# Flow Adjustment Inside and Around Large Finite-Size Wind Farms

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Large-Eddy Simulations

#### 2.1. LES Governing Equations and Modeling

#### 2.2. Numerical Setup

## 3. Suite of Simulations

^{−1}, and the Coriolis parameter is set to ${f}_{c}=1.195\times {10}^{-4}$ rad·s

^{−1}(i.e., equivalent to the latitude of the North Sea). The surface roughness is set to ${z}_{o}=0.05$ m and the reference potential temperature is ${\theta}_{0}=293$ K. The value of these parameters is set such that the inflow conditions generated for the wind farm simulations are comparable to those in the North Sea [40]. The simulations are initialized with a constant velocity in the streamwise direction of 10 m·s

^{−1}. The potential temperature is initialized with a prescribed constant temperature lapse rate $\Gamma $. Very small random perturbations are added to the velocity and the temperature fields at the lowest 100 m to induce the development of turbulence. The wind farms simulated are in both aligned and staggered configurations with a streamwise turbine spacing (${s}_{x}$) and a spanwise turbine spacing (${s}_{y}$) of 7 D. The simulations of the no-farm and the infinite wind farm cases are run for 8 h, while the simulations of the large finite-size wind farm cases are run for 6 h. The statistics presented in the latter sections are averaged over the last hour of the simulations, when quasi-steady flow conditions are guaranteed. Table 1 summarizes the key parameters of the various simulations carried out in this study.

#### 3.1. No-Farm Case

^{−1}and a turbulence intensity ($TI$) of 7% for $\Gamma =1$ K. The hub height velocity is 7.8 m·s

^{−1}and the turbulence intensity ($TI$) is 6.5% for $\Gamma =5$ K. The turbulence intensity is calculated as:

#### 3.2. Large Finite-Size Wind Farm Simulations

#### 3.3. Infinite Wind Farm Simulations

## 4. Results

- (i)
- The induction region, immediately upwind of the wind farm, where flow deceleration is induced by the wind farm due to the cumulated wind turbine blockage effect. If free-atmosphere stratification is strong enough, it could lead to a gravity-wave-induced blockage effect triggered by the farm. As a result, in that case, the velocity in this region would be significantly reduced, and the induction region could extend a few kilometers upwind of the farm.
- (ii)
- The entrance and development region, extending immediately downwind of the leading edge of the wind farm, where the flow is decelerated because of the momentum extraction by the wind turbines. The deceleration results in an upward mass flux from the wind farm top, which slows down the flow above the farm and causes an internal boundary layer (IBL) growth. For wind farms that are large enough, the IBL could reach the ABL height and result in a modification of the ABL depth.
- (iii)
- The fully-developed region, following the entrance and development region, in which changes of flow characteristics in the streamwise direction are negligible throughout the ABL, and thus the flow is considered to be fully developed. In this region, both the IBL and ABL depths are constant and equal to the height of the infinite wind farm ABL.
- (iv)
- The exit region, upwind of the trailing edge of the wind farm, where the flow accelerates and a downward mass flux is present, resulting in a decrease in the IBL and ABL heights. If free-atmosphere stratification is strong, the downward flow could trigger upwind propagating gravity waves and an advantageous pressure gradient at the trailing farm edge. As a consequence, in that case, the favorable gravity-wave-induced pressure gradient would lead to a substantial flow acceleration in the region. The exit region could extend a few kilometers upwind of the end of the farm, similar to the length of the induction region.
- (v)
- The wind farm wake region, downwind of the wind farm trailing edge, in which the flow recovers and returns to its undisturbed inflow velocity profile.

#### 4.1. Wind Farm Induction Region

#### 4.2. Wind Farm Flow Entrance and Development Region

#### 4.2.1. Velocity Adjustment

#### 4.2.2. IBL and ABL Growth

#### 4.2.3. Turbulent Shear Stress Adjustment

#### 4.2.4. TKE Adjustment

#### 4.2.5. Wind Farm Power Output

#### 4.3. Length of Flow Development Region

#### 4.4. Wind Farm Wakes

## 5. Summary

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Vertical profiles of the horizontally-averaged ABL flow characteristics in the no-farm cases: (

**a**) mean velocity magnitude $\overline{M}$; (

**b**) wind direction; (

**c**) potential temperature; (

**d**) total Reynolds shear stress ${({\left(\overline{{u}^{\prime}{w}^{\prime}}\right)}^{2}+{\left(\overline{{v}^{\prime}{w}^{\prime}}\right)}^{2})}^{1/2}$; and (

**e**) TKE. The black solid lines represent the top-tip, hub, and bottom-tip heights.

**Figure 3.**Flow adjustment regions in large finite-size wind farms in CNBLs with (

**a**) weak and (

**b**) strong free-atmosphere stratification. The flow can be divided into the following regions: (i) induction region; (ii) entrance and development region; (iii) fully-developed region; (iv) exit region; (v) wind farm wake.

