# Numerical and Experimental Investigations of an Elasto-Flexible Membrane Wing at a Reynolds Number of 280,000

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Configuration and Measurement Techniques

#### 2.1. Experimental Model

#### 2.2. Force Measurements

_{L}= ±0.8% and ∆C

_{D}= ±4%.

#### 2.3. Flow Field Measurements

#### 2.4. Membrane Deflections Measurements

^{2}.

#### 2.5. Test Cases and Conditions

## 3. Numerical Modeling

#### 3.1. Wing Geometry

#### 3.2. Grid Generation and Grid Independency Study

_{+}= 1 is used to assure an accurate resolution of the viscous sublayer, which corresponds to a distance of the first node of around 0.05 mm from the airfoil.

#### 3.3. Fluid Structure Interaction Set Up

#### 3.3.1. Structural Set Up

#### 3.3.2. Fluid Set Up

^{−4}. A second order upwind scheme for the convection terms was used and all the diffusion terms were discretized with the second order central difference scheme.

#### 3.4. Test Cases and Conditions

## 4. Results and Discussion

#### 4.1. The Elasto-Flexible Membrane Wing Compared to Its Rigid Counterpart

#### 4.2. Influence of the Transition Modeling

_{L}- and C

_{D}-AOA curves obtained with the k-ω SST model coupled with the γ-Reθ transition model in contrast to the k-ω SST model are symbolized in Figure 6 as blue diamonds and green stars, respectively. In both cases, the adaptivity of the elasto-flexible membrane wing was observed taking the above description into account.

_{L}-AOA curve: the lift coefficient of the wing is clearly higher for the transitional modeling than for the fully turbulent one. In order to better understand the difference between the models, Figure 8 and Figure 9 illustrate the pressure coefficient along the wing and the turbulence intensity Tu defined from the turbulence kinetic energy. In Figure 9, three different angles of attack, namely 0°, 6° and 15°, were chosen to analyze the difference between the models. Those three angles of attack were chosen taking into account the C

_{L}- and C

_{D}-AOA curves: 0° as both models have similar coefficients, 6° as the coefficients are significantly different and 15° as both models have once more similar coefficients. For AOA = 0°, it was observed that the boundary layer had the same nature in both models, which resulted directly in the same C

_{L}and C

_{D}(Figure 6). For AOA = 6°, the difference came from the laminar-turbulent transition situated on the upper side of the wing. The transition occurred around X/c = 55% and was directly observed in Figure 8 by means of the pressure coefficient. The value for -C

_{p}was higher for the transition model, which resulted in a higher lift coefficient and slightly higher deformation. For AOA = 15°, the transition point moved to the nose of the leading edge, and both boundary layers were similar. The pressure distributions along the wing for both models were therefore comparable, which resulted in C

_{L}values close to each other.

#### 4.3. Experimental Investigations Compared to FSI Simulations

_{L}and C

_{D}are plotted in Figure 6. Then, the deformation of the membrane is depicted in Figure 10; the upwind behind the model if depicted in Figure 11 and finally the flow field is shown in Figure 12 and Figure 13 for specific AOAs.

_{L}- and the C

_{D}-AOA curves. The results for the experiments are depicted as red circles. It can be seen that the aerodynamic coefficients obtained in the wind tunnel tests were different from the coefficients obtained with the U-RANS/FEM simulations. On the one hand, the experimental lift coefficient values were lower and, on the other hand, the experimental drag coefficient values were higher. One hypothesis to explain the first issue comes from the 3D effects during the experiments: the endplates should minimize the tip vortices of the wing but the flow observed was not completely uniform over the wingspan. Downwash was measured behind the model between the endplates using hot-wire anemometry and is depicted in Figure 11. It can be seen that 3D effects at the tips of the wing exist, which can explain a lower lift coefficient; see Figure 6. This phenomenon needs to be directly linked with the deformation of the membrane, which is analyzed in more detail in the following sections. The second issue is due to the dynamic calibration of the experimental model. As the drag of the structure was not negligible compared to the drag of the wing, the error of ∆C

_{D}= ±4% during the measurement had a significant effect on the results.

_{L}-AOA curve stayed constant until the smooth stall at AOA = 17°. Furthermore, the results obtained with the k-ω SST model were closer to the experiments. Therefore, in order to compare the FSI simulations with the experiments, only the results obtained with the k-ω SST model will be considered in the following.

_{L}-AOA curve. For AOA = 0°, the deformation obtained in the experiment was higher on the upper side of the membrane, which resulted in a higher lift coefficient. The opposite phenomenon was observed for AOAs = 6°, 10° and 15°: the deformation was higher for the FSI simulations, which caused a higher lift coefficient in the FSI results. The disparities were small but had a significant influence on the forces of the system. However, the general trend seen in the experiments can be reproduced by the FSI simulations: on the one hand, when the angle of attack becomes higher, the camber of the profile increases, which permits higher lift coefficients. On the other hand, the stall region appears smooth.

