# A Level Set-Based Actuator Disc Model for Turbine Realignment in Wind Farm Simulation: Meshing, Convergence and Applications

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## Abstract

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## 1. Introduction

## 2. Wind Farm CFD Model and Outline of the Simulation Framework

#### 2.1. Wind Farm CFD Model

#### 2.2. Outline of the Simulation Framework

## 3. Level Set and Mesh Adaptation for Actuator Disc Modeling of a Wind Farm

#### 3.1. Level Set Representation of an Actuator Disc Model of a Wind Farm

#### 3.2. Actuator Disc Discretization Using a Metric-Driven Adaptation Process

Algorithm 1 Generation of the a mesh adapted to the actuator disc representation of a wind farm. |

Input:
Target domain $\mathsf{\Omega}$, Actuator discs ${\left\{{\mathsf{\Omega}}_{i}\right\}}_{i=1}^{{n}_{t}}$ Number of desired nodes N, Maximum edge length ${h}_{max}$, Minimum edge length ${h}_{min}$Output:
Mesh adapted to discs M1: function GenerateDiscMesh($\mathsf{\Omega}$, ${\left\{{\mathsf{\Omega}}_{i}\right\}}_{i=1}^{{n}_{t}}$, N, ${h}_{max}$, ${h}_{min}$)2: $i\leftarrow 0$ 3: ${M}_{0}$← GenerateInitialMesh( $\mathsf{\Omega}$, ${\left\{{\mathsf{\Omega}}_{i}\right\}}_{i=1}^{{n}_{t}}$) 4: isConverged ← false5: while not isConverged do6: f← SetInterfaceFunction( $\mathsf{\Omega}$, ${\left\{{\mathsf{\Omega}}_{i}\right\}}_{i=1}^{{n}_{t}}$ ) 7: ${\mathcal{M}}_{\mathbf{f}}^{i}\leftarrow $ ComputeMetric( f, N, ${h}_{max}$, ${h}_{min}$ ) 8: ${M}_{i+1}$←AdaptMeshToMetric( ${M}_{i}$, ${\mathcal{M}}_{\mathbf{f}}^{i}$ ) 9: isConverged ← (${M}_{i}\simeq {M}_{i+1}$&$i<i{t}_{max}$) 10: $i\leftarrow i+1$ 11: end while12: $M\leftarrow {M}_{i}$ 13: return M14: end function |

#### 3.3. Selection of the Interface Parameter and Convergence to the Disc

## 4. Convergence Analysis of Standard and Level-Set-Based Approaches

## 5. Wind Farm Simulation Featuring Solution Adaptation and Turbine Realignment

#### 5.1. Adaptation to the Solution

Algorithm 2 Adaptive simulation of a wind farm. |

Input:
Target domain $\mathsf{\Omega}$, Actuator discs ${\left\{{\mathsf{\Omega}}_{k}\right\}}_{k=1}^{{n}_{t}}$, Number of desired nodes N, Maximum edge length ${h}_{max}$, Minimum edge length ${h}_{min}$Output:
Adapted mesh M, Velocity $\mathbf{u}$, Pressure p1: function AdaptiveWindFarmSimulation( $\mathsf{\Omega}$, ${\left\{{\mathsf{\Omega}}_{k}\right\}}_{k=1}^{{n}_{t}}$, N, ${h}_{max}$, ${h}_{min}$ )2: $i\leftarrow 0$ 3: ${M}_{0}$←GenerateDiscMesh( $\mathsf{\Omega}$, ${\left\{{\mathsf{\Omega}}_{k}\right\}}_{k=1}^{{n}_{t}}$, N, ${h}_{max}$, ${h}_{min}$ ) 4: isConverged ← false5: while not isConverged do6: $\mathbf{u},p$← SolveWindModel( ${M}_{i}$ ) 7: $\mathcal{M}\leftarrow $ ComputeTargetMetric( ${M}_{i}$, $\mathbf{u}$, p, ${\left\{{\mathsf{\Omega}}_{k}\right\}}_{k=1}^{{n}_{t}}$, N, ${h}_{max}$, ${h}_{min}$ ) 8: ${M}_{i+1}$← AdaptMeshToMetric( ${M}_{i}$, $\mathcal{M}$ ) 9: isConverged ← ( ${M}_{i}\simeq {M}_{i+1}$&$i<i{t}_{max}$ ) 10: $i\leftarrow i+1$ 11: end while12: $M\leftarrow {M}_{i}$ 13: return M, $\mathbf{u}$, p14: end function |

