# A Reduced Order Model to Predict Transient Flows around Straight Bladed Vertical Axis Wind Turbines

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. High Order Numerical Solver

#### 2.2. A Reduced Order Model for Data Prediction

#### 2.2.1. The Algorithm of HODMD

**Dimension reduction via SVD.**At this step, the spatial dimension J of the data collected is reduced to N linearly independent vectors using a singular value decomposition (SVD) [28]. By doing so, it is possible to clean the noise from the signal and to remove spatial redundancies. SVD is applied to the snapshots matrix (2):$${\mathit{V}}_{1}^{K}\simeq \mathit{U}\phantom{\rule{0.166667em}{0ex}}\Sigma \phantom{\rule{0.166667em}{0ex}}{\mathit{T}}^{\top},$$Then, the reduced snapshots matrix of dimension $N\times K$ can be written as$$\begin{array}{c}{\widehat{\mathit{V}}}_{1}^{K}=\Sigma \phantom{\rule{0.166667em}{0ex}}{\mathit{T}}^{\top},\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}{\mathit{V}}_{1}^{K}=\mathit{U}{\widehat{\mathit{V}}}_{1}^{K}.\hfill \end{array}$$The standard SVD-error, which determines the number of N SVD modes retained, is estimated for a certain tolerance ${\epsilon}_{1}$ (set by the user) as$$\frac{{\sigma}_{N+1}^{2}+\cdots +{\sigma}_{K}}{{\sigma}_{1}^{2}+\cdots +{\sigma}_{K}}\le {\epsilon}_{1}.$$**The DMD-d approximation.**The reduced snapshot matrix is used to construct the following matrix according to the higher order Koopman assumption$${\widehat{\mathit{V}}}_{d+1}^{K}\simeq {\widehat{\mathit{R}}}_{1}{\widehat{\mathit{V}}}_{1}^{K-d}+{\widehat{\mathit{R}}}_{2}{\widehat{\mathit{V}}}_{2}^{K-d+1}+\dots +{\widehat{\mathit{R}}}_{d}{\widehat{\mathit{V}}}_{d}^{K-1},$$$${a}_{m}/{a}_{1}<{\epsilon}_{2},\phantom{\rule{1.em}{0ex}}m=1,\dots ,M.$$The selection of the modes will be replaced by the selection criterion method [27] presented below for the construction of the HODMD-based ROM (expansion (1)).Once the DMD expansion (1) is approximated, the root mean square error (RMSE) is used to quantify the difference between the original data and the approximated solution (1), following$$RMSE\sim \frac{\left|\right|{\mathit{V}}_{original}-{\mathit{V}}_{DMD}{\left|\right|}_{F}}{\left|\right|{\mathit{V}}_{original}{\left|\right|}_{F}},$$

#### 2.2.2. Criterion Selection Method

## 3. Numerical Results

#### 3.1. Numerical Simulations for a One Bladed Vertical Axis Turbines

#### 3.2. Reduced Order Model for a One Bladed Vertical Axis Turbine

#### 3.3. Extensions for Three Bladed Vertical Axis Turbine

^{−2}rad/s), the errors induced by these modes in the construction of the ROM are negligible (as shown below). It is noticeable that the third mode retained by HODMDc represents the fourth harmonic of the fundamental frequency ${\omega}_{1}$. The last peculiarity was already observed in the one blade turbine case.

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

## Abbreviations

ROM | Reduced Order Model |

DMD | Dynamic Mode Decomposition |

HODMD | High Order Dynamic Mode Decomposition |

DG | Discontinuous Galerkin |

LES | Large Eddy Simulation |

uRANS | unsteady Reynolds Averaged Navier–Stokes |

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**Figure 1.**Vertical axis turbine flow fields: (

**a.1**) high order Discontinuous Galerkin mesh with $P=3$ (high order polynomial nodes are shown in red); (

**b.1**) low order mesh; (

**a.2**) high order DG numerical solution (LES) and (

**b.2**) low order numerical solution (uRANS). Contours show streamwise velocity (${u}_{x}$) with 10 levels from 0 to 0.2.

