# Study of a Bi-Vertical Axis Turbines Farm Using the Actuator Cylinder Method

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}; however, in extreme conditions and at specific locations, velocity can reach 5 m · s

^{−1}[4] with more than 10% of turbulent intensity [5,6], which makes the Alderney Race a promising site for tidal parcs installation. Considering the large range of current magnitudes and directions in the Alderney Race, the study of different configurations of tidal turbines arrangement within a farm submitted to different current conditions is of prime importance in order to better predict the energy capacity and to optimize the parc arrangement.

## 2. Materials and Methods

#### 2.1. The Actuator Cylinder Model: A Description

#### 2.2. Model Validation

#### 2.3. Hydroquest Turbine and Farm Configurations

^{−1}and three incidence angles $\gamma $ of 0, 10 and 20${}^{\circ}$ are tested. Those current conditions correspond to some flow regimes representative of a mean tide in the Alderney Race, extracted from regional model Telemac2D, and have been numerically tested on horizontal-axis tidal turbines arrays, both in aligned and staggered configurations by Nguyen et al. [16]. Power production and velocity profiles are extracted from each simulation. The tip speed ratio $\lambda $ is set at 2; therefore, rotor rotation speed is, respectively, $\omega =$ 0.75 and 1.5 rad · s

^{−1}for $\left|{U}_{\infty}\right|=$ 1.5 and 3.0 m · s

^{−1}. Turbulence intensity is fixed at 10% at the inlet, which is the order of turbulent intensity in the Alderney Race [5,6].

^{−1}. Objectives of those simulations are to determine and analyze interactions between turbines for different current directions, as well as the influences they have upon power production, in order to optimize their placement for a maximum electricity production.

## 3. Results

#### 3.1. Single Turbine

#### 3.2. Tidal Farm

#### 3.2.1. First Farm Configuration

#### 3.2.2. Second Farm Configuration

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${A}_{f}$ | projected area of the turbine |

${A}_{t}$ | inlet area of the domain |

C | blade chord |

${C}_{D},{C}_{L}$ | drag and lift coefficients of the blade |

D | rotor diameter |

$(\overrightarrow{{e}_{L}},\overrightarrow{{e}_{D}})$ | lift and drag forces coordinates system |

$(\overrightarrow{{e}_{N}},\overrightarrow{{e}_{T}})$ | blade coordinates system |

$(\overrightarrow{{e}_{X}},\overrightarrow{{e}_{Y}})$ | Cartesian coordinates system |

F | total force on the blade |

${F}_{D},{F}_{L}$ | drag and lift forces |

${F}_{N},{F}_{T}$ | normal and tangential forces |

${H}_{t}$ | rotor height |

N | rotor’s number of blades |

P | power production |

${P}_{cor},{P}_{uncor}$ | corrected and uncorrected pressure |

q | intensity of kinetic energy fluctuation |

R | rotor radius |

U | velocity |

${U}_{\infty}$ | inlet velocity |

W | relative flow velocity, $\overrightarrow{W}=\overrightarrow{U}-\omega R\overrightarrow{{e}_{T}}$ |

X | longitudinal distance downstream from the center of the turbine |

Y | lateral distance from the center of the turbine |

$\alpha $ | angle of attack |

$\gamma $ | current orientation relative to the $X-$axis |

$\Delta x$ | maximal cells size |

$\epsilon $ | blockage coefficient for vertical-axis turbine |

$\lambda $ | tip speed ratio, $\lambda ={\textstyle \frac{\omega R}{{U}_{\infty}}}$ |

