# The Efficiency of a Fence of Tidal Turbines in the Alderney Race: Comparison between Analytical and Numerical Models

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## Abstract

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## 1. Introduction

^{−2}, thus exhibiting the increase of this maximum power with the blockage ratio. Whelan et al. [6] then modified this model by integrating the change in water depth caused by the turbine operation. In addition, Nishino and Willden [7] used the analytical model of Garrett and Cummins [5] to investigate the flow through and around a fence of devices arrayed across part of a wide channel cross section.

## 2. Materials and Methods

#### 2.1. Site Description—Scenarios of Tidal Stream Energy Extraction

#### 2.2. Analytical Models

^{3}for the present purpose) is the fluid density, ${C}_{T}$ and ${C}_{P}$ are the thrust and power coefficients, $A$ is the area swept by the turbines’ blades, ${U}_{\infty}^{}$ is the velocity in the unperturbed flow (in a region of the flow that is not affected by the turbine), and ${U}_{D}^{}$ is the velocity in the disk.

^{−2}. Whelan et al. [6] rewrote the system of equations of G&C2007 and applied them to model the flow within a vertical plane passing through the turbine (Figure 3b). In doing so, they integrated the effect of the change in depth caused by the turbine (the rise of the free surface upstream of the turbine and drop of the free surface downstream of the turbine). Beyond confirming the effect of the blockage ratio, this new model, hereinafter denominated W2009, exhibited an increase in the extracted power with the Froude number. G&C2007 and W2009 were initially designed to analyze the performance of a single turbine in a channel. Nevertheless, those models are also applicable to multiple turbines spanned uniformly over the entire cross-section of a channel (as represented in Figure 3(a2) for G&C2007). However, in real-life applications, turbines are unlikely to be deployed over the entire channel cross-section because of either environmental and/or regulation constraints. Hence, Nishino and Willden [7] developed an analytical model, denoted N&W2012, to analyze the performance of a fence of turbines blocking partially a channel (Figure 3c). N&W2012 applied the G&C2007 model at two spatial scales: (i) at the “array scale” to analyze the flow through and around the fence of turbines, and (ii) at the “turbine scale” to analyze the flow through and around each turbine. For the present purpose, these three analytical models (G&C2007, W2009, and N&W2012) have been coded in Matlab in order to obtain, under the flow conditions synthetized in Table 1, the relationships between the power coefficient ${C}_{P}$, the thrust coefficient ${C}_{T}$, and the induction factor $a$. The results are presented and discussed in Section 3.

#### 2.3. Numerical Model

## 3. Results and Discussion

#### 3.1. Results Overview

#### 3.2. Inter-Comparison of Analytical Models

#### 3.3. Comparison between Analytical and Numerical Models

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Bathymetry of the study site with location of the fence of turbines. Depth-averaged current direction during peak (

**b**) ebb and (

**c**) flood of a mean spring tide. The lateral spacing (center to center) are indicated in number of turbine diameters D. For a lateral spacing of 1.5D, 11 turbines are active. For a lateral spacing of 3D, only the 6 turbines represented in black are active. The origin of the coordinates system (in meters) corresponds to the center of the fence of turbines (49°42.300′ N; 2°6.180′ W).

**Figure 2.**Time-series of (

**a**) depth-averaged velocity magnitude, (

**b**) depth-averaged velocity direction, and (

**c**) free surface elevation extracted from the Telemac3D model (simulation without turbine) in the center of the fence (49°42.300′ N; 2°6.180′ W). The red circles show times of peak ebb and flood retained to assess the array efficiency.

**Figure 3.**Schematic representation of the analytical models. (

**a1**) G&C2007 applied to a single turbine, (

**a2**) G&C2007 applied to a fence of turbines arrayed uniformly across the channel, (

**b**) W2009, and (

**c**) N&W2012. (

**c**) The dashed red curves represent the stream tube passing through the fence of turbines (surrounded by a red rectangle), and represent the flow at the “array scale”. The dashed black curves represent the stream tube passing through each turbine, and represent the flow at the “turbine scale”. ${U}_{D}$ is the velocity in the disk, ${U}_{A}$ is the velocity at the fence, and ${U}_{C}$ is the channel cross-sectional average of the streamwise velocity.

