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Data Driven Modal Decomposition of the Wake behind an NREL-5MW Wind Turbine^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

**u**$={(u,v,w)}^{T}$, and pressure, p, derived from the Navier–Stokes equations for incompressible flows read as follows:

#### 2.1. Proper Orthogonal Decomposition

#### 2.2. Sparsity-Promoting Dynamic Mode Decomposition

## 3. Simulation Setup

## 4. Modal Decomposition of the Wake

#### 4.1. POD Results

#### 4.2. SP-DMD Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Streamwise velocity contours of the snapshots’ ensemble mean. (

**a**) $x-y$ plane at $z=1.5$. (

**b**) $x-z$ plane at $y=0$. (

**c**) $z-y$ plane at $x=4$.

**Figure 3.**(

**a**) Singular values distribution with a close up for the first 20 modes. (

**b**) Cumulative turbulent kinetic energy distribution versus the fraction of POD modes.

**Figure 4.**Streamwise velocity iso-surfaces of the most energetic POD modes (red for $u=0.001$, blue for $u=-0.001$).

**Figure 5.**Vertical (

**left**) and transverse (

**right**) velocity iso-surfaces of the most energetic POD modes (magenta/green for $v,w=0.0008$, cyan/yellow for $v,w=-0.0008$).

**Figure 6.**Fourier transform of the time coefficients ${a}_{k}\left(t\right)$ associated with the most energetic POD modes shown in Figure 4.

**Figure 7.**The sparsity level $\mathbf{card}\left(\alpha \right)$ (

**a**) and the optimal performance loss $\%{\mathsf{\Pi}}_{loss}$ (

**b**) for different values of the sparsity parameter $\gamma $.

**Figure 8.**Eigenvalues resulting from the standard DMD algorithm (red crosses) and the sparsity-promoting algorithm (black circles). The right panel shows the logarithmic mapping of the eigenvalues, $\omega =-{\displaystyle \frac{log\left(\mu \right)}{i\Delta t}}$, where $\Delta t$ is the temporal separation between two consecutive snapshots and i the imaginary unit.

**Figure 9.**Streamwise velocity iso-surfaces (red for $u=0.001$, blue for $u=-0.001$ values) of the real part of the eight dynamic modes’ pairs selected by the sparsity-promoting algorithm, ordered according to their amplitude $\left|\alpha \right|$.

**Figure 10.**Vertical (

**left**) and transverse (

**right**) velocity iso-surfaces (magenta/green for $v,w=0.0008$, cyan/yellow for $v,w=-0.0008$ values) of the real part of the most relevant three dynamic modes’ pairs selected by the sparsity-promoting algorithm, ordered according to their amplitude $\left|\alpha \right|$.

**Table 1.**Frequencies and amplitudes of the selected complex conjugate dynamic modes’ pairs, computed with the standard and sparsity-promoting DMD.

$\mathfrak{R}\left(\mathit{\omega}\right)$-POD | $\mathfrak{R}\left(\mathit{\omega}\right)$-DMD | $\left|\mathit{\alpha}\right|$ (std. DMD) | $\left|\mathit{\alpha}\right|$ (SP-DMD) | |
---|---|---|---|---|

Pair 1 | 3.26 | 42.0 | 14.34 | 14.86 |

Pair 2 | 42.0 | 5.20 | 11.39 | 9.55 |

Pair 3 | 4.58 | 2.13 | 8.54 | 8.53 |

Pair 4 | 2.23 | 3.92 | 9.26 | 7.89 |

Pair 5 | 2.44 | 2.30 | 7.50 | 7.80 |

Pair 6 | 3.76 | 2.96 | 8.26 | 7.48 |

Pair 7 | 2.64 | 4.25 | 6.39 | 6.02 |

Pair 8 | 1.13 | 3.58 | 4.22 | 4.06 |

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**MDPI and ACS Style**

Cherubini, S.; De Cillis, G.; Semeraro, O.; Leonardi, S.; De Palma, P.
Data Driven Modal Decomposition of the Wake behind an NREL-5MW Wind Turbine. *Int. J. Turbomach. Propuls. Power* **2021**, *6*, 44.
https://doi.org/10.3390/ijtpp6040044

**AMA Style**

Cherubini S, De Cillis G, Semeraro O, Leonardi S, De Palma P.
Data Driven Modal Decomposition of the Wake behind an NREL-5MW Wind Turbine. *International Journal of Turbomachinery, Propulsion and Power*. 2021; 6(4):44.
https://doi.org/10.3390/ijtpp6040044

**Chicago/Turabian Style**

Cherubini, Stefania, Giovanni De Cillis, Onofrio Semeraro, Stefano Leonardi, and Pietro De Palma.
2021. "Data Driven Modal Decomposition of the Wake behind an NREL-5MW Wind Turbine" *International Journal of Turbomachinery, Propulsion and Power* 6, no. 4: 44.
https://doi.org/10.3390/ijtpp6040044