | 0.20 ($-74.0\%$) |

PL2 | 0.974 ($-0.8\%$) | 0.16 ($-79.0\%$) |

PL3 | 1.016 ($+3.4\%$) | 0.17 ($-78.1\%$) |

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## Share and Cite

**MDPI and ACS Style**

Delafin, P.-L.; Deniset, F.; Astolfi, J.A.; Hauville, F.
Performance Improvement of a Darrieus Tidal Turbine with Active Variable Pitch. *Energies* **2021**, *14*, 667.
https://doi.org/10.3390/en14030667

**AMA Style**

Delafin P-L, Deniset F, Astolfi JA, Hauville F.
Performance Improvement of a Darrieus Tidal Turbine with Active Variable Pitch. *Energies*. 2021; 14(3):667.
https://doi.org/10.3390/en14030667

**Chicago/Turabian Style**

Delafin, Pierre-Luc, François Deniset, Jacques André Astolfi, and Frédéric Hauville.
2021. "Performance Improvement of a Darrieus Tidal Turbine with Active Variable Pitch" *Energies* 14, no. 3: 667.
https://doi.org/10.3390/en14030667