**Figure 2.**Normal force coefficient for a vertical axis turbine: (

**a**) 3D high order DG simulation (LES turbulence model); (

**b**) low order fluent simulation (k-omega SST turbulence model. Both compared to experimental data [39] for four turbine revolutions.

**Figure 3.**Simulations of vertical axis turbine: Flow velocity components at lines: (

**a.1**) streamwise velocity ${u}_{x}$ at $y=-2$ m, (

**a.2**) normal velocity ${u}_{y}$ at $y=-2$ m; (

**b.1**) streamwise velocity ${u}_{x}$ at $y=-4$ m, (

**b.2**) normal velocity ${u}_{y}$ at $y=-4$ m. We include results from the 3D high order DG simulation (LES turbulence model) and from the low order fluent simulation k-omega SST turbulence model.

**Figure 4.**One-blade turbine case: temporal evolution of the streamwise (

**top**) and normal (

**bottom**) velocity fields, ${u}_{x}$ and ${u}_{x}$ for points $P({x}_{1},{y}_{1})=(0,-4)$ (

**left**) and $P({x}_{1},{y}_{1})=(2.8,-2)$ (

**right**) (see the computational domain in Figure 1).

**Figure 5.**One-blade turbine case: growth rates (

**left**) and amplitudes (

**right**) vs. frequencies of the DMD modes. Circles represent the modes selected by HODMDc for constructing the HODMD-based Reduced Order Model (ROM) and crosses represent the transient modes, omitted in the DMD expansion (1).

**Figure 6.**One-blade wind turbine case: streamwise (

**left**) and normal (

**right**) velocities, ${u}_{x}$ and ${u}_{y}$, of the instantaneous flow field at time instant 240 s. Two-dimensional $x-y$ cross section at z $=0.9375$ m.

**Figure 7.**One-blade wind turbine case: temporal evolution of streamwise (

**left**) and normal (

**right**) velocities in $y=-4$ m at time instant 240 s. From top to bottom: HODMD-based ROM constructed using M = 3, 5 and 21 DMD modes. Continuous black line and blue circles represent the original and the predicted solution, respectively.

**Figure 9.**Counterpart of Figure 4 in the three-blades turbine in points $P({x}_{1},{y}_{1})=(0,-4)$ (

**left**) and $P({x}_{1},{y}_{1})=(2\sqrt{2},2\sqrt{2})$ (

**right**).

**Figure 10.**Counterpart of Figure 5 in the three-blades turbine applying HODMD in 196 snapshots starting at time $80.1$ s.

**Figure 11.**Counterpart of Figure 6 in the three-blades turbine applying HODMD in 196 snapshots starting at time $80.1$ s.

**Figure 12.**Counterpart of Figure 7 in the three-blades turbine applying HODMD in 196 snapshots starting at time $80.1$ s. Data extracted at $y=2\sqrt{2}$ m.

**Figure 13.**Counterpart of Figure 7 in the three-blades turbine applying HODMD in 196 snapshots starting at time $80.1$ s. Data extracted at $y=2\sqrt{2}$ m.

**Figure 14.**Counterpart of Figure 5 the three-blades turbine applying HODMD in 145 snapshots starting at time $25.604$ s.

**Figure 15.**Counterpart of Figure 6 in the three-blades turbine applying HODMD in 145 snapshots starting at time $25.604$ s.

**Figure 16.**Counterpart of Figure 7 in the three-blades turbine applying HODMD in 145 snapshots starting at time $25.604$ s. Data extracted at $y=-4$ m.

**Figure 17.**Counterpart of Figure 7 in the three-blades turbine applying HODMD in 145 snapshots starting at time $25.604$ s. Data extracted at $y=2\sqrt{2}$ m.

**Table 1.**Numerical details of the Navier–Stokes computations for a one bladed rotating turbine. The out-of-plane length (marked *) for the high order mesh is discretised using 64 Fourier planes. The high order mesh (marked **) includes the number of elements and the polynomial order P = 3 and 64 Fourier planes for the 3D mesh. For the low order model (marked ***), almost identical results were obtained with a finer mesh of 31,584 elements.

