# Experiments on Flexible Filaments in Air Flow for Aeroelasticity and Fluid-Structure Interaction Models Validation

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Flexible Filament Manufacturing and Characterization

^{3}) using an additive manufacturing system (3D-Bioplotter by EnvisionTEC, https://envisiontec.com) following a rectilinear path during printing to avoid any curvature or deformation, and therefore produce a straight filament of uniform cross section and smooth surface finishing. The filament had rectangular cross section with width $w$ of 2.00 ± 0.05 mm and height $h$ of 0.40 ± 0.05 mm (measured with a digital caliper), and a linear mass density of 0.80 ± 0.16 g/m. Note that, in order to avoid the large error that would have arisen from measuring directly the mass of the filament (which was on the order of 0.1 g), the density of the silicon rubber provided above was deduced from measuring the mass of a bigger chunk of silicon rubber. The linear mass density of the filament provided above was therefore calculated as the product of the silicon rubber density times the width and height of the filament cross-section. One portion of the filament was used for the mechanical characterization described below, whilst another portion was used to realize the test piece for the FSI experiments (described in Section 2.2).

#### 2.2. Test Piece Description and Preliminary Characterization

#### 2.3. Wind Tunnel Flow Characterization

#### 2.4. Experimental Procedure

## 3. Results and Discussion

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Displacements and flapping frequencies measured for Filament 2. (

**a**) Displacement in the vertical plane; (

**b)**Displacement in the horizontal plane; (

**c**) Frequency in the vertical plane; (

**d**) Frequency in the horizontal plane.

**Figure A2.**Trajectory of the free-end of Filament 2 as seen by an observer located downstream of the filament and facing the flow (Y-Z plane in Figure 2c).

**Figure A3.**Autocorrelation function of the total displacement of the free-end of Filament 2 at different Reynolds number values (the x-axis scale is in seconds).

**Figure A4.**Reconstructed attractor in phase-space for the total displacement of the free-end of Filament 2 at different Reynolds number values.

**Figure A5.**Displacements and flapping frequencies measured for Filament 3. (

**a**) Displacement in the vertical plane; (

**b)**Displacement in the horizontal plane; (

**c**) Frequency in the vertical plane; (

**d**) Frequency in the horizontal plane.

**Figure A6.**Trajectory of the free-end of Filament 3 as seen by an observer located downstream of the filament and facing the flow (Y-Z plane in Figure 2c).

**Figure A7.**Autocorrelation function of the total displacement of the free-end of Filament 3 at different Reynolds number values (the x-axis scale is in seconds).

**Figure A8.**Reconstructed attractor in phase-space for the total displacement of the free-end of Filament 3 at different Reynolds number values.

**Figure A9.**Displacements and flapping frequencies measured for Filament 4. (

**a**) Displacement in the vertical plane; (

**b)**Displacement in the horizontal plane; (

**c**) Frequency in the vertical plane; (

**d**) Frequency in the horizontal plane.

**Figure A10.**Trajectory of the free-end of Filament 4 as seen by an observer located downstream of the filament and facing the flow (Y-Z plane in Figure 2c).

**Figure A11.**Autocorrelation function of the total displacement of the free-end of Filament 4 at different Reynolds number values (the x-axis scale is in seconds).

**Figure A12.**Reconstructed attractor in phase-space for the total displacement of the free-end of Filament 4 at different Reynolds number values.

**Figure A13.**Displacements and flapping frequencies measured for Filament 5. (

**a**) Displacement in the vertical plane; (

**b)**Displacement in the horizontal plane; (

**c**) Frequency in the vertical plane; (

**d**) Frequency in the horizontal plane.

**Figure A14.**Trajectory of the free-end of Filament 5 as seen by an observer located downstream of the filament and facing the flow (Y-Z plane in Figure 2c).

**Figure A15.**Autocorrelation function of the total displacement of the free-end of Filament 5 at different Reynolds number values (the x-axis scale is in seconds).

**Figure A16.**Reconstructed attractor in phase-space for the total displacement of the free-end of Filament 5 at different Reynolds number values.

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**Figure 1.**Mechanical characterization of the flexible filaments. (

**a**) Uniaxial tensile tests results; (

**b**) Stress-strain curve.

**Figure 2.**(

**a**) Schematic representation of the test piece; (

**b**) Filament orientation during the tests in the wind tunnel; (

**c**) Schematic representation of the experimental wind tunnel setup.

**Figure 3.**Preliminary characterization of the flexible filaments. (

**a**) First-mode natural vibration frequency; (

**b**) First-mode damping ratio in stagnant air.

**Figure 4.**Preliminary characterization of the free-stream wind tunnel flow. (

**a**) Streamwise turbulence intensity; (

**b**) Streamwise turbulence macroscale; (

**c**) Streamwise turbulence microscale.

**Figure 5.**Representative envelopes of motion for Filament 1 in the vertical plane (left) and horizontal plane (right) at three Reynolds numbers (top, middle and bottom). (

