# Validation of a Large-Eddy Simulation Approach for Prediction of the Ground Roughness Influence on Wind Turbine Wakes

^{*}

## Abstract

**:**

## 1. Background and Motivation

#### 1.1. Previous Work

#### 1.2. Goals and Structure of the Paper

## 2. Numerical Simulation Approach

#### 2.1. Flow Solver INCA

#### 2.2. Wall Modeling Methodology

#### 2.2.1. Standard Wall-Stress Model

#### 2.2.2. Assessment of the Standard Wall Model

#### 2.2.3. Grid Resolution Sensitivity

#### 2.2.4. Virtually Lifted-Wall Stress Model

#### 2.3. Actuator Line Modeling of the Rotor

#### 2.4. Blade Polar Identification

#### 2.5. Set-Up of the Full LES Simulation

#### 2.5.1. Boundary Conditions

#### 2.5.2. Parameters of Investigated Cases

## 3. Comparison of LES Results with Measurements

#### 3.1. Flow Evolution in the Empty Domain

#### 3.2. Rotor Performance

#### 3.3. Flow Field in the Wake Region

## 4. Summary and Conclusions

- In the fully coupled simulation approach, turbulent inflow is generated by an internal mapping approach (recycling technique) in an upstream segment of the computational domain with coarser grid resolution. Changes in the grid spacing significantly affect the axial evolution of turbulence intensities in case $\alpha =0.32$.
- The rough wall is approximated via a relation between instantaneous wall-shear stress and the tangential velocity at an off-wall position ${z}_{m}$. In accordance with [29], a better match with the logarithmic law of the wall is achieved when ${z}_{m}$ is chosen at least three cell layers away from the ground.
- For moderate roughness ($\alpha =0.16$), shear exponent, turbulence intensities and spectral densities of the approach flow match well with measurements in the lower half of the boundary layer.
- For high roughness ($\alpha =0.32$), a novel wall-model (virtually lifted wall) [14] was used: in this case, the match with the measured inflow is good at hub height, although not over the entire boundary layer.
- Erroneous values of turbulence intensities in wall-adjacent cell layers resulting from short-comings of the wall-stress model do not adversely affect the wake evolution.
- Actuator line (ALM) rotor modeling works best if the blade polar identification is based on an inverted blade element momentum theory in conjunction with 3D-RANS simulation of the blade geometry of the wind turbine model.
- In the near wake ($x=2R$), the mean flow matches well, whereas turbulence intensities in the LES do not reach the same maxima associated with the formation and break-down of tip vortices as the experimental data, probably due to limitations of the ALM method.
- In the far wake, $x>6R$, lateral profiles of mean flow, Reynolds shear stresses and turbulence intensities in a wall-parallel plane at hub height match better with measurements then profiles in a vertical plane through the wake center.
- An axial shift in the wake deficit $\Delta u$ is associated with overprediction of the wake depth. Nevertheless, the change rates $d\Delta u/dx$ are well reproduced. Whereas the influence of roughness on the vertical wake deflection is well captured, its impact on the wake width is larger in the LES than in the reference experiment.
- The remaining differences between simulation and measurements can not be tracked down to a single cause; they can be interpreted as a limit for the achievable accuracy with the current approach.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Non-dimensional mean axial velocity ${u}^{+}$ as function of normalized wall distance $z/\delta $ computed using the m-th grid point off the wall, indicated by colored dots as denoted in the legend. The black dashed straight line marks the logarithmic law for a rough wall with $\delta /{z}_{0}=4500$.

**Figure 2.**Reynolds stresses normalized by ${u}_{\tau}^{2}$: (

**a**) ${\overline{{u}^{\prime}{u}^{\prime}}}^{+}$, (

**b**) ${\overline{{w}^{\prime}{w}^{\prime}}}^{+}$, (

**c**) ${\overline{{v}^{\prime}{v}^{\prime}}}^{+}$ and (

**d**) ${\overline{{u}^{\prime}{w}^{\prime}}}^{+}$ as function of normalized wall distance $z/\delta $ for different off-wall position ${z}_{m}$.

**Figure 3.**Non-dimensional Reynolds stresses (

**a**) ${\overline{{u}^{\prime}{u}^{\prime}}}^{+}$, (

**b**) ${\overline{{w}^{\prime}{w}^{\prime}}}^{+}$, (

**c**) ${\overline{{v}^{\prime}{v}^{\prime}}}^{+}$ and (

**d**) ${\overline{{u}^{\prime}{w}^{\prime}}}^{+}$ as function of normalized wall distance $z/\delta $ for three different grid resolutions indicated in the legend.

**Figure 4.**Comparison of (

**a**–

**c**) thrust ${C}_{T}$ and (

**d**–

**f**) power coefficient ${C}_{P}$ as a function of tip speed ratio $\lambda $ obtained from measurement (EXP), 2D airfoil calculations (DSGN) and numerical simulation (RANS) for pitch angles ${\mathsf{\Theta}}_{p}={0}^{\circ},{1}^{\circ},{2}^{\circ}$.

**Figure 5.**Comparison of blade polars obtained from 2D airfoil calculations (DSGN), and numerical simulation (RANS): (

**a**–

**d**) lift ${C}_{l}$ and (

**e**–

**h**) drag ${C}_{d}$ coefficients as function of AoA $\theta $ at selected blade sections $\mu =r/R=0.25,0.5,0.75,0.9$.

**Figure 7.**(

**a**) Mean axial velocity $\overline{u}(x,{z}_{h})$ and (

**b**) turbulent kinetic energy $K(x,{z}_{h})$ as function of the axial distance to the rotor, both normalized to their respective values $\overline{u}({x}_{r},{z}_{h})$, $K({x}_{r},{z}_{h})$ at the rotor center along the entire domain without turbine. The dashed vertical lines indicate the borders between the regions and the legend denotes flow condition and grid resolution.

