# Hydrodynamic Efficiency Analysis of a Flexible Hydrofoil Oscillating in a Moderate Reynolds Number Fluid Flow

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

#### 2.1. Oscillating and Deformable Hydrofoil

#### 2.2. Fluids Dynamics Equations

#### 2.3. Structure Dynamics Equations

#### 2.4. Fluid-Structure Interaction Coupled Problem

## 3. Numerical Resolution

#### 3.1. Heaving Reference Frame Validation

#### 3.2. FSI Implicit Coupling Scheme Validation

#### 3.3. Mesh and Time Step Convergence Analysis

## 4. Results and Discussion

#### 4.1. Analysis of the Flow in the Wake of the Hydrofoil

#### 4.2. Flexibility Influence on the Hydrodynamic Forces

#### 4.3. Flexibility Influence on the Power Extraction Efficiency of the Hydrofoil

#### 4.4. Fluid Pressure and Vorticity Fields Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NACA | National Advisory Committee for Aeronautics |

LEV | Leading-edge vortex |

$\overline{a}$ | Mean value of a over one motion period |

$\mathit{u}=(u,v)$ | Flow velocity [m/s] |

${U}_{\infty}$ | Free-stream velocity [m/s] |

${\omega}_{z}$ | Vorticity in z-direction [1/s] |

p | Flow pressure [Pa] |

$\rho $ | Density [kg/m${}^{3}$] |

Re | Nombre de Reynolds [-] |

$\alpha $ | Angle of attack [°] |

h | Vertical position [m] |

T | Motion period [s] |

c | Hydrofoil chord length [-] |

${C}_{X}$ | Dimensionless x-projection of the hydrodynamic forces [-] |

${C}_{Y}$ | Dimensionless y-projection of the hydrodynamic forces [-] |

${C}_{M}$ | Dimensionless torque of the hydrodynamic forces [-] |

${C}_{P}$ | Dimensionless extracted power [-] |

$\eta $ | Efficiency [-] |

${f}^{\ast}$ | Dimensionless oscillating frequency [-] |

${t}^{\ast}$ | Dimensionless time [-] |

${\rho}_{s}$ | Mass density |

$\xi $ | Structure local displacements |

${E}_{Y}$ | Young’s modulus |

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**Figure 1.**Oscillating hydrofoil in power extraction regime. Both the heaving $h\left(t\right)$ and the pitching $\alpha \left(t\right)$ motions are forced.

**Figure 2.**Partly deformable hydrofoil geometry, fluid, and solid domains boundaries: ${\mathsf{\Omega}}_{LE}$ (gray) is non deformable and ${\mathsf{\Omega}}_{TE}$ (cyan) is deformable.

**Figure 5.**Instantaneous vorticity fields ${\omega}_{z}$ around the oscillating hydrofoil for the hydrofoils 1, 2, and 3, ${\alpha}_{0}=10$°, ${f}^{\ast}=0.025$, $\mathsf{\Delta}h=0.8$ m, and ${t}^{\ast}=0,\phantom{\rule{3.33333pt}{0ex}}0.25,\phantom{\rule{3.33333pt}{0ex}}0.5,\phantom{\rule{3.33333pt}{0ex}}0.75$.

**Figure 6.**Instantaneous vorticity fields ${\omega}_{z}$ around the oscillating hydrofoil 2, ${\alpha}_{0}=20,\phantom{\rule{3.33333pt}{0ex}}25$ and 30°, ${f}^{\ast}=0.025$, $\mathsf{\Delta}h=0.8$ m, and ${t}^{\ast}=0.75$.

**Figure 7.**Instantaneous vorticity fields ${\omega}_{z}$ around the oscillating hydrofoil for the hydrofoils 1, 2, and 3, ${\alpha}_{0}=40$°, ${f}^{\ast}=0.025$, $\mathsf{\Delta}h=0.8$ m, and ${t}^{\ast}=0,\phantom{\rule{3.33333pt}{0ex}}0.25,\phantom{\rule{3.33333pt}{0ex}}0.5,\phantom{\rule{3.33333pt}{0ex}}0.75$.

**Figure 8.**Hydrodynamic coefficients ${C}_{Y}$, ${C}_{M}$, and ${C}_{X}$ and hydrofoil vertical velocity $\dot{h}/{U}_{\infty}$ versus time, for ${\alpha}_{0}=$ 10°, 20°, 30°, and 40° and ${f}^{\ast}=0.025$, $\mathsf{\Delta}h=0.8$ m and for the three materials.

**Figure 9.**Mean hydrodynamic coefficient $\overline{{C}_{X}}$ versus pitching amplitude ${\alpha}_{0}$ for ${f}^{\ast}=0.025$, $\mathsf{\Delta}h=0.8$ m, and for the three materials.

**Figure 10.**Power coefficient ${C}_{P}$ and hydrofoil vertical velocity $\dot{h}/{U}_{\infty}$ versus time, for ${\alpha}_{10}$ to ${\alpha}_{40}$, ${f}^{\ast}=0.025$, $\mathsf{\Delta}h=0.8$ m, and for three materials.

