# The Effect of Free-Atmosphere Stratification on Boundary-Layer Flow and Power Output from Very Large Wind Farms

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## Abstract

**:**

## 1. Introduction

## 2. Large-Eddy Simulation Framework

#### 2.1. LES Governing Equations

#### 2.2. Wind-Turbine Parameterization

#### 2.3. Numerical Setup

Case | z_{o} (m) | Abbreviation | Γ (K/km) | Number of wind turbines (N_{tx} × N_{ty}) | s_{x} × s_{y} | Turbine arrangement | L_{x} × L_{y} × L_{z} (m^{3}) | N_{x} × N_{y} × N_{z} |
---|---|---|---|---|---|---|---|---|

A1 | 0.1 | − | 1 | No farm | − | − | 3906 × 3255 × 1045 | 126 × 175 × 80 |

A2 | 0.01 | − | 1 | No farm | − | − | 3906 × 3255 × 1045 | 126 × 175 × 80 |

A3 | 0.1 | − | 10 | No farm | − | − | 3906 × 3255 × 1045 | 126 × 175 × 80 |

A4 | 0.01 | − | 10 | No farm | − | − | 3906 × 3255 × 1045 | 126 × 175 × 80 |

B1 | 0.1 | s5 × 5 − Γ1 | 1 | 8 × 7 | 5 × 5 | Staggered | 3720 × 3255 × 1687 | 126 × 175 × 128 |

B2 | 0.1 | a5 × 5 − Γ1 | 1 | 8 × 7 | 5 × 5 | Aligned | 3720 × 3255 × 1687 | 126 × 175 × 128 |

C1 | 0.1 | s5 × 5 − Γ10 | 10 | 8 × 7 | 5 × 5 | Staggered | 3720 × 3255 × 1045 | 126 × 175 × 80 |

C2 | 0.1 | a5 × 5 − Γ10 | 10 | 8 × 7 | 5 × 5 | Aligned | 3720 × 3255 × 1045 | 126 × 175 × 80 |

D1 | 0.1 | s7 × 7 − Γ1 | 1 | 6 × 5 | 7 × 7 | Staggered | 3906 × 3255 × 1687 | 126 × 175 × 128 |

D2 | 0.1 | a7 × 7 − Γ1 | 1 | 6 × 5 | 7 × 7 | Aligned | 3906 × 3255 × 1687 | 126 × 175 × 128 |

E1 | 0.1 | s7 × 7 − Γ10 | 10 | 6 × 5 | 7 × 7 | Staggered | 3906 × 3255 × 1045 | 126 × 175 × 80 |

E2 | 0.1 | a7 × 7 − Γ10 | 10 | 6 × 5 | 7 × 7 | Aligned | 3906 × 3255 × 1045 | 126 × 175 × 80 |

## 3. LES Results

#### 3.1. No-Farm Case

**Figure 2.**Vertical profiles of horizontally-averaged velocity magnitude in M (

**a**) linear scale; and (

**b**) semi-log scale; (

**c**) wind direction; and (

**d**) horizontally-averaged potential temperature (Θ) inside the ABL for two different values of Γ and ${\text{z}}_{\mathrm{o}}$. The horizontal dotted lines show the top-tip and bottom-tip heights.

**Figure 3.**Vertical profile of total shear stress $\left(\sqrt{{\mathsf{\tau}}_{\text{xz}}^{2}+{\mathsf{\tau}}_{\text{yz}}^{2}}\right)$ for two different values of $\Gamma $ and ${\text{z}}_{\mathrm{o}}$.

#### 3.1.1. The Effect of Free-Atmosphere Stability on the ABL Depth

**Figure 4.**Correlation between the ABL heights: ${\mathsf{\delta}}_{\text{bl}}\text{}(\text{theory})$ calculated from Equation (6) (with ${\text{C}}_{\mathrm{R}}=0.5$ and ${\text{C}}_{\mathrm{N}}=0.11$) and ${\mathsf{\delta}}_{\text{bl}}\text{}(\text{LES})$ directly obtained from LES.

**Figure 5.**Correlation between ${\mathsf{\delta}}_{\text{bl}}^{\text{*}}\text{}(\text{theory})$ calculated from Equation (6) (with ${\text{C}}_{\mathrm{R}}=0.16$ and ${\text{C}}_{\mathrm{N}}=0.02$) and ${\mathsf{\delta}}_{\text{bl}}^{\text{*}}\text{}(\text{LES})$ directly obtained from LES.

