# Theoretical Modeling of Vertical-Axis Wind Turbine Wakes

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Wake Modeling

#### 2.1. Top-Hat Wake Model

#### 2.2. Gaussian Wake Model

## 3. Case Descriptions and the Results

## 4. Summary and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Time-averaged streamwise wind velocity, normalized with the hub-height speed, in the central vertical (

**left**) and lateral planes (

**right**). From top to the bottom: Case (1), Case (2) and Case (3). The turbines rotate counterclockwise.

**Figure 2.**Normalized velocity defect profiles ($\Delta U/{U}_{h}$), in wall-normal (

**left**) and spanwise (

**right**) directions, through the turbine center downstream for Case (1). LES data (open circle), top-hat model (solid red line), and Gaussian model (solid black line). The horizontal dotted lines denote the turbine extent.

**Figure 3.**Same as Figure 2 but for Case (2).

**Figure 4.**Same as Figure 2 but for Case (3).

**Figure 6.**Schematic of the two turbines immersed in the flow (red circles are the measurement locations) (

**left**), and wall-normal profiles of the normalized velocity defect (

**right**) for Case (5). The LES (open circle) and field measurement (asterisk) data were extracted from Hezaveh et al. [42].

**Figure 7.**Contours of the normalized velocity defect in the cross-sectional planes downwind of the turbine for Case (1) obtained from the LES. The turbine location is represented with the white rectangle.

**Figure 8.**Same as Figure 7 but obtained from the top-hat wake model.

**Figure 9.**Same as Figure 7 but obtained from the Gaussian-type wake model.

**Figure 10.**Uncertainty in the normalized velocity defect due to 10% change in the model coefficients in the top-hat (

**left**) and in the Gaussian (

**right**) wake models.

Cases | ${\mathit{z}}_{\mathit{h}}\left(\mathit{m}\right)$ | D (m) | H (m) | ${\mathit{\lambda}}_{\mathit{R}}$ | c (m) | ${\mathit{C}}_{\mathit{T}}$ | $\mathit{\xi}$ | ${\mathit{U}}_{\mathit{h}}$ (m/s) | ${\mathit{I}}_{\mathit{u}}(\%)$ |
---|---|---|---|---|---|---|---|---|---|

Case 1 | 40 | 26 | 24 | $3.8$ | $0.75$ | $0.65$ | $0.92$ | $7.0$ | $9.1$ |

Case 2 | 40 | 26 | 24 | $2.5$ | $0.75$ | $0.34$ | $0.92$ | $7.0$ | $9.1$ |

Case 3 | 40 | 26 | 48 | $3.8$ | $0.75$ | $0.64$ | $1.85$ | $7.0$ | $9.1$ |

Case 4 | 100 | 50 | 100 | $4.5$ | $1.5$ | $0.80$ | $2.0$ | $9.6$ | $8.3$ |

Case 5 | 6 | $1.2$ | $6.1$ | $2.2$ | $0.11$ | $0.47$ | $5.08$ | $10.9$ | $6.7$ |

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Abkar, M.
Theoretical Modeling of Vertical-Axis Wind Turbine Wakes. *Energies* **2019**, *12*, 10.
https://doi.org/10.3390/en12010010

**AMA Style**

Abkar M.
Theoretical Modeling of Vertical-Axis Wind Turbine Wakes. *Energies*. 2019; 12(1):10.
https://doi.org/10.3390/en12010010

**Chicago/Turabian Style**

Abkar, Mahdi.
2019. "Theoretical Modeling of Vertical-Axis Wind Turbine Wakes" *Energies* 12, no. 1: 10.
https://doi.org/10.3390/en12010010