#
Direct Scaling of Measure on Vortex Shedding through a Flapping Flag Device in the Open Channel around a Cylinder at Re∼10^{3}: Taylor’s Law Approach

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Thin Films-Based Device and Experimental Setup

#### 2.2. Strohual and Reynolds Numbers Relationship

#### 2.3. Direct Scaling Analysis on the Voltage Fluctuations: Taylor’s Law Approach

## 3. Experimental Results and Discussions

#### 3.1. Correlation Time

#### 3.2. Vortex Shedding Measures and Relationship Voltage-Pressure

#### 3.3. Taylor’s Law Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The sensor, mounted downstream of the bluff body, is affected by the vortexes and oscillates generating alternating voltage (

**a**). 3d view of the channel flume site in EUMER Lab of Unversity of Salento (

**b**).

**Figure 2.**The flexible structure of the piezoelectric devices was optimized to guarantee high flexibility and sensitivity. In addition all the materials used for its fabrication are completely biocompatible, suitable for environmental use (

**a**). The 3D sketch highlights the extremely compactness of the transducer (

**b**).

**Figure 4.**Stress–strain relationship of the composite flag obtained by Dynamic Mechanical Analysis (DMA).

**Figure 5.**Longitudinal section of the experimental setup with the flag of length l clamped at the cylinder of diameter D, inlet flow rate Q, and height of flow H in the open channel.

**Figure 6.**Domain of stability for the flapping flag marked by the shaded zone [30]. The red point represents our experimental data.

**Figure 8.**Strouhal number–Reynolds number relationship for circular cylinders taken from [5], the black crosses represent our experimental data.

**Figure 10.**Taylor’s law in the log–log plane at a time-scale equal to one minute for the first configuration.

**Figure 11.**Laws of variation of scaling exponent over the aggregation time: linear law (

**a**) and logarithmic law (

**b**).

**Figure 12.**Comparison between the observed frequency distribution and the Gaussian probability density function.

**Table 1.**Values of natural frequency of vibration, ${f}_{1}$, of the cantilever for the several length l.

l [cm] | 3.20 | 3.50 | 4.00 | 7.90 |

${f}_{1}$ [$\mathrm{Hz}$] | 8.40 | 7.38 | 5.40 | 1.10 |

**Table 2.**Values of peak frequency ${f}_{r}$ of the voltage fluctuations $\mathbf{v}\left(t\right)$ and corresponding Strouhal number $St$ for the different analyzed configurations characterized by lenght of flag l, heigth of flow h in the open channel, bulk velocity u and Reynolds number $Re$.

l | H | u | $\mathit{Re}$ | ${\mathit{f}}_{\mathit{r}}$ | $\mathit{St}$ |
---|---|---|---|---|---|

