# A Formulation of the Thrust Coefficient for Representing Finite-Sized Farms of Tidal Energy Converters

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Numerical Simulations

#### Model Validation

#### 2.2. Parameterization of Farms of TEC Devices

#### 2.2.1. Resultant Force for a Turbine Farm

#### 2.2.2. Thrust Coefficient for a Turbine Farm

#### 2.2.3. Setup for Numerical Simulations

## 3. Results and Discussion

#### 3.1. Formulation of the Thrust Coefficient for Farms of Turbines

- (I)
- For farms with two rows, the expression for ${C}_{tFarm}$ is inversely proportional to the lateral distance between devices, ${S}_{y}/D$. This value can go from ${S}_{y}/D=1$, when the turbines are adjacent to each other, to ${S}_{y}/D\to \infty $ for very laterally spaced farms, where one would expect the drag force to be additive since the wakes do not interact.
- (II)
- For farms with more than two rows, the dependence of ${C}_{tFarm}$ is still inversely proportional to ${S}_{y}/D$, but also decays exponentially with the distance between the devices in the stream-wise direction ${S}_{x}/D$. An exponential decay is proposed because it tends to zero as ${S}_{x}/D\to 0$ (i.e., the devices get closer), which captures the fact that the flow cannot penetrate the farm. On the other hand, when the stream-wise distance between the devices increases, the exponential term tends towards unity, which is equivalent to saying that ${C}_{tFarm}$ becomes independent of ${S}_{x}/D$. This expression is similar to the one proposed by Simón-Moral et al. [37] for parameterizing canopies of vegetation.

#### 3.2. Comparison of ${C}_{tFarm}$ Parameterization with Previous Work

## 4. Conclusions

- It is designed for staggered farms, where all the turbines occupy the same ground area.
- It does not consider a significant misalignment between the mean flow direction and the turbine axes.
- It is designed for devices that are installed at the bottom of the sea, and which do not interact with the free-surface.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DES | Detached-Eddy Simulations |

LES | Large-Eddy Simulations |

TEC | Tidal Energy Converter |

RANS | Reynolds-Averaged Navier–Stokes |

RMSE | Root Mean Square Error |

RSR | Root mean square error-observations Standard deviation Ratio |

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**Figure 2.**Mean streamwise velocity normalized by ${U}_{0}$ (the inlet velocity at the hub height, ${Z}_{hub}$) as a function of $Z/{Z}_{hub}$, downstream of the 1st, 5th, and 11th rows of turbines. Circles show measurements (Data from: Markfort et al. [30]). Continuous lines are the results from the DES simulations coupled with the actuator disk approach. Dashed lines mark the bottom, center, and top of the disks.

**Figure 3.**Turbulence intensity in the streamwise direction (${\sigma}_{u}/{U}_{0}$) profile as a function of $Z/{Z}_{hub}$, downstream of the 1st, 5th, and 11th rows of turbines. Circles show measurements (Data from: Markfort et al. [30]). Continuous lines are the results from the DES simulations coupled with the actuator disk approach. Dashed lines mark the bottom, center, and top of the disks.

**Figure 4.**Schematic of a representative control volume used to calculate the resultant force for an array of devices. Here, ${L}_{x}$ and ${L}_{y}$ are the length and the width of the control volume, respectively; meanwhile, the height, ${L}_{z}$, is the same as the channel. In this example, the control volume encloses 18 devices, and comprises a volume that includes from the first to the seventh row of turbines.

**Figure 5.**Resultant force calculated by using control volumes that go from the first row of turbines through the last one (see Figure 4), per unit of mass ($\rho {L}_{x}{L}_{y}{L}_{z}$); ▪ analytic force calculated as the force of one actuator disk times the total number of devices in the farm; ⧫ analytic force calculated by using the average velocity at the location of the disks, ${U}_{d}$, instead of the undisturbed velocity, ${U}_{\infty}$; • resultant force obtained from DES numerical simulations.

**Figure 6.**$\left(\mathbf{a}\right)$ variation of ${C}_{tFarm}$ due to changes in the size of the farm in the spanwise direction, ${L}_{y}$. The ratio ${L}_{y}/{S}_{y}$ can be interpreted as the number of columns of turbines; $\left(\mathbf{b}\right)$ schematic of the control volumes used for calculating ${C}_{tFarm}$ for two and four rows of turbines.

**Figure 7.**$\left(\mathbf{a}\right)$ variation of ${C}_{tFarm}$ due to changes in the size of the farm in the streamwise direction, ${L}_{x}$. The ratio ${L}_{x}/{S}_{x}$ can be interpreted as the number of rows of turbines; $\left(\mathbf{b}\right)$ schematic of the control volumes used for calculating ${C}_{tFarm}$ for various numbers of rows.