**Figure 4.**Contours of time-averaged horizontal velocity magnitude M on the xz plane through the center of a wind turbine column for the case (

**a**) FS−a$\Gamma 1$; (

**b**) FS−s$\Gamma 1$; (

**c**) FS−a$\Gamma 5$; (

**d**) FS−s$\Gamma 5$. The IBL height (thick blue line), the CNBL height (white line), the CNBL height of the inflow (dashed black line), Elliot’s 0.8 power law (bright green line) and the CNBL height with an infinite wind farm (solid black line) are also included. The thin blue lines are the velocity streamlines (i.e., the vertical component is made larger by a scale factor of 5). The black vertical solid lines indicate the start and end of different flow regions.

**Figure 5.**Time- and spanwise-averaged velocity magnitude at the hub height in the wind farm induction region, normalized by the inflow velocity magnitude at hub height.

**Figure 6.**Contours of time- and spanwise-averaged vertical potential temperature gradient $d\theta /dz$ for the case (

**a**) FS−s$\Gamma 1$; (

**b**) FS−s$\Gamma 5$. The dotted black lines indicate the bottom and the top heights of the capping inversion. Contours of time- and spanwise-averaged modified pressure for the case (

**c**) FS−s$\Gamma 1$; (

**d**) FS−s$\Gamma 5$. The vertical solid black lines indicate the streamwise positions of the beginning and the end of the wind farm. Results are not shown for the aligned farms as they are similar to the staggered farms.

**Figure 7.**Vertical profiles of time- and spanwise-averaged velocity magnitude at the 1st, 5th, 12th and 36th turbine rows for the cases: (

**a**) FS−a$\Gamma 1$; (

**b**) FS−s$\Gamma 1$; (

**c**) FS−a$\Gamma 5$; and (

**d**) FS−s$\Gamma 5$. The black solid profile represents the velocity magnitude of the ABL inflow, and the black dashed profile represents the time- and horizontal-averaged velocity magnitude of the infinite wind farm with the same wind farm configuration. The black horizontal solid lines represent the turbine top-tip, hub and bottom-tip heights.

**Figure 8.**Time- and horizontally-averaged (the distance of streamwise averaging is 7D, with the turbine row placed at the center) velocity magnitude at the hub height at different wind turbine rows for the wind farm cases under (

**a**) $\Gamma =1$ K/km; and (

**b**) $\Gamma =5$ K/km. The black horizontal dotted and solid lines are the time- and horizontally-averaged velocity magnitude at the hub height for the corresponding aligned and staggered infinite wind farm cases, respectively

**Figure 9.**Vertical profiles of time- and spanwise-averaged total shear stress at the 1st, 5th, 12th and 36th turbine rows for the cases: (

**a**) FS−a$\Gamma 1$; (

**b**) FS−s$\Gamma 1$; (

**c**) FS−a$\Gamma 5$; and (

**d**) FS−s$\Gamma 5$. The black solid profile represents the total shear stress of the ABL inflow, and the black dashed profile represents the time- and horizontal-averaged total shear stress of the infinite wind farm with the same wind farm configuration. The black horizontal solid lines represents the top-tip, hub and bottom-tip heights.

**Figure 10.**Vertical profiles of time- and spanwise-averaged TKE at the 1st, 2nd, 7th, and 36th turbine rows for the cases: (

**a**) FS−a$\Gamma 1$; (

**b**) FS−s$\Gamma 1$; (

**c**) FS−a$\Gamma 5$; and (

**d**) FS−s$\Gamma 5$. The black solid profile represents the TKE of the ABL inflow, and the black dashed profile represents the time- and horizontal-averaged TKE of the infinite wind farm with the same wind farm configuration. The black horizontal solid lines represents the top-tip, hub and bottom-tip heights.

**Figure 11.**Power production at different turbine rows inside the wind farms for the (

**a**) $\Gamma =1$ K/km cases and (

**b**) $\Gamma =5$ K/km cases, normalized by the power production of a single turbine operating under the same lapse rate. The black horizontal dotted and solid lines are the power production of a wind turbine for the corresponding aligned and staggered infinite wind farm cases, respectively.

**Figure 12.**Vertical profiles of time- and spanwise-averaged velocity magnitude at 1 km, 5 km and 10 km downwind of the wind farm, and at the 36th (last) turbine row for the cases: (

**a**) FS−a$\Gamma 1$; (

**b**) FS−s$\Gamma 1$; (

**c**) FS−a$\Gamma 5$; and (

**d**) FS−s$\Gamma 5$. The solid black profile represents the velocity magnitude of the ABL inflow, and the black dashed profile represents the time- and horizontal-averaged velocity magnitude of the infinite wind farm with the same wind farm configuration. The black horizontal solid lines represent the top-tip, hub and bottom-tip heights.