#### 3.1.2. The Effect of Free-Atmosphere Stability on the Surface-Layer Parameterization

#### 3.2. Wind-Farm Case

**Figure 6.**Vertical profiles of horizontally-averaged velocity magnitude in (

**a**) linear scale; and (

**b**) semi-log scale; (

**c**) wind direction; and (

**d**) horizontally-averaged potential temperature (Θ) through very large wind farms for two different wind-turbine spacings, and two different values of Γ.

**Figure 7.**Vertical profile of total shear stress $\left(\sqrt{{\mathsf{\tau}}_{\text{xz}}^{2}+{\mathsf{\tau}}_{\text{yz}}^{2}}\right)$ for two different wind-turbine spacings, and two different values of $\Gamma $ through very large wind farms.

**Figure 8.**Correlation between ${\mathsf{\delta}}_{\text{bl}}$ and ${\mathsf{\delta}}_{\text{bl}}^{\text{*}}$ theory and their counterparts directly obtained from LES.

#### 3.3. Layout Effect

**Figure 9.**Vertical profiles of (

**a**) horizontally-averaged velocity magnitude M; and (

**b**) total shear stress for two different values of and two different layouts.

Case | Power/Turbine (MW) |
---|---|

s5 × 5 − Γ1 | 0.3069 |

a5 × 5 − Γ1 | 0.2835 |

s5 × 5 − Γ10 | 0.1995 |

a5 × 5 − Γ10 | 0.1841 |

s7 × 7 − Γ1 | 0.4303 |

a7 × 7 − Γ1 | 0.3811 |

s7 × 7 − Γ10 | 0.2993 |

a7 × 7 − Γ10 | 0.2690 |

**Figure 10.**Contours of mean (

**a**) and instantaneous (

**b**) streamwise velocity (m/s) at the hub-height level for the different cases. Only a section of the domain is shown.

## 4. One-Dimensional (1D) Modeling of Very Large Wind Farms in Conventionally Neutral Conditions

**Table 3.**The values of $a$, ${\mathrm{C}}_{\text{T}}^{\prime}$, and ${\mathrm{C}}_{\text{T}}$ for the different cases.

Case | a | ${C}_{T}^{\prime}$ | ${C}_{T}$ |
---|---|---|---|

$\text{s}5\times 5-\Gamma 1$ | $0.197$ | $0.98$ | $0.63$ |

$\text{a}5\times 5-\Gamma 1$ | $0.199$ | $1.00$ | $0.64$ |

$\text{s}5\times 5-\Gamma 10$ | $0.203$ | $1.02$ | $0.64$ |

$\text{a}5\times 5-\Gamma 10$ | $0.207$ | $1.04$ | $0.65$ |

$\text{s}7\times 7-\Gamma 1$ | $0.195$ | $0.97$ | $0.63$ |

$\text{a}7\times 7-\Gamma 1$ | $0.195$ | $0.97$ | $0.63$ |

$\text{s}7\times 7-\Gamma 10$ | $0.192$ | $0.95$ | $0.62$ |

$\text{a}7\times 7-\Gamma 10$ | $0.195$ | $0.97$ | $0.63$ |

**Figure 14.**Influence of free-atmosphere stability on the friction velocity at the turbine-top level.

**Figure 16.**Effect of underlying surface roughness and free-atmosphere stability on the power output for $\text{s}=7.$

## 5. Conclusions

## Acknowledgments

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**MDPI and ACS Style**

Abkar, M.; Porté-Agel, F.
The Effect of Free-Atmosphere Stratification on Boundary-Layer Flow and Power Output from Very Large Wind Farms. *Energies* **2013**, *6*, 2338-2361.
https://doi.org/10.3390/en6052338

**AMA Style**

Abkar M, Porté-Agel F.
The Effect of Free-Atmosphere Stratification on Boundary-Layer Flow and Power Output from Very Large Wind Farms. *Energies*. 2013; 6(5):2338-2361.
https://doi.org/10.3390/en6052338

**Chicago/Turabian Style**

Abkar, Mahdi, and Fernando Porté-Agel.
2013. "The Effect of Free-Atmosphere Stratification on Boundary-Layer Flow and Power Output from Very Large Wind Farms" *Energies* 6, no. 5: 2338-2361.
https://doi.org/10.3390/en6052338