$\rho $ | fluid density |

$\omega $ | rotor rotational velocity |

## References

- Agence de l’Environnement et de la Maîtrise de l’Énergie (ADEME). Étude Stratégique de la FilièRe Hydrolien Marin. Technical Report. Available online: https://librairie.ademe.fr/energies-renouvelables-reseaux-et-stockage/3624-etude-strategique-de-la-filiere-hydrolien-marin.html (accessed on 5 November 2020).
- Nguyen, V.; Guillou, S.S.; Thiébot, J.; Santa Cruz, A. A methodology for representing the effect of vertical-axis turbines on the flow. In Proceedings of the 2nd International Conference on Offshore Renewable Energy, Glasgow, UK, 12–14 September 2016. [Google Scholar]
- Thiébaut, M.; Sentchev, A.; Bailly du Bois, P. Merging velocity measurements and modeling to improve understanding of tidal stream resource in Alderney Race. Energy
**2019**, 178, 460–470. [Google Scholar] [CrossRef] [Green Version] - Bahaj, A.; Myers, L. Analytical estimates of the Energy Yield Potential from the Alderney Race (Channel Islands) Using Marine Current Energy Converters. Renew. Energy
**2004**, 29, 1931–1945. [Google Scholar] [CrossRef] - Thiébaut, M.; Filipot, J.F.; Maisondieu, C.; Damblans, G.; Jochum, C.; Kilcher, L.; Guillou, S. Characterization of the vertical evolution of the three-dimensional turbulence for fatigue design of tidal turbines. Philos. Trans. R. Soc. Math. Phys. Eng. Sci.
**2020**, 378, 20190495. [Google Scholar] [CrossRef] - Thiébaut, M.; Filipot, J.; Maisondieu, C.; Damblans, G.; Duarte, R.; Droniou, E.; Chaplain, N.; Guillou, S. A comprehensive assessment of turbulence at a tidal-stream energy site influenced by wind-generated ocean waves. Energy
**2019**, 191, 116550. [Google Scholar] [CrossRef] - Strickland, J.H.; Webster, B.T.; Nguyen, T.A. Vortex Model of the Darrieus Turbine: An Analytical and Experimental Study. Trans. ASME J. Fluids Eng.
**1979**, 101, 500–505. [Google Scholar] [CrossRef] - Templin, R.J. Aerodynamic Performance Theory of the NRC Vertical Axis Wind Turbine; Technical Report LTR-LA-160; National Research Council of Canada: Ottawa, ON, Canada, 1973. [Google Scholar]
- Strickland, J. The Darrieus Turbine: A Performance Prediction Model Using Multiple Streamtubes; Technical Report SAND-75-0431; Sandia Laboratory: Albuberque, NM, USA, 1975. [Google Scholar]
- Paraschivoiu, I. Aerodynamic Loads and Performance of the Darrieus Turbine. Am. Inst. Aeronaut. Astronaut. J. Energy
**1981**, 6, 406–412. [Google Scholar] - Poguluri, S.K.; Lee, H.; Bae, Y.H. An investigation on the aerodynamic performance of a co-axial contra-rotating vertical-axis wind turbine. Energy
**2021**, 219, 119547. [Google Scholar] [CrossRef] - Shamsoddin, S.; Porté-Agel, F. Large Eddy Simulation of Vertical Axis Wind Turbine Wakes. Energies
**2014**, 7, 890–912. [Google Scholar] [CrossRef] - Grondeau, M.; Guillou, S.; Mercier, P.; Poizot, E. Wake of a Ducted Vertical Axis Tidal Turbine in Turbulent Flows, LBM Actuator-Line