#### 5.2. Adaptation and Realignment with the Flow

Algorithm 3 Adaptive simulation of a wind farm with turbine realignment. |

Input:
Target domain $\mathsf{\Omega}$, Actuator discs ${\left\{{\mathsf{\Omega}}_{k}\right\}}_{k=1}^{{n}_{t}}$, Number of desired nodes N, Maximum edge length ${h}_{max}$, Minimum edge length ${h}_{min}$Output:
Adapted mesh M, Velocity $\mathbf{u}$, Pressure p1: function WindFarmSimulationWithRealignment($\mathsf{\Omega}$, ${\left\{{\mathsf{\Omega}}_{k}\right\}}_{k=1}^{{n}_{t}}$, N, ${h}_{max}$, ${h}_{min}$)2: $j\leftarrow 0$ 3: ${M}_{0}$←GenerateDiscMesh($\mathsf{\Omega}$, ${\left\{{\mathsf{\Omega}}_{k}\right\}}_{k=1}^{{n}_{t}}$, N, ${h}_{max}$, ${h}_{min}$) 4: isConvergedDiscModel ← false5: while not isConvergedDiscModel do6: $i\leftarrow 0$ 7: isConvergedCFD ← false8: while not isConvergedCFD do9: $\mathbf{u},p$← SolveWindModel(${M}_{i}$) 10: $\mathcal{M}\leftarrow $ ComputeTargetMetric(${M}_{i}$, $\mathbf{u}$, p, ${\left\{{\mathsf{\Omega}}_{k}\right\}}_{k=1}^{{n}_{t}}$, N, ${h}_{max}$, ${h}_{min}$) 11: ${M}_{i+1}$← AdaptMeshToMetric(${M}_{i}$,$\mathcal{M}$) 12: isConvergedCFD ← (${M}_{i}\simeq {M}_{i+1}$&$i<i{t}_{max}^{CFD}$) 13: $i\leftarrow i+1$ 14: end while15: ${\left\{{\mathsf{\Omega}}_{k}^{new}\right\}}_{k=1}^{{n}_{t}}$ ← UpdateDiscModel(${\left\{{\mathsf{\Omega}}_{k}\right\}}_{k=1}^{{n}_{t}}$,$\mathbf{u}$) 16: isConvergedDiscModel ←$\left({\left\{{\mathsf{\Omega}}_{k}^{new}\right\}}_{k=1}^{{n}_{t}}\phantom{\rule{4pt}{0ex}}\simeq \phantom{\rule{4pt}{0ex}}{\left\{{\mathsf{\Omega}}_{k}\right\}}_{k=1}^{{n}_{t}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\&\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4pt}{0ex}}j<i{t}_{max}^{disc}\right)$ 17: ${\left\{{\mathsf{\Omega}}_{k}\right\}}_{k=1}^{{n}_{t}}$←${\left\{{\mathsf{\Omega}}_{k}^{new}\right\}}_{k=1}^{{n}_{t}}$ 18: $j\leftarrow j+1$ 19: end while20: $M\leftarrow {M}_{i}$ 21: return M, $\mathbf{u}$, p22: end function |

## 6. Results

#### 6.1. Convergence Analysis of the Adaptive Simulation Process

#### 6.2. Simulation of Four Neighboring Wind Farms: 219 Turbines

#### 6.3. Validation of the Turbine Realignment Process

#### 6.4. Actuator Disc Realignment in a Wind Farm Featuring 115 Turbines

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

CAD | Computer-Aided Design |

STL | STereoLithography |

PLC | Piecewise Linear Complex |

ALE | Arbitrary Lagrangian–Eulerian |

FEM | Finite Element Method |

FVM | Finite Volume Method |

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**Figure 1.**Wind speedup of the model for (

**a**) a two turbine configuration and (

**b**) the Sisante wind farm (115 turbines), illustrating in white the actuator disc boundaries.

**Figure 3.**Level set function for a test case featuring one turbine. The range of the illustrated level set field is $[0,700]$ m.