(cm) | (cm) | (m s${}^{-1})$ | (-) | (s${}^{-1})$ | (-) |

3.2 | 4.6 | 0.293 | 5859 | 2.50 | 0.17 |

3.2 | 4.0 | 0.337 | 6738 | 2.35 | 0.14 |

3.5 | 6.9 | 0.197 | 3934 | 2.09 | 0.21 |

3.5 | 6.8 | 0.198 | 3963 | 2.20 | 0.22 |

3.5 | 6.8 | 0.198 | 3934 | 1.93 | 0.19 |

3.5 | 6.7 | 0.201 | 4022 | 2.11 | 0.21 |

3.5 | 6.7 | 0.201 | 4022 | 2.12 | 0.21 |

3.5 | 6.6 | 0.204 | 4083 | 1.98 | 0.19 |

3.5 | 6.5 | 0.207 | 4146 | 2.05 | 0.20 |

3.5 | 6.5 | 0.207 | 4146 | 2.00 | 0.19 |

3.5 | 6.4 | 0.211 | 4211 | 2.13 | 0.20 |

3.5 | 6.3 | 0.214 | 4278 | 2.22 | 0.21 |

3.5 | 6.2 | 0.217 | 4347 | 2.45 | 0.23 |

3.5 | 6.2 | 0.217 | 4347 | 2.45 | 0.23 |

4.0 | 10.7 | 0.126 | 2519 | 1.45 | 0.23 |

4.0 | 6.8 | 0.198 | 3963 | 2.01 | 0.20 |

4.0 | 5.6 | 0.241 | 4813 | 2.54 | 0.21 |

4.0 | 5.4 | 0.250 | 4991 | 2.33 | 0.19 |

4.0 | 5.2 | 0.259 | 5183 | 2.86 | 0.22 |

4.0 | 5.1 | 0.264 | 5284 | 2.54 | 0.19 |

4.0 | 4.7 | 0.287 | 5734 | 2.72 | 0.19 |

4.0 | 4.5 | 0.299 | 5989 | 3.01 | 0.20 |

4.0 | 4.4 | 0.306 | 6125 | 3.50 | 0.23 |

4.0 | 4.3 | 0.313 | 6167 | 3.67 | 0.23 |

4.0 | 4.0 | 0.337 | 6738 | 3.90 | 0.23 |

4.0 | 3.4 | 0.396 | 7926 | 4.54 | 0.23 |

7.9 | 4.6 | 0.293 | 5859 | 2.71 | 0.19 |

7.9 | 4.0 | 0.337 | 6738 | 3.54 | 0.21 |

**Table 3.**Values of scaling exponent of Taylor’s law b for each configuration at several aggregation time ranges.

b | ||||
---|---|---|---|---|

$\Delta \mathit{t}$ | First | Second | Third | Fourth |

(s) | $\mathit{l}=\mathbf{3.2}$ cm | $\mathit{l}=\mathbf{3.2}$ cm | $\mathit{l}=\mathbf{7.9}$ cm | $\mathit{l}=\mathbf{7.9}$ cm |

$\mathit{Re}$ = 5944 | $\mathit{Re}$ = 6835 | $\mathit{Re}$ = 5944 | $\mathit{Re}$ = 6835 | |

15 | 1.773 | 1.128 | 1.049 | 0.875 |

30 | 1.951 | 1.177 | 1.162 | 0.957 |

60 | 2.197 | 1.363 | 1.331 | 1.055 |

90 | 2.262 | 1.420 | 1.326 | 1.031 |

120 | 2.330 | 1.572 | 1.438 | 1.071 |

150 | 2.440 | 1,717 | 1.547 | 1.244 |

180 | 2.496 | 1.735 | 1.644 | 1.148 |

**Table 4.**Values of ${r}^{2}$ related to previous fitting laws for each configuration at several aggregation time ranges.

${\mathit{r}}^{2}$ | ||||
---|---|---|---|---|

$\Delta \mathit{t}$ | First | Second | Third | Fourth |

(s) | $\mathit{l}$= 3.2 cm | $\mathit{l}$= 3.2 cm | $\mathit{l}$= 7.9 cm | $\mathit{l}$= 7.9 cm |

$\mathit{Re}=\mathbf{5944}$ | $\mathit{Re}=\mathbf{6835}$ | $\mathit{Re}=\mathbf{5944}$ | $\mathit{Re}=\mathbf{6835}$ | |

15 | 0.953 | 0.964 | 0.971 | 0.970 |

30 | 0.952 | 0.967 | 0.978 | 0.975 |

60 | 0.952 | 0.964 | 0.985 | 0.978 |

90 | 0.975 | 0.964 | 0.983 | 0.978 |

120 | 0.977 | 0.966 | 0.985 | 0.977 |

150 | 0.977 | 0.973 | 0.989 | 0.973 |

180 | 0.974 | 0.971 | 0.991 | 0.981 |

First | Second | Third | Fourth | |
---|---|---|---|---|

${b}_{0}$ | 0.989 | 0.353 | 0.416 | 0.544 |

k | 0.287 | 0.257 | 0.222 | 0.120 |

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**MDPI and ACS Style**

De Bartolo, S.; Vittorio, M.D.; Francone, A.; Guido, F.; Leone, E.; Mastronardi, V.M.; Notaro, A.; Tomasicchio, G.R. Direct Scaling of Measure on Vortex Shedding through a Flapping Flag Device in the Open Channel around a Cylinder at *Re*∼10^{3}: Taylor’s Law Approach. *Sensors* **2021**, *21*, 1871.
https://doi.org/10.3390/s21051871

**AMA Style**

De Bartolo S, Vittorio MD, Francone A, Guido F, Leone E, Mastronardi VM, Notaro A, Tomasicchio GR. Direct Scaling of Measure on Vortex Shedding through a Flapping Flag Device in the Open Channel around a Cylinder at *Re*∼10^{3}: Taylor’s Law Approach. *Sensors*. 2021; 21(5):1871.
https://doi.org/10.3390/s21051871

**Chicago/Turabian Style**

De Bartolo, Samuele, Massimo De Vittorio, Antonio Francone, Francesco Guido, Elisa Leone, Vincenzo Mariano Mastronardi, Andrea Notaro, and Giuseppe Roberto Tomasicchio. 2021. "Direct Scaling of Measure on Vortex Shedding through a Flapping Flag Device in the Open Channel around a Cylinder at *Re*∼10^{3}: Taylor’s Law Approach" *Sensors* 21, no. 5: 1871.
https://doi.org/10.3390/s21051871