# Blockage Corrections for Tidal Turbines—Application to an Array of Turbines in the Alderney Race

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## Abstract

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## 1. Introduction

## 2. Materials and Method

#### 2.1. Study Site and Turbine Array

#### 2.2. Analytical Model

#### 2.3. Numerical Model

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$\alpha $ | Ratio of the downstream to the upstream velocity, dimensionless |

$\beta $ | Ratio of the velocity in the disc to the upstream velocity, dimensionless |

$\gamma $ | Bypass induction factor, dimensionless |

$\epsilon $ | Blockage ratio, dimensionless |

$\rho $ | Water density, kg/m${}^{3}$ |

${\nu}_{eff}$ | Effective kinematic viscosity, m${}^{2}$ s${}^{-1}$ |

$\delta p$ | Pressure drop across the disc, kg m${}^{-1}$ s${}^{-2}$ |

$\delta z$ | Free surface elevation, m |

A | Area swept by the blades, m${}^{2}$ |

a | Axial induction factor, dimensionless |

${C}_{T}$ | Thrust coefficient, dimensionless |

${C}_{P}$ | Power coefficient, dimensionless |

e | Thickness of the disc, m |

F | Thrust force, kg m s${}^{-2}$ |

$Fr$ | Froude number, dimensionless |

g | Acceleration of the earth’s gravity, m/s${}^{2}$ |

h | Water depth, m |

K | Resistance coefficient, dimensionless |

l | Transverse width of the flow, m |

${P}_{a}$ | Atmospheric pressure, kg m${}^{-1}$ s${}^{-2}$ |

${s}_{1}$ | Upstream area of the actuator disk per unit transverse width of the flow, m |

${s}_{d}$ | Area of the actuator disk per unit transverse width of the flow, m |

${s}_{w}$ | Far downstream area of the actuator disk per unit transverse width of the flow, m |

${S}_{x}$, S${}_{y}$, S${}_{z}$ | Source terms, kg m${}^{-2}$ s${}^{-2}$ |

t | Time, s |

${U}_{\infty}$ | Upstream velocity, m s${}^{-1}$ |

${U}_{d}$ | Velocity in the disc, m s${}^{-1}$ |

${U}_{2}$ | Velocity in the far downstream, m s${}^{-1}$ |

${U}_{w}$ | Velocity in the far wake, m s${}^{-1}$ |

u, v, w | Three components of the velocity, m s${}^{-1}$ |

${z}_{2}$ | Water depth, m |

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**Figure 2.**(

**a**) Location of the study zone. (

**b**) English Channel computational domain and bathymetry. (

**c**) The red cross represents the location of the array.

**Figure 3.**Schematic representation of the two configurations. (

**a**) Aligned layout and (

**b**) staggered layout. The rows are numbered in the direction of the flow. Thus, line 1 stands for the row located in the free stream.

**Figure 4.**Schematic representation of the flow around an AD in a channel with free surface deformation. The numbers 1–3 represent particular locations (upstream, at the disk, downstream).

**Figure 6.**Evolution of the thrust coefficient as a function of the induction factor for the case with blockage correction (red curve) and for the case without blockage corrections (blue curve). The values of $Fr$ and B are calculated from the averaged values of velocities and water depth retained for our application in the Alderney Race (Section 3, Table 1) ($Fr=0.152$ and $B=0.09$).

**Figure 7.**Time-series of the resistance coefficient K during a mean spring tide on 9 August 2014. The peak ebb and flood are represented by blue and red circles, respectively.

**Figure 8.**Time-series of (

**a**) free surface elevation (with respect to the mean sea level), (

**b**) depth-averaged velocity magnitude and (

**c**) depth-averaged velocity direction (nautical convention) extracted from the Telemac3D model (simulation without turbines) in the centre of the tidal farm (49.705${}^{\circ}$ N; 2.103${}^{\circ}$ W) during a mean spring tide on 9 August 2014. The peak ebb and flood are represented by blue and red circles, respectively.

**Figure 9.**Spatial distribution of depth-averaged velocities (m/s) at (

**a**) peak ebb and (

**b**) flood during a mean spring tide in the Alderney Race (simulation without turbines). The cross represents the centre of the tidal farm. The origin of the reference frame (coordinates in m) corresponds to the centre of the farm.

**Figure 10.**Time-series of output per row of turbines with (solid lines, curves captioned “w/”) and without blockage corrections (dotted lines, curves captioned “w/o”). (

**a**) Aligned and (

**b**) staggered layouts. Note that rows/lines of turbines are ordered with respect to the incoming flow. Thus, line 1 refers to the line of devices facing the ebb and peak flow whereas line 5 refers to the last line of the turbine array.

**Figure 11.**The mean production per row. The numbers of rows is N. For the aligned configuration, the lime green bar (AL${}_{w}$) corresponds to the case with blockage correction and the green bar (AL${}_{w/o}$) to the case without blockage correction. For the staggered configuration, the purple bar (ST${}_{w}$) corresponds to the case with blockage correction and the red bar (ST${}_{w/o}$) to the case without blockage correction.

**Figure 12.**(

**a**) Aligned arrangement and (

**b**) staggered arrangement: difference in velocity magnitude with and without blockage corrections (in m/s). The results were extracted along a horizontal plane (located at the hub height) at peak flood of a mean spring tidal cycle (the seabed is quasi-horizontal).

**Table 1.**Characteristics of the flow during peak ebb and flood in the centre of the tidal farm (49.705${}^{\circ}$ N; 2.103${}^{\circ}$ W).

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

Velocity magnitude (m/s) | $3.02$ | $2.76$ |

Velocity direction (${}^{\circ}$/North) | 206 | 32 |

Water depth (m) | $41.05$ | $46.52$ |

Froude number (dimensionless) | $0.150$ | $0.129$ |

Local blockage ratio (%) | $8.93$ | $7.88$ |

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**MDPI and ACS Style**

Djama Dirieh, N.; Thiébot, J.; Guillou, S.; Guillou, N. Blockage Corrections for Tidal Turbines—Application to an Array of Turbines in the Alderney Race. *Energies* **2022**, *15*, 3475.
https://doi.org/10.3390/en15103475

**AMA Style**

Djama Dirieh N, Thiébot J, Guillou S, Guillou N. Blockage Corrections for Tidal Turbines—Application to an Array of Turbines in the Alderney Race. *Energies*. 2022; 15(10):3475.
https://doi.org/10.3390/en15103475

**Chicago/Turabian Style**

Djama Dirieh, Nasteho, Jérôme Thiébot, Sylvain Guillou, and Nicolas Guillou. 2022. "Blockage Corrections for Tidal Turbines—Application to an Array of Turbines in the Alderney Race" *Energies* 15, no. 10: 3475.
https://doi.org/10.3390/en15103475