Approach. Energies
**2019**, 12, 497–512. [Google Scholar] [CrossRef] [Green Version] - Dabiri, J. Potential order-of-magnitude enhancement of wind farm power density via counter-rotating vertical-axis wind turbine arrays. J. Renew. Sustain. Energy
**2011**, 3, 043104. [Google Scholar] [CrossRef] [Green Version] - Palm, M.; Huijsmans, R.; Pourquie, M. The Applicability of Semi-Empirical Wake Models for Tidal Farms. In Proceedings of the European Wave and Tidal Energy Conference, Southampton, UK, 5–9 September 2011. [Google Scholar]
- Nguyen, V.T.; Santa Cruz, A.; Guillou, S.; Shiek Elsouk, M.; Thiébot, J. Effects of the Current Direction on the Energy Production of a Tidal Farm: The Case of Raz Blanchard (France). Energies
**2019**, 12, 2478. [Google Scholar] [CrossRef] [Green Version] - Clary, V.; Oudart, T.; Larroudé, P.; Sommeria, J.; Maître, T. An optimally-controlled RANS Actuator force model for different computations of tidal turbine arrays. Ocean. Eng.
**2020**, 212, 107677. [Google Scholar] [CrossRef] - Madsen, H.A. The Actuator Cylinder—A Flow Model for Vertical Axis Wind Turbines. Ph.D. Thesis, Aalborg University Centre, Aalborg, Denmark, 1982. [Google Scholar]
- Sheldahl, R.; Klimas, P. Aerodynamic Characteristics of Seven Symmetrical Airfoil Sections through 180-Degree Angle of Attack for Use in Aerodynamic Analysis of Vertical Axis Wind Turbines; Technical Report SAND-80-2114; Sandia Laboratory: Albuquerque, NM, USA, 1981. [Google Scholar]
- Brochier, G. Étude Numérique de la Couche Limite Instationnaire sur un Profil d’Aile en Mouvement, Application et Expérimentation à l’Éolienne Darrieus. Ph.D. Thesis, Université d’Aix-Marseille II, Marseille, France, 1986. [Google Scholar]
- Nguyen, V.T.; Guillou, S.S.; Thiébot, J.; Santa Cruz, A. Modelling turbulence with an Actuator Disk representing a tidal turbine. Renew. Energy
**2016**, 97, 625–635. [Google Scholar] [CrossRef] - Maitre, T.; Amet, E.; Pellone, C. Modeling of the flow in a Darrieus water turbine: Wall grid refinement analysis and comparison with experiments. Renew. Energy
**2012**, 51, 497–512. [Google Scholar] [CrossRef] - Howell, R.; Qin, N.; Edwards, J.; Durrani, N. Wind tunnel and numerical study of a small vertical axis wind turbine. Renew. Energy
**2009**, 35, 412–422. [Google Scholar] [CrossRef] [Green Version] - Hilewit, D.; Matida, E.; Fereidooni, A.; Aboel Ella, H.; Nitzsche, F. Power coefficient measurements of a novel vertical axis wind turbine. Energy Sci. Eng.
**2019**, 7, 2373–2382. [Google Scholar] [CrossRef] - Ross, I. Wind Tunnel Blockage Corrections: An Application to Vertical-Axis Wind Turbines. Master’s Thesis, School of Engineering of the University of Dayton, Dayton, OH, USA, 2010. [Google Scholar]
- Agence de l’Environnement et de la Maîtrise de l’Énergie (ADEME). Available online: www.ademe.fr (accessed on 5 November 2020).
- Hydroquest. Available online: www.hydroquest.fr (accessed on 5 November 2020).