**Figure 8.**Changes in ${C}_{tFarm}$ due to the variation of: $\left(\mathbf{a}\right)$ the streamwise distance between devices, ${S}_{x}/D$; $\left(\mathbf{b}\right)$ the spanwise distance between devices, ${S}_{y}/D$; and $\left(\mathbf{c}\right)$ the ratio between the depth of the channel and the hub height, $H/{Z}_{hub}$. The results are divided into two cases: farms with two rows of turbines (light blue), and farms with more than two rows (purple). Since we do not observe a significant influence of ${S}_{x}/D$ on ${C}_{tFarm}$ for farms with two rows, we perform extra simulations (marked with dashed lines), where we see ${C}_{tFarm}$ remains insensitive to the distance in the streamwise direction.

**Figure 9.**${C}_{tFarm}/{C}_{t}^{\prime}$, for farms with exactly two rows of turbines, versus the disk separation in: $\left(\mathbf{a}\right)$ the streamwise direction, and $\left(\mathbf{b}\right)$ in the spanwise direction. Continuous line: Empirical solution proposed in Equation (9). Dots: Results from DES numerical simulations.

**Figure 10.**${C}_{tFarm}/{C}_{t}^{\prime}$, for farms with more than two rows of turbines, versus the disks separation in: $\left(\mathbf{a}\right)$ the streamwise direction, and $\left(\mathbf{b}\right)$ in the spanwise direction. Continuous line: Empirical solution proposed in Equation (9). Dots: Results from DES numerical simulations.

Case | ${\mathit{S}}_{\mathit{x}}/\mathit{D}$ | ${\mathit{S}}_{\mathit{y}}/\mathit{D}$ | $\mathit{H}/{\mathit{Z}}_{\mathit{hub}}$ |
---|---|---|---|

C.1 | 5 | 4 | 4.2 |

C.2 | 7 | 4 | 4.2 |

C.3 | 3 | 4 | 4.2 |

C.4 | 5 | 2 | 4.2 |

C.5 | 5 | 6 | 4.2 |

C.6 | 5 | 4 | 3.3 |

C.7 | 5 | 4 | 5.0 |

Parameter | Value |
---|---|

Turbines diameter (D) | 10 m |

Hub height (${z}_{hub}$) | 12 m |

Thrust coefficient $\overline{{C}_{t}^{\prime}}$ | 0.85 |

Channel length (${L}_{x}$) | 350 m |

Channel width (${L}_{y}$) | 240 m |

Grid resolution ($im\times jm\times km$) | 268 × 192 × 128 |

Reynolds number based on the velocity at the hub ($R{e}_{{Z}_{hub}}$) | $7.5\times {10}^{6}$ |

Lateral boundary conditions | Symmetric |

**Table 3.**Comparison of the results of the $\xi $ parameter (Data from: Abkar and Porté-Agel [14]) with $\xi $ calculated by using our parameterization of ${C}_{tFarm}/{C}_{t}^{\prime}$.

${\mathit{S}}_{\mathit{x}}/\mathit{D}$ | ${\mathit{S}}_{\mathit{y}}/\mathit{D}$ | ${\mathit{C}}_{\mathbf{tFarm}}/{\mathit{C}}_{\mathit{t}}^{\prime}$ | $\mathit{\xi}$ Calculated | $\mathit{\xi}$ Proposed by Abkar and Porté-Agel [14] | Error $(\%)$ |
---|---|---|---|---|---|

5 | 5 | 0.71 | 1.12 | 1.13 | 0.6 |

7 | 7 | 0.69 | 1.10 | 1.07 | 3.2 |

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**MDPI and ACS Style**

Soto-Rivas, K.; Richter, D.; Escauriaza, C.
A Formulation of the Thrust Coefficient for Representing Finite-Sized Farms of Tidal Energy Converters. *Energies* **2019**, *12*, 3861.
https://doi.org/10.3390/en12203861

**AMA Style**

Soto-Rivas K, Richter D, Escauriaza C.
A Formulation of the Thrust Coefficient for Representing Finite-Sized Farms of Tidal Energy Converters. *Energies*. 2019; 12(20):3861.
https://doi.org/10.3390/en12203861

**Chicago/Turabian Style**

Soto-Rivas, Karina, David Richter, and Cristian Escauriaza.
2019. "A Formulation of the Thrust Coefficient for Representing Finite-Sized Farms of Tidal Energy Converters" *Energies* 12, no. 20: 3861.
https://doi.org/10.3390/en12203861