# Validation of a Model for Estimating the Strength of a Vortex Created from the Bound Circulation of a Vortex Generator

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Experimental Setup

_{h}= U

_{∞}h/ν = 8.5 ms

^{−1}× 0.06 m/15 × 10

^{−6}m

^{2}s

^{−1}= 34,000, where U

_{∞}= 8.5 ms

^{−1}was the freestream velocity and h = 0.06 m was the VG height.

^{2}(corresponding to approximately 2.3 δ × δ) [20,21]. The cameras were placed outside the wind-tunnel.

#### 2.2. Numerical Model

^{6}and 8.2 × 10

^{6}polyhedral cells for the rectangular and triangular set-ups, respectively. The polyhedral meshing option was chosen, since it provides a detailed and balanced solution for complicated studies. Further, it requires lesser computational power since it generates five times fewer cells than the equivalent tetrahedral. Another advantage was that it does not require more surface preparation [22]. The characteristic dimension called the Base size was set equal to 0.05 m for generating the mesh discretization of the overall domain. Specifically, for the rectangular configuration, two relative sizes were chosen for refining the areas that required a detailed resolution. A relative size was chosen for discretizing the region close to the VGs and a second refined region was modeled for capturing the far wake region. A different approach was implemented for discretizing the triangular geometry.

^{−6}. The convergence of the solution was also verified by monitoring the behavior of the residuals. For the rectangular configuration, the residuals reached the tolerance limit and remained almost constant after the first 2000 iteration steps, while for the triangular set-up 1200 iteration steps were needed to meet the same limit.

## 3. Estimating the Bound and Trailed Circulation from CFD

_{b}(y) is the bound circulation at each location. The bound circulation Γ

_{b}(y), thus, varied with the VG geometry and the incoming flow as a function of the wall-normal distance y. The velocity profile shown in Figure 6 corresponds to the computed streamwise average velocity profile at 3h upstream i.e., x/h = −3h, where the flow was undisturbed by the vanes.

_{s}, to the tip y = h. If it had been a pure airplane wing, one should start the integration at y = 0, but to avoid interference with the horizontal wall at y = 0, the starting value was set to a fraction of the VG height and the value chosen was 0.2h. From quantitative estimation, one can observe that the circulation for the innermost part of the VG did not contribute significantly to the tip vortex. This can also be understood from the flow visualization presented in at the end of the results section. To estimate the strength of the tip vortex from the computed flow in the wake of the VGs, one could apply the best fit to a Lamb–Oseen Vortex (LOV) (in polar coordinates).

## 4. Results and Discussion

_{t}/U∙h = 0.65, which was almost identical with the value obtained from the line integration and the LOV, see Figure 8. The trailed circulation for the triangular VG from the integration of the bound circulation was equal to Γ

_{t}/U∙h = 0.61 and was still quite close to the value of 0.55 estimated from the wake flow using LOV and line integration, which further supported the hypothesis.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

VG | Vortex Generator |

RANS | Reynolds-Averaged Navier–Stokes |

LOV | Lamb-Oseen Vortex |

PIV | Particle Image Velocimetry |

RST | Reynolds Stress Transport Turbulence Model |

S-A | Spalart–Allmaras Transport Turbulence Model |

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**Figure 1.**A sketch of the Laboratoire de Mécanique des Fluides de Lille (LMFL) wind tunnel (figure originally extracted from http://lml.univ-lille1.fr/?page=144&menu_curr_set=144).

**Figure 2.**Top-view of the configuration design showing the geometric parameters (

**top**). A 3-D drawing of the set-up (

**bottom**).

**Figure 4.**Example of the ensemble averaged velocity field at 3h downstream of the rectangular vortex generators (VGs).

**Figure 5.**Prism layers attached to the rectangular VGs (

**top**). Top-view of the mesh discretization of the rectangular VGs (

**bottom**).

**Figure 6.**Streamwise average velocity profile at 3h upstream, Reynolds Stress Turbulence (RST) turbulence model.

**Figure 7.**Integrating the pressure jump at the height y from the leading edge s = 0 to the trailing edge s = c(y) for determining the load f

_{n}(y) [N/m] normal to the VGs surface.

**Figure 8.**Circulation results for the rectangular configuration testing Reynolds Stress, k-ω SST, Spalart–Allmaras turbulence models in five downstream planes; (

**left**) Reynolds stress model, (

**middle**) k-ω SST, and (

**right**) Spalart–Allmaras.

**Figure 9.**Circulation results for the triangular configuration testing Reynolds Stress, k-ω SST, Spalart–Allmaras turbulence models in two downstream planes; (

**left**) Reynolds stress model, (

**middle**) k-ω SST, and (

**right**) Spalart–Allmaras.

**Figure 10.**Determination of Γ with the use of Lamb–Oseen Vortex, (

**left**) rectangular VG and (

**right**) triangular VG, RST turbulence model.

**Figure 12.**Distribution of the trailed circulation along VGs height. Rectangular VG (

**left**) and triangular VG (

**right**).

**Figure 13.**Trailing vortex associated streamlines visualized for the (

**left**) rectangular and (

**right**) triangular vortex generators, RST turbulence model.

Polyhedral Mesh | Rectangular | Triangular |
---|---|---|

Base size [m] | 0.05 | 0.05 |

Relative size close to VGs [%] | 9 | |

Relative size in the wake region [%] | 11 | - |

Prism layer mesh | ReD | TrD |

Number of prism layers | 20 | 20 |

Prism layer stretching | 1.5 | 1.5 |

Prism layer thickness [% of the base size] | 5 | 5 |

Total amount of cells | 11.3 × 10^{6} | 8.2 × 10^{6} |

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**MDPI and ACS Style**

Hansen, M.O.L.; Charalampous, A.; Foucaut, J.-M.; Cuvier, C.; Velte, C.M. Validation of a Model for Estimating the Strength of a Vortex Created from the Bound Circulation of a Vortex Generator. *Energies* **2019**, *12*, 2781.
https://doi.org/10.3390/en12142781

**AMA Style**

Hansen MOL, Charalampous A, Foucaut J-M, Cuvier C, Velte CM. Validation of a Model for Estimating the Strength of a Vortex Created from the Bound Circulation of a Vortex Generator. *Energies*. 2019; 12(14):2781.
https://doi.org/10.3390/en12142781

**Chicago/Turabian Style**

Hansen, Martin O. L., Antonis Charalampous, Jean-Marc Foucaut, Christophe Cuvier, and Clara M. Velte. 2019. "Validation of a Model for Estimating the Strength of a Vortex Created from the Bound Circulation of a Vortex Generator" *Energies* 12, no. 14: 2781.
https://doi.org/10.3390/en12142781