## 5. Conclusions

_{L}and higher C

_{D}.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamic |

U-RANS | Unsteady Reynolds-Averaged Navier-Stokes |

FSI | Fluid Structure Interaction |

AR | Aspect Ratio, - |

c | Chord, m |

E | Young’s Modulus, MPa |

Re | Reynolds Number, - |

AOA | Angle of Attack, ° |

X, Y, Z | Dimensionless System Coordinates, - |

y_{+} | Dimensionless Wall Distance, - |

U_{∞} | Freestream Velocity, m/s |

U | x Component of the Velocity, m/s |

W | z Component of the Velocity, m/s |

∆c_{r} | Grid Resolution, mm |

q_{∞} | Freestream Dynamic Pressure, Pa |

C_{P}; C_{L}; C_{D} | Pressure Coefficient, -; Lift Coefficient, -; Drag Coefficient, - |

Tu | Turbulence Intensity, - |

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**Figure 1.**(

**a**) Experimental model installed in the wind tunnel test section; (

**b**) Experimental model; (

**c**) Sketch of the experimental model (unit in mm).

**Figure 2.**Description of the membrane material: stress-strain curves obtained from unidirectional tensile tests, from [10].

**Figure 3.**(

**a**) Experimental plane for the hot-wire measurements (upper side) and the photogrammetry; (

**b**) Set up of the photogrammetry with the four cameras circled in red (2 on the upper side, 2 on the lower side); (

**c**) Set up of the experimental model installed in the wind tunnel test section with the hot-wire anemometry system.

**Figure 4.**(

**a**) Quasi-2D fluid mesh; (

**b**) Details of the O-grid around the airfoil; (

**c**) Boundary conditions used for the fluid set up.

**Figure 6.**(

**a**) Lift coefficient and (

**b**) drag coefficient of the elasto-flexible membrane wing, its rigid counterpart and the experiments at Re = 280,000.

**Figure 7.**(

**a**) Deformed geometry of the elasto-flexible membrane wing for AOA = 12°, 14°, 15°, 16° and 18°; (

**b**) Pressure distribution of the elasto-flexible membrane wing for AOA = 12°, 14°, 15°, 16° and 18°; (

**c**) Geometry of the rigid membrane wing; (

**d**) Pressure distribution of the rigid geometry for AOA = 13°, 14° and 15°.

**Figure 8.**(

**a**) Deformed geometry of the elasto-flexible membrane wing for AOA = 6°; (

**b**) Pressure distribution of the elasto-flexible membrane wing for AOA = 6°; (

**c**) Deformed geometry of the elasto-flexible membrane wing for AOA = 15°; (

**d**) Pressure distribution of the elasto-flexible membrane wing for AOA = 15°.

**Figure 9.**Turbulence intensity Tu [-] for AOA = 0° (

**a**,

**b**), 6° (

**c**,

**d**) and 15° (

**e**,

**f**) for the k-ω SST model without (

**left**) and with transition (

**right**).

**Figure 10.**Deformation of the membrane for the different cases at AOA of 0° (

**a**), 6° (

**b**), 10° (

**c**) and 15° (

**d**) at U∞ = 20 m/s. The experimental tests and the FSI simulations are shown.

**Figure 11.**Downwash angle β measurement using a hot wire anemometry system for AOA = 6° at Re = 280,000 at 220 mm behind the trailing edge.

**Figure 12.**U and W velocities obtained in (

**a**) for the experiment simulations and in (

**b**) for the FSI simulations at AOA = 6° and U∞ = 20 m/s.

**Figure 13.**U and W velocities obtained in (

**a**) for the experiment simulations and in (

**b**) for the FSI simulations at AOA = 15° and U∞ = 20 m/s.

Measurement Techniques | Flow Conditions | |||
---|---|---|---|---|

U_{∞} [m/s] | q_{∞} [Pa] | Re | α [°] | |

Force measurement | 20 | 230 | 280,000 | 0, 2, 4, 6, 8, 10, 12, 14, 15, 16, 17, 18 |

Hot-wire anemometry | 20 | 230 | 280,000 | 6, 15 |

Stereo-photogrammetry | 20 | 230 | 280,000 | 0, 6, 10, 15 |

Parameters | Coarse | Medium | Fine | Extra Fine |
---|---|---|---|---|

Total nodes number | 39,226 | 80,374 | 159,774 | 299,824 |

Normal layer | 62 | 90 | 120 | 175 |

Circumferential layer | 2 × 75 | 2 × 100 | 2 × 140 | 2 × 185 |

Minimal angle | 26.4 | 26.8 | 27.3 | 27.6 |

Aspect ratio | 5840 | 3180 | 1930 | 1400 |

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**MDPI and ACS Style**

Piquee, J.; Breitsamter, C.
Numerical and Experimental Investigations of an Elasto-Flexible Membrane Wing at a Reynolds Number of 280,000. *Aerospace* **2017**, *4*, 39.
https://doi.org/10.3390/aerospace4030039

**AMA Style**

Piquee J, Breitsamter C.
Numerical and Experimental Investigations of an Elasto-Flexible Membrane Wing at a Reynolds Number of 280,000. *Aerospace*. 2017; 4(3):39.
https://doi.org/10.3390/aerospace4030039

**Chicago/Turabian Style**

Piquee, Julie, and Christian Breitsamter.
2017. "Numerical and Experimental Investigations of an Elasto-Flexible Membrane Wing at a Reynolds Number of 280,000" *Aerospace* 4, no. 3: 39.
https://doi.org/10.3390/aerospace4030039