**Figure 4.**Thrust (

**a**,

**c**) and power (

**b**,

**d**) coefficients as a function of the induction factor. Estimations are shown at peak (

**top**) ebb and (

**bottom**) flood of the mean spring tide represented in Figure 2.

**Figure 5.**Vertical distribution of the velocity magnitude at different positions along the wake of the turbine located in the middle of the fence (sixth turbine from the left for Δ = 1.5D and third turbine from the left for Δ = 3D, Figure 1b,c). Blue lines represent the simulations without turbines. Black lines represent the simulations with turbines. Continuous and dashed lines correspond to a lateral spacing of 1.5D and 3D, respectively. The horizontal lines indicate the top and the bottom of the turbines. The velocities have been extracted, at peak flood tide, from the simulation with AD and $K$ = 3 ($a$ = 0.33 and 0.34 for Δ = 1.5D and Δ = 3D, respectively).

**Figure 6.**Horizontal distribution of the velocity deficit (velocity magnitude with turbines minus velocity magnitude without turbines) at the hub height (15 m above the flat seabed) at different streamwise positions from the fence of devices. Continuous and dashed lines correspond to a lateral spacing of 1.5D and 3D, respectively. The velocities have been extracted, at peak flood tide, from the simulation with AD with $K$ = 3 ($a$ = 0.33 and 0.34 for Δ = 1.5D and Δ = 3D, respectively). The orientation of the axis $y$ differs from that retained in Figure 1.

**Figure 7.**Horizontal distribution of the turbulent kinetic energy (TKE) at the hub height (15 m above the flat seabed) at different streamwise positions from the fence of devices. Continuous and dashed lines correspond to a lateral spacing of 1.5D and 3D, respectively. The values have been extracted, at peak flood tide, from the simulation with AD with $K$ = 3 ($a$ = 0.33 and 0.34 for Δ = 1.5D and Δ = 3D, respectively). The orientation of the axis $y$ differs from that retained in Figure 1.

**Table 1.**Characteristics of the flow during peak ebb and flood in the center of the fence (49°42.300 ′N; 2°6.180′ W) without the effects of turbines. Δ corresponds to the lateral spacing between devices (1.5D or 3D). The blockage ratio is equal to the surface area swept by the blades ($\pi {D}^{2}/4$ ) divided by the cross-section of the flow, that is to say the water depth multiplied by the distance between two consecutive turbines (1.5D or 3D).

Peak Ebb | Peak Flood | |
---|---|---|

Water depth (m) | 41.05 | 46.52 |

Depth-averaged velocity magnitude (m/s) | 3.02 | 2.76 |

Depth-averaged current direction (°/North) | 206 | 32 |

Blockage ratio (%) Δ = 1.5D/3D | 17.86/8.93 | 15.76/7.88 |

Froude number (dimensionless) | 0.150 | 0.129 |

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**MDPI and ACS Style**

Thiébot, J.; Djama Dirieh, N.; Guillou, S.; Guillou, N.
The Efficiency of a Fence of Tidal Turbines in the Alderney Race: Comparison between Analytical and Numerical Models. *Energies* **2021**, *14*, 892.
https://doi.org/10.3390/en14040892

**AMA Style**

Thiébot J, Djama Dirieh N, Guillou S, Guillou N.
The Efficiency of a Fence of Tidal Turbines in the Alderney Race: Comparison between Analytical and Numerical Models. *Energies*. 2021; 14(4):892.
https://doi.org/10.3390/en14040892

**Chicago/Turabian Style**

Thiébot, Jérôme, Nasteho Djama Dirieh, Sylvain Guillou, and Nicolas Guillou.
2021. "The Efficiency of a Fence of Tidal Turbines in the Alderney Race: Comparison between Analytical and Numerical Models" *Energies* 14, no. 4: 892.
https://doi.org/10.3390/en14040892