# Interaction of Wind Turbine Wakes under Various Atmospheric Conditions

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. LES Framework

_{0}is the constant density. The mean pressure term is ${p}_{0}\left(x,y\right)$ whose spatial gradient acts to drive the flow convection. The last term, ${\rho}_{0}gz$, represents the hydrostatic pressure. The deviatoric part of the fluid stress tensor is ${\tau}_{ij}^{D}={\tau}_{ij}-{\tau}_{kk}{\delta}_{ij}/3$ where ${\delta}_{ij}$ is the Kronecker delta. The subgrid-scale (SGS) stresses are included in the τ

_{ij}term. The SGS flux was computed using the Smagorinsky [25] model with the Smagorinsky constant of 0.13. Using the local cell dimensions, $\Delta x$, $\Delta y$ and $\Delta z$, the filter length scale was based on $\Delta ={\left(\Delta x\Delta y\Delta z\right)}^{1/3}$. The latitude ($\varphi $) of 45° and the planetary rotation rate ($\omega )$ of 7.27 × 10

^{−5}rad/s served as inputs to the Coriolis force defined as f = 2$\omega $[0, cos($\varphi $), sin($\varphi $)]. The buoyancy effect was calculated using the Boussinesq approximation, where g is the gravity, $\theta $ is the resolved potential temperature, ${\theta}_{0}$ is the reference temperature taken to be 300 K. The aerodynamic force computed by the actuator line method is denoted as F

_{i}. The temperature flux, q

_{j}, used in the transport equation for the resolved potential temperature equation (Equation (3)) is defined as below:

#### 2.2. Atmospheric Boundary Layer

_{i}= 750 m, is defined as the boundary layer height. The horizontal average wind speed of 8 m/s was fixed at the hub-height, Z = 90 m. The streamwise mean velocity profiles from the NLABL (

**N**eutrally stable case,

**L**ow surface roughness

**A**tmospheric

**B**oundary

**L**ayer, Table 1) case with 5 m and 10 m resolutions are shown in Figure 1a. For both resolutions, reasonable agreement with the log-law can be seen. The large peaks at Z/Z

_{0}~ 7 × 10

^{5}were caused by the temperature capping inversion in which the velocity profile inflected towards higher speeds above the boundary layer height. The resolution effects on the boundary layer flow were more pronounced in the velocity spectra as shown in Figure 1b. While the inertial range that matches the −5/3 power law curve was similar in both cases, higher wave lengths were better resolved for the 5 m case before smaller turbulence scales dissipate via the sub-grid scale closure. Therefore, the 5 m grid resolution was chosen for the background mesh as well as the precursor inflow boundary condition. Total computational resources of approximately one million computational hours were required for this study.

#### 2.3. Computational Domain

#### 2.4. Wind Turbine Model

_{i}in Equation (2) in the atmospheric flow solver, were fully coupled with FAST which employed a combined modal and multi-body dynamics formulation for the flexible blades and the tower, assuming small deflections. A typical blade tip deflection in the present work is shown in Figure 3 where the maximum deflection of 3.5 m was observed, which corresponds to 2.8% of the rotor diameter. This length would only incur, at most, 1.6° deflection angle at the blade root. The time step size was restricted for the blade tip to traverse no more than a grid cell per time-step (Δt = 0.02 s). The actuator line representation consisted of a two-way coupling in which the flow information was used as an input to compute the aerodynamic forces which fed back to the flow with new positions. The peak magnitude of the forces at the actuator points and the projection width was controlled by the Gaussian width [12] which was set as equal to twice the grid cell length. The resolution of 5 m used in the precursor domain (Figure 1b) to generate the turbulent inflow conditions captured a wider inertial range that enabled us to resolve higher wave lengths in comparison to the precursor domain with a 10 m resolution used in a previous study by Churchfield et al. [32]. His work demonstrated that the power predictions for individual turbines, using a similar grid nesting procedure and Gaussian width, yielded good agreement with the field measurements. In the present study, the turbulent kinetic energy (TKE) was measured at various locations downstream of the turbine using a coarse test grid (2$\Delta x$, 2$\Delta y,$ 2$\Delta z$) and a maximum of 2.3% difference was observed when compared with the TKE results obtained from the original high-resolution grid (Figure 2b). Therefore, no further grid independence study was performed in this investigation.

## 3. Results

#### 3.1. Velocity and Turbulent Kinetic Energy

#### 3.2. Power Generation

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) Mean streamwise velocity profile and (

**b**) energy spectra for the NLABL (refer to Table 1) conditions.

**Figure 2.**Schematic of the computational domain: (

**a**) baseline 5 m resolution mesh, and with (

**b**) local refined mesh at 2.5 m resolution near the turbines.

**Figure 4.**Top view of the instantaneous (

**a**,

**c**,

**e**,

**g**) and time-averaged (

**b**,

**d**,

**f**,

**h**) streamwise velocities for NLABL, NHABL, ULABL, and UHABL (refer to Table 1) at hub-height (Z = 90 m).

**Figure 5.**Top view of the instantaneous (

**a**,

**c**,

**e**,

**g**) and time-averaged (

**b**,

**d**,

**f**,

**h**) vertical velocities for NLABL, NHABL, ULABL, and UHABL (refer to Table 1) at hub-height (Z = 90 m).

**Figure 6.**Streamwise view of the instantaneous and time-averaged streamwise velocities for NLABL (

**a**,

**b**), NHABL (

**c**,

**d**), ULABL (

**e**,

**f**), and UHABL (

**g**,

**h**) along the centerline. Refer to Table 1 for acronyms.

**Figure 7.**Streamwise cut of the turbulent kinetic energy (TKE) contours for (

**a**) NLABL, (

**b**) NHABL, (

**c**) ULABL, and (

**d**) UHABL along the centerline. Refer to Table 1 for acronyms.

**Figure 10.**Reynolds shear stress for (

**a**) horizontal profiles of <u′v′> and (

**b**) vertical profiles of <u′w′> normalized by the mean velocity at hub-height (8 m/s). Turbines positions are indicated with black dashed-lines.

**Figure 11.**Time series of the power signals for upstream (WT1) and downstream (WT2) turbines: (

**a**) NLABL, (

**b**) NHABL, (

**c**) ULABL, and (

**d**) UHABL. Refer to Table 1 for acronyms.

**Figure 12.**(

**a**) Power root mean square (RMS) and (

**b**) time-averaged power for NLABL, NHABL, ULABL, and UHABL. Refer to Table 1.

**Table 1.**Matrix of atmospheric boundary layer (ABL) test cases depending on surface roughness and atmospheric stability.

Atmospheric Stability | Surface Roughness | |
---|---|---|

Low | High | |

Neutral | NLABL | NHABL |

Unstable | ULABL | UHABL |

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**MDPI and ACS Style**

Lee, S.; Vorobieff, P.; Poroseva, S. Interaction of Wind Turbine Wakes under Various Atmospheric Conditions. *Energies* **2018**, *11*, 1442.
https://doi.org/10.3390/en11061442

**AMA Style**

Lee S, Vorobieff P, Poroseva S. Interaction of Wind Turbine Wakes under Various Atmospheric Conditions. *Energies*. 2018; 11(6):1442.
https://doi.org/10.3390/en11061442

**Chicago/Turabian Style**

Lee, Sang, Peter Vorobieff, and Svetlana Poroseva. 2018. "Interaction of Wind Turbine Wakes under Various Atmospheric Conditions" *Energies* 11, no. 6: 1442.
https://doi.org/10.3390/en11061442