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

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 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 |

## References

- IEA and World Bank. Sustainable Energy For All. Available online: https://datacatalog.worldbank.org/dataset/sustainable-energy-all (accessed on 18 March 2019).
- Hagerman, G.; Polagye, B.; Bedard, R.; Previsic, M. EPRI Guideline Methodology for Estimating Tidal Current Energy Resources and Power Production by Tidal In-Stream Energy Conversion (TISEC) Devices; Technical Report; Electric Power Research Institute: Palo alto, CA, USA, 2006. [Google Scholar]
- Tarbotton, M.; Larson, M. Canada Ocean Energy Atlas Phase 1: Potential Tidal Current Energy Resources Analysis Background; Canadian Hydraulics Centre: Ottawa, ON, Canada, 2006. [Google Scholar]
- Robins, P.E.; Neill, S.P.; Lewis, M.J.; Ward, S.L. Characterising the spatial and temporal variability of the tidal-stream energy resource over the northwest European shelf seas. Appl. Energy
**2015**, 147, 510–522. [Google Scholar] [CrossRef][Green Version] - Li, D.; Yao, Y.; Chen, Q.; Ye, Z. Numerical simulation of tidal current energy in Yangtze Estuary-Hangzhou Bay, China. In Proceedings of the OCEANS 2015, Genova, Italy, 18–21 May 2015; pp. 1–6. [Google Scholar] [CrossRef]
- Yang, Z.; Wang, T.; Copping, A.; Geerlofs, S. Modeling of in-stream tidal energy development and its potential effects in Tacoma Narrows, Washington, USA. Ocean Coast. Manag.
**2014**, 99, 52–62. [Google Scholar] [CrossRef] - Nash, S.; O’Brien, N.; Olbert, A.; Hartnett, M. Modelling the far field hydro-environmental impacts of tidal farms—A focus on tidal regime, inter-tidal zones and flushing. Comput. Geosci.
**2014**, 71, 20–27. [Google Scholar] [CrossRef] - Wang, T.; Yang, Z. A modeling study of tidal energy extraction and the associated impact on tidal circulation in a multi-inlet bay system of Puget Sound. Renew. Energy
**2017**, 114, 204–214. [Google Scholar] [CrossRef] - Piano, M.; Robins, P.E.; Davies, A.G.; Neill, S.P. The Influence of Intra-Array Wake Dynamics on Depth-Averaged Kinetic Tidal Turbine Energy Extraction Simulations. Energies
**2018**, 11, 2852. [Google Scholar] [CrossRef] - Myers, L.; Bahaj, A. An experimental investigation simulating flow effects in first generation marine current energy converter arrays. Renew. Energy
**2012**, 37, 28–36. [Google Scholar] [CrossRef][Green Version] - Nishino, T.; Willden, R.H.J. The efficiency of an array of tidal turbines partially blocking a wide channel. J. Fluid Mech.
**2012**, 708, 596–606. [Google Scholar] [CrossRef] - Stansby, P.; Stallard, T. Fast optimisation of tidal stream turbine positions for power generation in small arrays with low blockage based on superposition of self-similar far-wake velocity deficit profiles. Renew. Energy
**2016**, 92, 366–375. [Google Scholar] [CrossRef] - Fitch, A.C.; Olson, J.B.; Lundquist, J.K.; Dudhia, J.; Gupta, A.K.; Michalakes, J.; Barstad, I. Local and Mesoscale Impacts of Wind Farms as Parameterized in a Mesoscale NWP Model. Mon. Weather Rev.
**2012**, 140, 3017–3038. [Google Scholar] [CrossRef] - Abkar, M.; Porté-Agel, F. A new wind-farm parameterization for large-scale atmospheric models. J. Renew. Sustain. Energy
**2015**, 7, 013121. [Google Scholar] [CrossRef] - Calaf, M.; Meneveau, C.; Meyers, J. Large eddy simulation study of fully developed wind-turbine array boundary layers. Phys. Fluids
**2010**, 22, 015110. [Google Scholar] [CrossRef][Green Version] - Calaf, M.; Parlange, M.B.; Meneveau, C. Large eddy simulation study of scalar transport in fully developed wind-turbine array boundary layers. Phys. Fluids
**2011**, 23, 126603. [Google Scholar] [CrossRef][Green Version] - Porté-Agel, F.; Lu, H.; Wu, Y.T. Interaction between Large Wind Farms and the Atmospheric Boundary Layer. Procedia IUTAM
**2014**, 10, 307–318. [Google Scholar] [CrossRef][Green Version] - Aghsaee, P.; Markfort, C.D. Effects of flow depth variations on the wake recovery behind a horizontal-axis hydrokinetic in-stream turbine. Renew. Energy
**2018**, 125, 620–629. [Google Scholar] [CrossRef] - Kolekar, N.; Banerjee, A. Performance characterization and placement of a marine hydrokinetic turbine in a tidal channel under boundary proximity and blockage effects. Appl. Energy
**2015**, 148, 121–133. [Google Scholar] [CrossRef] - Spalart, P.; Allmaras, S. A one-equation turbulence model for aerodynamic flows. Rech. Aerosp.