**a**) Vertical plane, $Re=202$; (

**b**) Horizontal plane, $Re=202$; (

**c**) Vertical plane, $Re=410$; (

**d**) Horizontal plane, $Re=410$; (

**e**) Vertical plane, $Re=804$; (

**f**) Horizontal plane, $Re=804$ (x-axis and y-axis scales are in mm).

**Figure 6.**Displacements and flapping frequencies measured for Filament 1. (

**a**) Displacement in the vertical plane; (

**b)**Displacement in the horizontal plane; (

**c**) Frequency in the vertical plane; (

**d**) Frequency in the horizontal plane.

**Figure 7.**Trajectory of the free-end of Filament 1 as seen by an observer located downstream of the filament and facing the flow (Y-Z plane in Figure 2c).

**Figure 8.**Autocorrelation function of the total displacement of the free-end of Filament 1 at different Reynolds number values (the x-axis scale is in seconds).

**Figure 9.**Reconstructed attractor in phase-space for the total displacement of the free-end of Filament 1 at different Reynolds number values.

**Figure 11.**Dimensionless displacements and frequencies for Filaments 1−5 vs. Reynolds number. (

**a**) Dimensionless displacement in the vertical plane; (

**b)**Dimensionless displacement in the horizontal plane; (

**c**) Dimensionless frequency in the vertical plane; (

**d**) Dimensionless frequency in the horizontal plane. Legend: Filament 1 (white △); Filament 2 (red □); Filament 3 (green ◁); Filament 4 (blue ○); Filament 5 (black ▽).

**Figure 12.**Condensed representation of the observed structural response of the flexible filaments. (

**a**) Dynamic map; (

**b**) Stability map. Legend: SD = static deflection; SAV = small-amplitude vibration; LCO = large-amplitude limit-cycle oscillation with two-dimensional trajectory; LCO-8 = large-amplitude limit-cycle oscillation with three-dimensional figure-eight-shaped trajectory; NPO = large-amplitude non-periodic oscillation; TR = transition.

Reference | Fluid | Structure | Motion |
---|---|---|---|

Pereira Gomez et al. [25] | Polyethylene glycol syrup in laminar flow | Flexible metal plate with rear mass at the trailing edge, clamped behind a rigid circular cylinder free to rotate around its axis, oriented in cross-flow | 2D |

Pereira Gomes and Lienhart [26] | Water and polyethylene glycol syrup in laminar and turbulent flow | Flexible metal plate with rear mass at the trailing edge, clamped behind a rigid cylindrical body (circular or rectangular cross section) oriented in cross-flow | 2D |

Kalmbach and Breuer [27] | Water in turbulent flow | Flexible rubber membrane with rear mass at the trailing edge, clamped behind a rigid and fixed circular cylinder oriented in cross-flow | 2D |

De Nayer et al. [28] | Water in turbulent flow | Flexible rubber membrane clamped behind fixed rigid circular cylinder oriented in cross-flow | 2D/3D |

Hessenthaler et al. [29] | Aqueous glycerol solution in laminar flow | Flexible cantilever-beam plate in merging flow from two inlets | 3D |

This study | Air in laminar and turbulent flow | Flexible cantilever-beam filaments of variable length in uniform flow | 3D |

Filament No. | $\mathit{L}$ (mm) | ${\mathit{f}}_{1}\left(\mathbf{Hz}\right)$ | ${\mathit{\zeta}}_{1}(-)$ |
---|---|---|---|

1 | 60.0 ± 0.5 | 2.5 ± 0.1 | 0.012 ± 0.002 |

2 | 50.0 ± 0.5 | 2.8 ± 0.1 | 0.015 ± 0.002 |

3 | 40.0 ± 0.5 | 3.3 ± 0.2 | 0.017 ± 0.002 |

4 | 30.0 ± 0.5 | 3.9 ± 0.3 | 0.021 ± 0.003 |

5 | 20.0 ± 0.5 | 6.3 ± 1.0 | 0.028 ± 0.004 |

6 | 10.0 ± 0.5 | 16 ± 3 | 0.035 ± 0.005 |

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**MDPI and ACS Style**

Silva-Leon, J.; Cioncolini, A. Experiments on Flexible Filaments in Air Flow for Aeroelasticity and Fluid-Structure Interaction Models Validation. *Fluids* **2020**, *5*, 90.
https://doi.org/10.3390/fluids5020090

**AMA Style**

Silva-Leon J, Cioncolini A. Experiments on Flexible Filaments in Air Flow for Aeroelasticity and Fluid-Structure Interaction Models Validation. *Fluids*. 2020; 5(2):90.
https://doi.org/10.3390/fluids5020090

**Chicago/Turabian Style**

Silva-Leon, Jorge, and Andrea Cioncolini. 2020. "Experiments on Flexible Filaments in Air Flow for Aeroelasticity and Fluid-Structure Interaction Models Validation" *Fluids* 5, no. 2: 90.
https://doi.org/10.3390/fluids5020090