**Figure 13.**Vertical profiles of time- and spanwise-averaged TKE at 1 km, 5 km and 10 km downwind of the wind farm, and at the 36th (last) turbine row for the cases: (

**a**) FS−a$\Gamma 1$; (

**b**) FS−s$\Gamma 1$; (

**c**) FS−a$\Gamma 5$; and (

**d**) FS−s$\Gamma 5$. The solid black profile represents the TKE of the ABL inflow, and the black dashed profile represents the time- and horizontal-averaged total shear stress of the infinite wind farm with the same turbine configuration. The black horizontal solid lines represent the top-tip, hub and bottom-tip heights.

Case | $\mathbf{\Gamma}$ (K/km) | Number of Turbines | ${\mathit{s}}_{\mathit{x}}\times {\mathit{s}}_{\mathit{y}}$ | Turbine Arrangement | ${\mathit{L}}_{\mathit{x}}\times {\mathit{L}}_{\mathit{y}}\times {\mathit{L}}_{\mathit{z}}$ (km ^{3})
| ${\mathit{N}}_{\mathit{x}}\times {\mathit{N}}_{\mathit{y}}\times {\mathit{N}}_{\mathit{z}}$ |
---|---|---|---|---|---|---|

No-Farm Case | ||||||

NF$-\Gamma 1$ | 1 | - | - | - | $14\times 2.8\times 2.4$ | $350\times 175\times 240$ |

NF$-\Gamma 5$ | 5 | - | - | - | $14\times 2.8\times 2.4$ | $350\times 175\times 240$ |

Large Finite-Size Wind Farm Case | ||||||

FS−s$\Gamma 1$ | 1 | $36\times 5$ | $7D\times 7D$ | Staggered | $42\times 2.8\times 2.4$ | $1050\times 175\times 240$ |

FS−a$\Gamma 1$ | 1 | $36\times 5$ | $7D\times 7D$ | Aligned | $42\times 2.8\times 2.4$ | $1050\times 175\times 240$ |

FS−s$\Gamma 5$ | 5 | $36\times 5$ | $7D\times 7D$ | Staggered | $42\times 2.8\times 2.4$ | $1050\times 175\times 240$ |

FS−a$\Gamma 5$ | 5 | $36\times 5$ | $7D\times 7D$ | Aligned | $42\times 2.8\times 2.4$ | $1050\times 175\times 240$ |

Infinite Wind Farm Case | ||||||

Inf−s$\Gamma 1$ | 1 | $8\times 4$ | $7D\times 7D$ | Staggered | $4.5\times 2.8\times 2.4$ | $112\times 140\times 240$ |

Inf−a$\Gamma 1$ | 1 | $8\times 4$ | $7D\times 7D$ | Aligned | $4.5\times 2.8\times 2.4$ | $112\times 140\times 240$ |

Inf−s$\Gamma 5$ | 5 | $8\times 4$ | $7D\times 7D$ | Staggered | $4.5\times 2.8\times 2.4$ | $112\times 140\times 240$ |

Inf−a$\Gamma 5$ | 5 | $8\times 4$ | $7D\times 7D$ | Aligned | $4.5\times 2.8\times 2.4$ | $112\times 140\times 240$ |

$\mathbf{\Gamma}$ (K/km) | $\mathit{U}\phantom{\rule{0.166667em}{0ex}}$ (m·s${}^{-1}$) | ${\mathit{z}}_{\mathit{i}}$ (m) | N (s${}^{-1})$ | ${\mathit{D}}_{\mathit{h}}$ (m) | $\mathit{\theta}\left({\mathit{D}}_{\mathit{h}}\right)$ (K) | $\mathit{Fr}$ |
---|---|---|---|---|---|---|

$\Gamma =1$ | 9.63 | 710 | 0.0057 | 2370 | 295 | 0.94 |

$\Gamma =5$ | 8.05 | 510 | 0.013 | 1220 | 299 | 1.31 |

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**MDPI and ACS Style**

Wu, K.L.; Porté-Agel, F.
Flow Adjustment Inside and Around Large Finite-Size Wind Farms. *Energies* **2017**, *10*, 2164.
https://doi.org/10.3390/en10122164

**AMA Style**

Wu KL, Porté-Agel F.
Flow Adjustment Inside and Around Large Finite-Size Wind Farms. *Energies*. 2017; 10(12):2164.
https://doi.org/10.3390/en10122164

**Chicago/Turabian Style**

Wu, Ka Ling, and Fernando Porté-Agel.
2017. "Flow Adjustment Inside and Around Large Finite-Size Wind Farms" *Energies* 10, no. 12: 2164.
https://doi.org/10.3390/en10122164