**Figure 11.**Vorticity and pressure fields around the oscillating hydrofoils (1–3) for ${\alpha}_{0}=40$°, ${f}^{\ast}=0.025$, and $\mathsf{\Delta}h=0.8$ m, at times ${t}^{\ast}=0.0625,\phantom{\rule{3.33333pt}{0ex}}0.125,\phantom{\rule{3.33333pt}{0ex}}0.1875$, and 0.25.

**Table 1.**Mechanical characteristics of the hydrofoil material for the domain ${\mathsf{\Omega}}_{TE}$.

${\mathit{Mat}}_{1}$ | ${\mathit{Mat}}_{2}$ | ${\mathit{Mat}}_{3}$ | |
---|---|---|---|

Designation (−) | hydrofoil 1 | hydrofoil 2 | hydrofoil 3 |

Density ${\rho}_{s}$ (kg/m${}^{3}$) | 1420 | 1420 | 1420 |

Young’s Modulus ${E}_{Y}$ (GPa) | 1.0 | 0.1 | 0.01 |

Poisson’s Coefficient $\nu $ (−) | 0.35 | 0.35 | 0.35 |

Solvers | Cells | Time Steps/Cycle | $\u2329{\mathit{C}}_{\mathit{D}}\u232a$ | $\widehat{{\mathit{C}}_{\mathit{L}}}$ |
---|---|---|---|---|

Blackburn & Henderson [34] | 422 | 2000 | 1.414 | 1.776 |

Kinsey & Dumas [2] | 65,600 | 2000 | 1.412 | 1.755 |

Present | 75,000 | 2000 | 1.467 | 1.761 |

Relative deviations with [34] | - | - | 3.75% | 0.84% |

**Table 3.**Numerical and experimental hydrodynamic coefficients ${C}_{L}$ et ${C}_{D}$, angle of attack deviation $\mathsf{\Delta}\alpha $, and maximal strain ${d}_{y}$ for a 3D deformable cantilevered NACA0015. The relative deviations are based on the numerical results.

${\mathit{C}}_{\mathit{L}}$ (-) | ${\mathit{C}}_{\mathit{D}}$ (-) | $\mathbf{\Delta}\mathit{\alpha}$ (°) | ${\mathit{d}}_{\mathit{y}}$ [mm] | |
---|---|---|---|---|

Experiment [41] | 0.80 | 0.045 | 0.2 | 1.48 |

Numerical—Ansys [23] | 0.85 | 0.058 | 0.17 | 1.5 |

Numerical—OpenFoam | 0.74 | 0.058 | 0.22 | 1.35 |

Relative deviations with [41] | 7.5% | 22% | 10% | 8.5% |

**Table 4.**Power efficiency $\eta $ (%) for coarse to refine mesh for hydrofoil 1, ${f}^{\ast}=0.025$ and $\mathsf{\Delta}h=0.8$ m, as a function of pitching amplitude ${\alpha}_{0}$.

Mesh 1 | Mesh 2 | Mesh 3 | Mesh 4 | |
---|---|---|---|---|

${\alpha}_{0}=10$° | 0.189 | 0.220 | 0.219 | 0.219 |

${\alpha}_{0}=15$° | 2.26 | 2.26 | 2.27 | 2.27 |

${\alpha}_{0}=20$° | 3.91 | 3.72 | 3.75 | 3.75 |

**Table 5.**Power extraction efficiency $\eta $ (%) for ${f}^{\ast}=0.025$, $\mathsf{\Delta}h=0.8$ m, and for the three materials.

Hydrofoil 1 | Hydrofoil 2 | Hydrofoil 3 | |
---|---|---|---|

${\alpha}_{0}=10$° | 0.2197 | 0.2567 | 0.3166 |

${\alpha}_{0}=15$° | 2.279 | 2.442 | 2.878 |

${\alpha}_{0}=20$° | 3.746 | 4.026 | 4.374 |

${\alpha}_{0}=25$° | 5.088 | 6.119 | 6.070 |

${\alpha}_{0}=30$° | 8.258 | 8.655 | 13.72 |

${\alpha}_{0}=35$° | 12.87 | 15.75 | 21.66 |

${\alpha}_{0}=40$° | 13.99 | 17.84 | 20.45 |

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## Share and Cite

**MDPI and ACS Style**

Brousseau, P.; Benaouicha, M.; Guillou, S.
Hydrodynamic Efficiency Analysis of a Flexible Hydrofoil Oscillating in a Moderate Reynolds Number Fluid Flow. *Energies* **2021**, *14*, 4370.
https://doi.org/10.3390/en14144370

**AMA Style**

Brousseau P, Benaouicha M, Guillou S.
Hydrodynamic Efficiency Analysis of a Flexible Hydrofoil Oscillating in a Moderate Reynolds Number Fluid Flow. *Energies*. 2021; 14(14):4370.
https://doi.org/10.3390/en14144370

**Chicago/Turabian Style**

Brousseau, Paul, Mustapha Benaouicha, and Sylvain Guillou.
2021. "Hydrodynamic Efficiency Analysis of a Flexible Hydrofoil Oscillating in a Moderate Reynolds Number Fluid Flow" *Energies* 14, no. 14: 4370.
https://doi.org/10.3390/en14144370