**1994**, 1, 5–21. [Google Scholar] - Escauriaza, C.; Sotiropoulos, F. Reynolds Number Effects on the Coherent Dynamics of the Turbulent Horseshoe Vortex System. Flow Turbul. Combust.
**2011**, 86, 231–262. [Google Scholar] [CrossRef] - Escauriaza, C.; Sotiropoulos, F. Initial stages of erosion and bed form development in a turbulent flow around a cylindrical pier. J. Geophys. Res.-Earth
**2011**, 116. [Google Scholar] [CrossRef] - Escauriaza, C.; Sotiropoulos, F. Lagrangian model of bed-load transport in turbulent junction flows. J. Fluid Mech.
**2011**, 666, 36–76. [Google Scholar] [CrossRef] - Gajardo, D.; Escauriaza, C.; Ingram, D.M. Capturing the development and interactions of wakes in tidal turbine arrays using a coupled BEM-DES model. Ocean Eng.
**2019**, 181, 71–88. [Google Scholar] [CrossRef][Green Version] - Burton, T.; Sharpe, D.; Jenkins, N.; Bossanyi, E. Wind Energy Handbook; John Wiley & Sons, Ltd.: Chichester West Sussex, UK, 2011; ISBN 0-471-48997-2. [Google Scholar]
- Blackmore, T.; Batten, W.M.J.; Bahaj, A.S. Influence of turbulence on the wake of a marine current turbine simulator. Proc. R. Soc. A-Math. Phys.
**2014**, 470, 20140331. [Google Scholar] [CrossRef] [PubMed] - Lloyd, T.P.; Turnock, S.R.; Humphrey, V.F. Assessing the influence of inflow turbulence on noise and performance of a tidal turbine using large eddy simulations. Renew. Energy
**2014**, 71, 742–754. [Google Scholar] [CrossRef] - Smirnov, A.; Shi, S.; Celic, I. Random flow generation technique for Large eddy simulations and particle-dynamics modeling. J. Fluids Eng.
**2001**, 123, 359–371. [Google Scholar] [CrossRef] - Chamorro, L.P.; Porté-Agel, F. Turbulent Flow Inside and Above a Wind Farm: A Wind-Tunnel Study. Energies
**2011**, 4, 1916–1936. [Google Scholar] [CrossRef] - Markfort, C.D.; Zhang, W.; Porté-Agel, F. Turbulent flow and scalar transport through and over aligned and staggered wind farms. J. Turbul.
**2012**, 13. [Google Scholar] [CrossRef] - Wu, Y.T.; Porté-Agel, F. Simulation of Turbulent Flow Inside and Above Wind Farms: Model Validation and Layout Effects. Bound.-Layer Meteorol.
**2013**, 146, 181–205. [Google Scholar] [CrossRef] - Moriasi, D.N.; Arnold, J.G.; Van Liew, M.W.; Bingner, R.L.; Harmel, R.D.; Veith, T.L. Model evaluation guidelines for systematic quantification of accuracy in watershed simulations. Trans. ASABE
**2007**, 50, 885–900. [Google Scholar] [CrossRef] - Stallard, T.; Collings, R.; Feng, T.; Whelan, J. Interactions between tidal turbine wakes: Experimental study of a group of three-bladed rotors. Philos. Trans. R. Soc. A
**2013**, 371, 20120159. [Google Scholar] [CrossRef] - Frost, C.H.; Evans, P.S.; Harrold, M.J.; Mason-Jones, A.; O’Doherty, T.; O’Doherty, D.M. The impact of axial flow misalignment on a tidal turbine. Renew. Energy
**2017**, 113, 1333–1344. [Google Scholar] [CrossRef] - Hill, C.; Musa, M.; Guala, M. Interaction between instream axial flow hydrokinetic turbines and uni-directional flow bedforms. Renew. Energy
**2016**, 86, 409–421. [Google Scholar] [CrossRef] - Musa, M.; Hill, C.; Sotiropoulos, F.; Guala, M. Performance and resilience of hydrokinetic turbine arrays under large migrating fluvial bedforms. Nat. Energy
**2018**, 3, 839–846. [Google Scholar] [CrossRef] - Simón-Moral, A.; Santiago, J.L.; Krayenhoff, E.S.; Martilli, A. Streamwise Versus Spanwise Spacing of Obstacle Arrays: Parametrization of the Effects on Drag and Turbulence. Bound.-Layer Meteorol.
**2014**, 151, 579–596. [Google Scholar] [CrossRef] - Sørensen, J.N.; Shen, W.Z.; Munduate, X. Analysis of wake states by a full-field actuator disc model. Wind Energy
**1998**, 1, 73–88. [Google Scholar] [CrossRef] - Mikkelsen, R.F. Actuator Disc Methods Applied to Wind Turbines. Ph.D. Thesis, Technical University of Denmark, Kongens Lyngby, Denmark, 2004. [Google Scholar]
- Chen, Y.; Lin, B.; Lin, J.; Wang, S. Experimental study of wake structure behind a horizontal axis tidal stream turbine. Appl. Energy
**2017**, 196, 82–96. [Google Scholar] [CrossRef] - Chen, C.; Liu, H.; Beardsley, R.C. An Unstructured Grid, Finite-Volume, Three-Dimensional, Primitive Equations Ocean Model: Application to Coastal Ocean and Estuaries. J. Atmos. Ocean. Technol.
**2003**, 20, 159–186. [Google Scholar] [CrossRef]

**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 |

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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