**Figure 4.**(

**a**) Interface function for a test case featuring one actuator disc, and (

**b**) closeup to the top of the disc. The range of the interface function is [0, 1].

**Figure 5.**Wind farm test case featuring 219 turbines: (

**a**) level set, (

**b**) level set in logarithmic scale.

**Figure 6.**Interface function for a wind farm test case featuring 219 turbines: (

**a**) overview, and (

**b**) closeup to four turbines. The range of the interface function is [0, 1].

**Figure 7.**(

**a**) Overview and (

**b**) closeup of the adapted disc mesh for the case of a single turbine, and (

**c**) adapted disc mesh for four wind farms on the North sea.

**Figure 9.**Target cost function with respect to E for (

**a**) one actuator disc, and (

**b**) set of four wind farms on the North Sea (219 turbines).

**Figure 10.**${\mathcal{L}}^{2}\u2014$error on the actuator disc area vs. size indicator (logarithmic scale): (

**a**) one turbine and (

**b**) set of 4 wind farms on the North Sea.

**Figure 12.**${\mathcal{L}}^{2}-$error vs. size indicator (logarithmic scale) of the boundary tight and level set approaches.

**Figure 13.**Boundary tight vs. level set meshes. On the top half we have the boundary tight generated mesh, whereas in the top bottom we have the level set mesh.

**Figure 14.**${\mathcal{L}}^{2}-$error vs. size indicator (logarithmic scale) of the conformal, level set and adaptive level set approaches.

**Figure 15.**Adaptive simulation process for a set of 4 wind farms on the North Sea (219 turbines): (

**a**) initial, (

**b**) intermediate and (

**c**) final meshes, with (

**d**–

**f**) the corresponding closeups. The meshes are colored according to the wind speedup with respect to the velocity upstream of the wind farm.

**Figure 16.**Final computed speedup for the adaptive simulation process for a set of four wind farms on the North Sea (219 turbines).

**Figure 17.**Turbine re-alignment simulation. Adapted meshes for the: (

**a**) initial configuration, (

**b**) intermediate correction, (

**c**) final configuration. The meshes are colored according to the wind speedup with respect to the velocity at the hub of the turbine.

**Figure 18.**Actuator disc realignment process on the Sisante wind farm (115 turbines): (

**a**) initial configuration and (

**b**) final realigned configuration. The mesh is colored with respect to the wind speedup, and streamlines are also displayed.

**Figure 19.**Angle between the inflow direction and the computed flow direction at the Sisante wind farm (115 turbines).

**Figure 20.**Closeup of the actuator disc realignment process on the Sisante wind farm (115 turbines) in the areas with higher realignment: (

**a**,

**b**) initial configuration and (

**c**,

**d**) final realigned configuration. The mesh is colored with respect to the wind speedup, and streamlines are also displayed.

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## Share and Cite

**MDPI and ACS Style**

Gargallo-Peiró, A.; Revilla, G.; Avila, M.; Houzeaux, G.
A Level Set-Based Actuator Disc Model for Turbine Realignment in Wind Farm Simulation: Meshing, Convergence and Applications. *Energies* **2022**, *15*, 8877.
https://doi.org/10.3390/en15238877

**AMA Style**

Gargallo-Peiró A, Revilla G, Avila M, Houzeaux G.
A Level Set-Based Actuator Disc Model for Turbine Realignment in Wind Farm Simulation: Meshing, Convergence and Applications. *Energies*. 2022; 15(23):8877.
https://doi.org/10.3390/en15238877

**Chicago/Turabian Style**

Gargallo-Peiró, Abel, Gonzalo Revilla, Matias Avila, and Guillaume Houzeaux.
2022. "A Level Set-Based Actuator Disc Model for Turbine Realignment in Wind Farm Simulation: Meshing, Convergence and Applications" *Energies* 15, no. 23: 8877.
https://doi.org/10.3390/en15238877