**Figure 2.**Actuator Cylinder model scheme: The rotor is represented as a hollow cylinder which external and internal radius are determined by the blade path.

**Figure 3.**Normal (

**left**) and tangential (

**right**) forces in function of the azimuthal position $\theta $, comparison between the experience of Strickland for three blades and the numerical results of Nguyen et al. [2].

**Figure 4.**Transverse profiles of adimensional velocity $U/{U}_{\infty}$ and intensity of kinetic energy fluctuation $q/{U}_{\infty}$ downstream the turbine; X represents the distance downstream (black lines on the velocity field figure).

**Figure 6.**Meshed turbine and tunnel on Ansys Meshing: (

**a**) Mesh around the turbine; (

**b**) Mesh inflation around the rotor cylinder.

**Figure 7.**Computational domain for the single turbine; center of the domain is located at the center of the central duct of the turbine.

**Figure 8.**Mesh convergence test in terms of adimensional velocity profiles (see probes location in Figure 9).

**Figure 9.**Transverse and longitudinal profiles of adimensional velocity intensity in the wake of the Hydroquest turbine for $\gamma =0$${}^{\circ}$; probes locations at the upstream distance X from the turbine center are shown in black lines.

**Figure 10.**Turbines arrangement within the farm (

**a**) with turbines name; zoom near the four turbines (

**b**).

**Figure 11.**Flow around a single turbine: Non-dimensional velocity (${U}_{a}=U/{U}_{\infty}$) field for ${U}_{\infty}$ = 1.5 $\mathrm{m}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$, (

**a**–

**c**), ${U}_{\infty}$ = 3.0 $\mathrm{m}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ (

**d**–

**f**), and ${U}_{\infty}$ = −3.0 $\mathrm{m}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$, (

**g**–

**i**).

**Figure 12.**Non-dimensional velocity (${U}_{a}=U/{U}_{\infty}$) field for the first farm configuration: ${U}_{\infty}>0$ on the left side, and ${U}_{\infty}<0$ on the right side.

**Figure 13.**Flow acceleration depending on the velocity sign, in the case of a flow over a single turbine (positive velocity on the left, negative velocity on the right; red lines indicate an accelerated flow, and green lines indicate a decelerated flow).

**Figure 14.**Non-dimensional velocity (${U}_{a}=U/{U}_{\infty}$) field for the second farm configuration: ${U}_{\infty}>0$ on the left side, and ${U}_{\infty}<0$ on the right side.

Characteristic | Value | Unit |
---|---|---|

Rotor radius | 0.061 | $\mathrm{m}$ |

Blade number | 1, 2, and 3 | - |

Blade profile | NACA 0012 | - |

Chord | 0.0914 | $\mathrm{m}$ |

Inlet velocity | 0.091 | m · s ^{−1} |

TSR | 5 | - |

Characteristic | Value | Unit |
---|---|---|

Rotor radius | 0.06 | $\mathrm{m}$ |

Blade number | 2 | - |

Blade profile | NACA 0018 | - |

Chords | 0.02 | $\mathrm{m}$ |

Inlet velocity | 0.02 | m · s ^{−1} |

Turbulent intensity | 5 | % |

TSR | 3.85 | - |

Rotation speed | 83 | $\mathrm{tr}/min$ |

Velocity | ${\mathit{U}}_{\mathit{\infty}}$ = 1.5 $\mathit{m}\phantom{\rule{0.166667em}{0ex}}\mathit{\xb7}\phantom{\rule{0.166667em}{0ex}}{\mathit{s}}^{-1}$ | ${\mathit{U}}_{\mathit{\infty}}$ = 3 $\mathit{m}\phantom{\rule{0.166667em}{0ex}}\mathit{\xb7}\phantom{\rule{0.166667em}{0ex}}{\mathit{s}}^{-1}$ | ${\mathit{U}}_{\mathit{\infty}}$ = −3 $\mathit{m}\phantom{\rule{0.166667em}{0ex}}\mathit{\xb7}\phantom{\rule{0.166667em}{0ex}}{\mathit{s}}^{-1}$ | ||||
---|---|---|---|---|---|---|---|

Incidence | Rotor 1 | Rotor 2 | Rotor 1 | Rotor 2 | Rotor 1 | Rotor 2 | |

$\gamma =$ 0 ${}^{\circ}$ | 0.080 | 0.092 | 0.650 | 0.745 | 0.650 | 0.745 | |

$\gamma =$ 10 ${}^{\circ}$ | 0.079 | 0.081 | 0.632 | 0.644 | 0.648 | 0.674 | |

$\gamma =$ 20 ${}^{\circ}$ | 0.088 | 0.083 | 0.706 | 0.662 | 0.693 | 0.701 |

**Table 4.**Power produced ($\mathrm{M}\mathrm{W}$) by every rotors of every turbines in the first farm configuration for the different flow conditions tested.