High Order Solver | Low Order Fluent | |
---|---|---|

Blade/Mesh movement | Sliding mesh | Sliding mesh |

Numerical scheme | 3D DG-Fourier | 2D Finite Volume |

Out-of-plane length | 0.5 m * | 0 m |

Turbulence model | Large Eddy Simulation | k-omega SST (uRANS) |

Numerical details | polynomial order P = 3 | Second order Upwind |

Order of accuracy | 3 | 2 |

Mesh size | 1.1 millions ** | 7896 *** |

Time advancement | 2nd order semi-implicit | 1st order implicit |

Time step | 0.001 s | 0.2 s |

**Table 2.**Vertical axis turbine conditions. All in international metric system. $TSR=\omega D/2U$ denotes the tip-speed ratio, $R{e}_{c}=\frac{c}{\nu}\sqrt{{(\omega D/2)}^{2}+{U}^{2}}$ is the Reynolds number based on the chord and rotating speed whilst $R{e}_{D}=\frac{DU}{\nu}$ is based on the turbine diameter and free stream velocity .

Blade | Turb. | Length | Stream | Rot. | Tip Speed | Kin. | Reynolds | ||
---|---|---|---|---|---|---|---|---|---|

Chord | Diameter | Ratio | Vel. | Speed | Ratio | Visc | |||

Symbol | c | D | $c/D$ | U | $\omega $ | $TSR$ | $\nu $ | $R{e}_{c}$ | $R{e}_{D}$ |

Units | m | m | - | m/s | rad/s | - | m^{2}/s | - | - |

Oler et al. [39] | 0.1524 | 1.22 | 0.125 | 0.091 | 0.749 | 5 | $0.1\times {10}^{-5}$ | $6.83\times {10}^{4}$ | $1.07\times {10}^{5}$ |

Simulations | 1.0 | 8.0 | 0.125 | 0.088 | 0.11 | 5 | $5.0\times {10}^{-5}$ | $0.90\times {10}^{4}$ | $0.14\times {10}^{5}$ |

**Table 3.**Frequencies selected by HODMDc (in rad/s) presented in Figure 5.

m | ${\mathit{\delta}}_{\mathit{m}}$ | ${\mathit{\omega}}_{\mathit{m}}$ | ${\mathit{a}}_{\mathit{m}}$ | M |
---|---|---|---|---|

1 | −9.5158 × ${10}^{-3}$ | 1.0418 × ${10}^{-1}$ | 8.3676 × ${10}^{-2}$ | 3 |

2 | −1.3190 × ${10}^{-2}$ | 2.0196 × ${10}^{-1}$ | 4.3080 × ${10}^{-2}$ | 5 |

3 | −5.5556 × ${10}^{-3}$ | 4.2243 × ${10}^{-1}$ | 4.5354 × ${10}^{-2}$ | 7 |

4 | 8.5601 × ${10}^{-2}$ | 1.0403 × 10 | 1.1669 × 10 | 9 |

5 | 9.8704 × ${10}^{-2}$ | 8.6673 | 2.1948 × 10 | 11 |

6 | 8.7053 × ${10}^{-2}$ | 1.0250 × 10 | 9.6744 | 13 |

7 | −2.8024 × ${10}^{-2}$ | 3.0565 × ${10}^{-1}$ | 9.0310 × ${10}^{-3}$ | 15 |

8 | 4.7800 × ${10}^{-2}$ | 4.5260 | 6.1397 × ${10}^{-1}$ | 17 |

9 | −1.9750 × ${10}^{-2}$ | 5.2824 × ${10}^{-1}$ | 1.2018 × ${10}^{-2}$ | 19 |

10 | 8.5236 × ${10}^{-2}$ | 8.8956 | 6.9477 | 21 |

**Table 4.**Frequencies selected by HODMDc (in rad/s) presented in Figure 10.

m | ${\mathit{\delta}}_{\mathit{m}}$ | ${\mathit{\omega}}_{\mathit{m}}$ | ${\mathit{a}}_{\mathit{m}}$ | M |
---|---|---|---|---|