**Figure 8.**Vertical profiles at selected locations in the wake region of the empty domain for intermediate roughness ($\alpha =0.16$, top row) and large roughness ($\alpha =0.32$, bottom row): (

**a**) mean axial velocity $\overline{u}$; (

**,**e**b**) axial intensity $\overline{{u}^{\prime}{u}^{\prime}}$; (

**,**f**c**) Reynolds shear stress $\overline{{u}^{\prime}{w}^{\prime}}$; (

**,**g**d**) turbulent kinetic energy K; all normalized with the reference speed ${\overline{u}}_{h}=\overline{u}({x}_{r},{z}_{h})$.

**,**h**Figure 9.**Normalized energy spectral density of the three velocity components at the rotor center ${S}_{{u}_{i}{u}_{j}}\left({z}_{h}\right)/\overline{{u}_{i}^{\prime}{u}_{j}^{\prime}}\left({z}_{h}\right)$ with $i\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}j$, within simulated boundary layers and experimental SBL configuration: (

**a**,

**d**) $u$-component; (

**b**,

**e**) $v$-component and (

**c**,

**f**) $w$-component. The legend denotes flow conditions, dashed lines denote the slope of the Kolmogorov cascade.

**Figure 10.**Comparison of measurements and numerical results for (

**a**) power ${C}_{P}$ and (

**b**) thrust coefficient ${C}_{T}$ as function of yaw angle $\gamma $. The legend states the flow conditions.

**Figure 11.**Profiles of the normalized mean axial velocity component $\overline{u}/{\overline{u}}_{bl}\left({z}_{h}\right)$ in the (

**a**) mid-vertical and (

**b**) mid-horizontal plane at axial positions $x/R=$ 2, 6, 10, 14, 18. Symbols denote measurements, whereas lines correspond to LES results. The legend denotes the case. The origin of local coordinates in the upper subfigure is located at $x=0.8$ and in the lower subfigure at $x=1$.

**Figure 12.**Profiles of the normalized Reynolds shear stress (

**a**) $\overline{{u}^{\prime}{w}^{\prime}}/{\overline{u}}_{h}^{2}$ in the mid-vertical and (

**b**) $\overline{{u}^{\prime}{v}^{\prime}}/{\overline{u}}_{h}^{2}$ the mid-horizontal plane at axial positions $x/R=$ 2, 6, 10, 14, 18. Symbols denote measurements, whereas lines correspond to LES results. The legend denotes the case.

**Figure 13.**Profiles of the turbulence intensities in axial direction $\overline{{u}^{\prime}{u}^{\prime}}$ (top), in lateral direction $\overline{{v}^{\prime}{v}^{\prime}}$ and in vertical (wall-normal) direction $\overline{{w}^{\prime}{w}^{\prime}}$ (bottom), normalized by ${\overline{u}}_{h}^{2}$ in (

**a**) mid-vertical and (

**,**c**,**e**b**) mid-horizontal planes at axial positions $x/R=$ 2, 6, 10, 14, 18. Circles denote measurements, whereas lines correspond to LES results. The color code denotes the shear exponent $\alpha =0.16$ (blue) and $\alpha =0.32$ (red).

**,**d**,**f**Figure 14.**Profiles of the normalized turbulent kinetic energy $K/{\overline{u}}_{h}^{2}$ in the (

**a**) mid-vertical and (

**b**) mid-horizontal plane at axial positions $x/R=$ 2, 6, 10, 14, 18. Symbols denote measurements, whereas lines correspond to LES results. The legend denotes the case.

**Figure 15.**Wake parameters of the model wind turbine operating at design tip-speed ratio above moderate (SBL) and high (RBL) ground roughness: (

**a**) centerline velocity deficit $\Delta {u}_{C}$, (

**b**) lateral ${\sigma}_{y}$ and (

**c**) vertical half width ${\sigma}_{z}$, and (

**d**) vertical centerline position ${z}_{C}$. The legend indicates the flow condition.

**Table 1.**Characteristics of the turbulent boundary layers in the reference experiment [15].

Name | Acronym | ${\mathit{z}}_{0}$ [mm] | $\mathit{\alpha}$ | ${\mathit{u}}_{\mathit{\tau}}$ [m/s] | ${\mathit{I}}_{\mathit{u}}\left({\mathit{z}}_{\mathit{h}}\right)$ | ${\mathit{z}}_{0}^{+}={\mathit{u}}_{\mathit{\tau}}{\mathit{z}}_{0}/\mathit{\nu}$ |
---|---|---|---|---|---|---|

SBL | sf016 | 0.51 | 0.16 | 0.65 | 8.3% | 19 |

RBL | sf032 | 5.06 | 0.32 | 0.89 | 13.8% | 190 |

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**MDPI and ACS Style**

Stein, V.P.; Kaltenbach, H.-J.
Validation of a Large-Eddy Simulation Approach for Prediction of the Ground Roughness Influence on Wind Turbine Wakes. *Energies* **2022**, *15*, 2579.
https://doi.org/10.3390/en15072579

**AMA Style**

Stein VP, Kaltenbach H-J.
Validation of a Large-Eddy Simulation Approach for Prediction of the Ground Roughness Influence on Wind Turbine Wakes. *Energies*. 2022; 15(7):2579.
https://doi.org/10.3390/en15072579

**Chicago/Turabian Style**

Stein, Victor P., and Hans-Jakob Kaltenbach.
2022. "Validation of a Large-Eddy Simulation Approach for Prediction of the Ground Roughness Influence on Wind Turbine Wakes" *Energies* 15, no. 7: 2579.
https://doi.org/10.3390/en15072579