Machine | M11 | M21 | M22 | M31 | ${\mathit{P}}_{\mathbf{tot}}$ | |||||
---|---|---|---|---|---|---|---|---|---|---|

Case | Rotor 1 | Rotor 2 | Rotor 1 | Rotor 2 | Rotor 1 | Rotor 2 | Rotor 1 | Rotor 2 | ||

γ = 0∘ | ${\mathit{U}}_{\mathit{\infty}}>0$ | 0.572 | 0.711 | 0.561 | 0.670 | 0.521 | 0.625 | 0.232 | 0.245 | 4.137 |

${U}_{\infty}<0$ | 0.229 | 0.237 | 0.572 | 0.661 | 0.573 | 0.600 | 0.544 | 0.588 | 4.004 | |

γ = 10∘ | ${U}_{\infty}>0$ | 0.655 | 0.715 | 0.637 | 0.682 | 0.550 | 0.608 | 0.670 | 0.711 | 5.228 |

${U}_{\infty}<0$ | 0.774 | 0.788 | 0.647 | 0.698 | 0.608 | 0.598 | 0.630 | 0.617 | 5.360 | |

γ = 20∘ | ${U}_{\infty}>0$ | 0.732 | 0.713 | 0.727 | 0.756 | 0.647 | 0.628 | 0.741 | 0.769 | 5.713 |

${U}_{\infty}<0$ | 0.847 | 0.834 | 0.736 | 0.763 | 0.672 | 0.639 | 0.691 | 0.659 | 5.841 |

**Table 5.**Adimensional power produced by every rotors of every turbines in the first farm configuration for the different flow conditions tested, adimensioned by the header turbine (M11 for a ${U}_{\infty}>0$, M31 for ${U}_{\infty}<0$).

Machine | M11 | M21 | M22 | M31 | |||||
---|---|---|---|---|---|---|---|---|---|

Case | Rotor 1 | Rotor 2 | Rotor 1 | Rotor 2 | Rotor 1 | Rotor 2 | Rotor 1 | Rotor 2 | |

γ = 0∘ | ${\mathit{U}}_{\mathit{\infty}}>0$ | 1.00 | 1.00 | 0.98 | 0.94 | 0.91 | 0.88 | 0.41 | 0.34 |

${U}_{\infty}<0$ | 0.42 | 0.40 | 1.05 | 1.12 | 1.05 | 1.02 | 1.00 | 1.00 | |

γ = 10∘ | ${U}_{\infty}>0$ | 1.00 | 1.00 | 0.97 | 0.95 | 0.84 | 0.85 | 1.02 | 0.99 |

${U}_{\infty}<0$ | 1.23 | 1.28 | 1.03 | 1.13 | 0.97 | 0.97 | 1.00 | 1.00 | |

γ = 20∘ | ${U}_{\infty}>0$ | 1.00 | 1.00 | 0.99 | 1.06 | 0.88 | 0.88 | 1.01 | 1.08 |

${U}_{\infty}<0$ | 1.23 | 1.27 | 1.07 | 1.16 | 0.97 | 0.97 | 1.00 | 1.00 |

**Table 6.**Power produced ($\mathrm{M}\mathrm{W}$) by every rotor of every turbine in the second farm configuration for the different flow conditions tested.

Machine | M11 | M21 | M22 | M31 | ${\mathit{P}}_{\mathbf{tot}}$ | |||||
---|---|---|---|---|---|---|---|---|---|---|

Case | Rotor 1 | Rotor 2 | Rotor 1 | Rotor 2 | Rotor 1 | Rotor 2 | Rotor 1 | Rotor 2 | ||

γ = 0∘ | ${\mathit{U}}_{\mathit{\infty}}>0$ | 0.589 | 0.645 | 0.572 | 0.684 | 0.547 | 0.647 | 0.222 | 0.259 | 4.165 |

${U}_{\infty}<0$ | 0.249 | 0.279 | 0.576 | 0.682 | 0.600 | 0.660 | 0.508 | 0.546 | 4.100 | |

γ = 10∘ | ${U}_{\infty}>0$ | 0.677 | 0.682 | 0.635 | 0.679 | 0.596 | 0.629 | 0.676 | 0.718 | 5.292 |

${U}_{\infty}<0$ | 0.775 | 0.809 | 0.658 | 0.730 | 0.617 | 0.625 | 0.568 | 0.611 | 5.474 | |

γ = 20∘ | ${U}_{\infty}>0$ | 0.729 | 0.679 | 0.170 | 0.130 | 0.658 | 0.649 | 0.430 | 0.168 | 3.613 |

${U}_{\infty}<0$ | 0.543 | 0.200 | 0.215 | 0.161 | 0.679 | 0.640 | 0.618 | 0.642 | 3.698 |

**Table 7.**Adimensional power produced by every rotors of every turbines in the second farm configuration for the different flow conditions tested, adimensioned by the header turbine.

