# Modelling of Cantilever-Based Flow Energy Harvesters Featuring C-Shaped Vibration Inducers: The Role of the Fluid/Beam Interaction

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. CFD Modelling of the FSI for the VI

#### 2.2. The Beam Model: Modified Euler-Bernoulli Equation

#### 2.3. The Quasi Steady Forcing

#### 2.4. The Experimental Setup

#### 2.5. Finite Dimensional Approach and Galerkin Expansion

## 3. Results

#### 3.1. Model Calibration against Experimental Results

#### 3.2. Evaluation of the Work Done by Fluid Forces on the VI

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

FEH | Flow Energy Harvester |

VI | Vibration Inducer |

AoA | Angle of Attack |

VIV | Vortex-Induced Vibration |

CFD | Computational Fluid Dynamics |

EB | Euler-Bernoulli |

## Appendix A. Equivalent Beam Model

- The first natural angular frequency ${\omega}_{10}$ of the equivalent beam must match ${\omega}_{exp}$, namely:$${\omega}_{10}=a\phantom{\rule{0.166667em}{0ex}}{\left(1.875/L\right)}^{2}={\omega}_{exp},$$
- The tip deflection must match the measured one, namely:$${w}_{L}={p}_{L}{L}^{3}/\left(3\phantom{\rule{0.166667em}{0ex}}E\phantom{\rule{0.166667em}{0ex}}J\right)={w}_{exp};$$

Steel | Equivalent | Units |
---|---|---|

${\rho}_{0}=7860$ | $\rho =7171$ | kg/m^{3} |

${E}_{0}=208$ | $E=234$ | GPa |

${h}_{0}=0.25$ | $h=0.316$ | mm |

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**Figure 1.**Picture of the experimental FEH (

**a**); the red VI in the foreground is mounted on the tip of the beam, the latter being coated with black paint. Sketch of the FEH (

**b**); all dimensions are in mm.

**Figure 2.**3D and 2D view (lower inset) of the non-uniform grid employed in the CFD code. Upper inset: example of flow field (velocity magnitude in m/s). The VI is depicted in red.

**Figure 4.**Experimental signal: transverse displacement of beam tip. The inset shows the overlapping of the harmonic fitting function with the experimental recording for the time interval 5 s < t < 9 s.

**Figure 5.**Comparison between experimental (black line) and numerical results (red line): time history of the tip motion (

**a**) and frequency spectrum of the tip motion (

**b**).

**Figure 6.**Power flows between the fluid and the FEH. To assess the accuracy of the modelling of the power transfer from the fluid to the beam, it is compared to the one transferred from the VI to the beam, the latter emerging as an internal power flow once the FEH is thought as composed by the VI and the beam.

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**MDPI and ACS Style**

Sciortino, G.; Lombardi, V.; Prestininzi, P. Modelling of Cantilever-Based Flow Energy Harvesters Featuring C-Shaped Vibration Inducers: The Role of the Fluid/Beam Interaction. *Appl. Sci.* **2023**, *13*, 416.
https://doi.org/10.3390/app13010416

**AMA Style**

Sciortino G, Lombardi V, Prestininzi P. Modelling of Cantilever-Based Flow Energy Harvesters Featuring C-Shaped Vibration Inducers: The Role of the Fluid/Beam Interaction. *Applied Sciences*. 2023; 13(1):416.
https://doi.org/10.3390/app13010416

**Chicago/Turabian Style**

Sciortino, Giampiero, Valentina Lombardi, and Pietro Prestininzi. 2023. "Modelling of Cantilever-Based Flow Energy Harvesters Featuring C-Shaped Vibration Inducers: The Role of the Fluid/Beam Interaction" *Applied Sciences* 13, no. 1: 416.
https://doi.org/10.3390/app13010416