1 | −2.4974 × ${10}^{-3}$ | 1.0372 × ${10}^{-1}$ | 1.4785 × ${10}^{-1}$ | 3 |

2 | −5.6401 × ${10}^{-3}$ | 2.1195 × ${10}^{-1}$ | 9.5278 × ${10}^{-2}$ | 5 |

3 | −2.6524 × ${10}^{-2}$ | 3.3103 × ${10}^{-1}$ | 2.9438 × ${10}^{-2}$ | 7 |

4 | −2.9580 × ${10}^{-2}$ | 4.3577 × ${10}^{-1}$ | 2.2503 × ${10}^{-2}$ | 9 |

5 | −3.2188 × ${10}^{-2}$ | 5.4754 × ${10}^{-1}$ | 1.8342 × ${10}^{-2}$ | 11 |

6 | −2.8720 × ${10}^{-2}$ | 6.5329 × ${10}^{-1}$ | 1.7896 × ${10}^{-2}$ | 13 |

7 | −2.7517 × ${10}^{-2}$ | 7.8174 × ${10}^{-1}$ | 1.3842 × ${10}^{-2}$ | 15 |

8 | −2.8857 × ${10}^{-2}$ | 1.0277 | 1.2363 × ${10}^{-2}$ | 17 |

9 | −4.0700 × ${10}^{-2}$ | 1.1119 | 1.1227 × ${10}^{-2}$ | 19 |

10 | −4.2714 × ${10}^{-2}$ | 1.2517 | 8.6121 × ${10}^{-3}$ | 21 |

**Table 5.**Frequencies selected by HODMDc (in rad/s) presented in Figure 14.

m | ${\mathit{\delta}}_{\mathit{m}}$ | ${\mathit{\omega}}_{\mathit{m}}$ | ${\mathit{a}}_{\mathit{m}}$ | M |
---|---|---|---|---|

1 | −3.0199 × ${10}^{-2}$ | 1.3516 × ${10}^{-1}$ | 8.5508 × ${10}^{-2}$ | 3 |

2 | −4.0940 × ${10}^{-2}$ | 2.6042 × ${10}^{-1}$ | 4.9633 × ${10}^{-2}$ | 5 |

3 | −3.0227 × ${10}^{-2}$ | 4.2156 × ${10}^{-1}$ | 4.3403 × ${10}^{-2}$ | 7 |

4 | −4.0559 × ${10}^{-2}$ | 5.9441 × ${10}^{-1}$ | 3.7762 × ${10}^{-2}$ | 9 |

5 | −5.0647 × ${10}^{-2}$ | 8.5848 × ${10}^{-1}$ | 3.0700 × ${10}^{-2}$ | 11 |

6 | −9.1483 × ${10}^{-2}$ | 1.4785 | 2.2538 × ${10}^{-2}$ | 13 |

7 | −3.8967 × ${10}^{-2}$ | 1.1690 | 3.1258 × ${10}^{-2}$ | 15 |

8 | −1.0826 × ${10}^{-1}$ | 1.5864 | 1.8197 × ${10}^{-2}$ | 17 |

9 | −6.2864 × ${10}^{-2}$ | 1.3355 | 2.3832 × ${10}^{-2}$ | 19 |

10 | −7.8737 × ${10}^{-2}$ | 7.2259 × ${10}^{-1}$ | 2.0244 × ${10}^{-2}$ | 21 |

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## Share and Cite

**MDPI and ACS Style**

Le Clainche, S.; Ferrer, E. A Reduced Order Model to Predict Transient Flows around Straight Bladed Vertical Axis Wind Turbines. *Energies* **2018**, *11*, 566.
https://doi.org/10.3390/en11030566

**AMA Style**

Le Clainche S, Ferrer E. A Reduced Order Model to Predict Transient Flows around Straight Bladed Vertical Axis Wind Turbines. *Energies*. 2018; 11(3):566.
https://doi.org/10.3390/en11030566

**Chicago/Turabian Style**

Le Clainche, Soledad, and Esteban Ferrer. 2018. "A Reduced Order Model to Predict Transient Flows around Straight Bladed Vertical Axis Wind Turbines" *Energies* 11, no. 3: 566.
https://doi.org/10.3390/en11030566