Machine | M11 | M21 | M22 | M31 | |||||
---|---|---|---|---|---|---|---|---|---|

Case | Rotor 1 | Rotor 2 | Rotor 1 | Rotor 2 | Rotor 1 | Rotor 2 | Rotor 1 | Rotor 2 | |

γ = 0∘ | ${\mathit{U}}_{\mathit{\infty}}>0$ | 1.00 | 1.00 | 0.97 | 1.06 | 0.93 | 1.00 | 0.38 | 0.40 |

${U}_{\infty}<0$ | 0.49 | 0.51 | 1.13 | 1.25 | 1.18 | 1.21 | 1.00 | 1.00 | |

γ = 10∘ | ${U}_{\infty}>0$ | 1.00 | 1.00 | 0.94 | 1.00 | 0.88 | 0.92 | 1.00 | 1.05 |

${U}_{\infty}<0$ | 1.36 | 1.32 | 1.16 | 1.19 | 1.09 | 1.02 | 1.00 | 1.00 | |

γ = 20∘ | ${U}_{\infty}>0$ | 1.00 | 1.00 | 0.23 | 0.19 | 0.90 | 0.96 | 0.59 | 0.25 |

${U}_{\infty}<0$ | 0.88 | 0.31 | 0.35 | 0.25 | 1.10 | 1.00 | 1.00 | 1.00 |

**Table 8.**Ratio of power production of second configuration over first configuration; cells in red show that configuration 2 produces at least 5 less than configuration 1, and cells in green that power produced by configuration 2 is at least 5 more than configuration 1.

Machine | M11 | M21 | M22 | M31 | ${\mathit{P}}_{\mathbf{tot}}$ | |||||
---|---|---|---|---|---|---|---|---|---|---|

Case | Rotor 1 | Rotor 2 | Rotor 1 | Rotor 2 | Rotor 1 | Rotor 2 | Rotor 1 | Rotor 2 | ||

γ = 0∘ | ${\mathit{U}}_{\mathit{\infty}}>0$ | 1.03 | 0.91 | 1.02 | 1.02 | 1.05 | 1.04 | 0.96 | 1.06 | 1.01 |

${U}_{\infty}<0$ | 1.09 | 1.18 | 1.01 | 1.03 | 1.05 | 1.10 | 0.93 | 0.93 | 1.02 | |

γ = 10∘ | ${U}_{\infty}>0$ | 1.03 | 0.95 | 1.00 | 1.00 | 1.08 | 1.03 | 1.01 | 1.01 | 1.01 |

${U}_{\infty}<0$ | 1.00 | 1.03 | 1.02 | 1.05 | 1.01 | 1.05 | 0.90 | 0.99 | 1.01 | |

γ = 20∘ | ${U}_{\infty}>0$ | 1.00 | 0.95 | 0.23 | 0.17 | 1.02 | 1.03 | 0.58 | 0.22 | 0.63 |

${U}_{\infty}<0$ | 0.64 | 0.24 | 0.29 | 0.21 | 1.01 | 1.00 | 0.89 | 0.97 | 0.63 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jégo, L.; Guillou, S.S.
Study of a Bi-Vertical Axis Turbines Farm Using the Actuator Cylinder Method. *Energies* **2021**, *14*, 5199.
https://doi.org/10.3390/en14165199

**AMA Style**

Jégo L, Guillou SS.
Study of a Bi-Vertical Axis Turbines Farm Using the Actuator Cylinder Method. *Energies*. 2021; 14(16):5199.
https://doi.org/10.3390/en14165199

**Chicago/Turabian Style**

Jégo, Laurie, and Sylvain S. Guillou.
2021. "Study of a Bi-Vertical Axis Turbines Farm Using the Actuator Cylinder Method" *Energies* 14, no. 16: 5199.
https://doi.org/10.3390/en14165199