# High-Efficiency Can Be Achieved for Non-Uniformly Flexible Pitching Hydrofoils via Tailored Collective Interactions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Methods

## 3. Results

#### 3.1. Isolated Flexible Foil Performance

#### 3.2. Case I: Flexible Foils in In-Line Arrangements

#### 3.3. Case II: Flexible Foils in Staggered Arrangements

#### 3.4. Case III: Flexible Foils Pitching at Different Amplitudes

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Fully Rigid Foil Isolated Thrust and Efficiency

**Figure A1.**Isolated rigid foil (

**a**) thrust coefficient and (

**b**) efficiency as a function of dimensionless amplitude. The colors from blue to red are mapped from the lowest to highest frequencies as indicated in the legend.

## Appendix B. Weighted Average

## Appendix C. Normalized Collective Performance Metrics

## Appendix D. Leader and Follower Foil Performance

#### Appendix D.1. Flexible Foils in In-Line Arrangements

**Figure A2.**Normalized (

**a**,

**b**) thrust, (

**c**,

**d**) power and (

**e**,

**f**) efficiency of leader (left column) and follower (right column) in in-line arrangements as a function of synchrony and streamwise foil spacing.

#### Appendix D.2. Flexible Foils in Staggered Arrangements

**Figure A3.**Normalized (

**a**,

**b**) thrust, (

**c**,

**d**) power and (

**e**,

**f**) efficiency of the leader (left column) and the follower (right column) foils in staggered arrangements for a fixed streamwise spacing of ${X}^{*}=0.5$ as a function of synchrony and cross-stream foil spacing.

#### Appendix D.3. Flexible Foils Pitching at Different Amplitudes

**Figure A4.**Normalized (

**a**,

**b**) thrust, (

**c**,

**d**) power and (

**e**,

**f**) efficiency of leader (left column) and follower (right column) foils as a function of synchrony and follower-to-leader amplitude ratio for an in-line arrangement at $({X}^{*},{Y}^{*})=(0.5,0)$.

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**Figure 1.**The current study charts a path to exceeding 60% collective efficiency while pitching foil studies typically never break 40% efficiency. The markers are colored yellow for single rigid foil studies, blue for studies of a pair of interacting rigid foils, red for single flexible foil studies, and purple for our current study where we examine a pair of interacting flexible foils. Not all data was extracted from each study. The highest efficiency cases were chosen for two-dimensional foils and three-dimensional foils with $A\phantom{\rule{-4.27pt}{0ex}}R\approx 2$ [13,19,20,25,32,33,34,35].

**Figure 2.**(

**a**) Schematic of the water channel setup, (

**b**) single pitching mechanism, (

**c**) the flexible foils in a staggered arrangement, and (

**d**) photograph of a non-uniformly flexible foil.

**Figure 3.**(

**a**) Thrust coefficient and (

**b**) efficiency as a function of the dimensionless amplitude, ${A}_{0}^{*}$, for different pitching frequencies, f. The lowest to the highest frequencies within the range of $1\le f\le 1.5$ are mapped from blue to red.

**Figure 4.**Normalized collective (

**a**) thrust, (

**b**) power, and (

**c**) efficiency as a function of synchrony and streamwise spacing between the leader and follower foils.

**Figure 5.**Normalized collective (

**a**) thrust, (

**b**) power and (

**c**) efficiency as a function of synchrony and cross-stream spacing between the leader and follower.

**Figure 6.**Normalized collective (

**a**) thrust, (

**b**) power and (

**c**) efficiency as a function of synchrony and the follower-to-leader amplitude ratio for an in-line arrangement at $({X}^{*},{Y}^{*})=(0.5,0)$.

Case I | Case II | Case III | |
---|---|---|---|

${X}^{*}$ | 0.25–1.25 | 0.5 | 0.5 |

${Y}^{*}$ | 0 | 0–0.4 | 0 |

${R}_{{A}^{*}}$ | 1 | 1 | 1–1.48 |

${A}_{0,L}^{*}$ | 0.25 | 0.25 | 0.25 |

f [Hz] | 1.3 | 1.3 | 1.3 |

$\varphi $ | 0–2$\pi $ with $\pi /12$ increments | ||

U [m/s] | 0.094 | 0.094 | 0.094 |

**Table 2.**Time-averaged net thrust, power and drag coefficients, as well as propulsive efficiency of an isolated rigid foil at ${A}_{0}^{*}=0.25$ and $f=1.5$ Hz and an isolated flexible foil at ${A}_{0}^{*}=0.25$ and $f=1.3$ Hz. The reported values for the rigid and flexible foils are taken at their peak efficiency. $\pm (\xb7)$ represents the standard deviation calculated from 5 experimental trials.

Performance Coefficients at Peak Efficiency | |
---|---|

${C}_{T,\mathrm{iso}}^{\mathrm{rigid}}$ | $0.16\pm 0.01$ |

${C}_{P,\mathrm{iso}}^{\mathrm{rigid}}$ | $0.55\pm 0.006$ |

${C}_{D,\mathrm{iso}}^{\mathrm{rigid}}$ | $0.04\pm 0.009$ |

${\eta}_{\mathrm{iso}}^{\mathrm{rigid}}$ | $0.29\pm 0.018$ |

${C}_{T,\mathrm{iso}}$ | $0.25\pm 0.01$ |

${C}_{P,\mathrm{iso}}$ | $0.53\pm 0.001$ |

${C}_{D,\mathrm{iso}}$ | $0.056\pm 0.008$ |

${\eta}_{\mathrm{iso}}$ | $0.48\pm 0.018$ |

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**MDPI and ACS Style**

Kurt, M.; Mivehchi, A.; Moored, K.
High-Efficiency Can Be Achieved for Non-Uniformly Flexible Pitching Hydrofoils via Tailored Collective Interactions. *Fluids* **2021**, *6*, 233.
https://doi.org/10.3390/fluids6070233

**AMA Style**

Kurt M, Mivehchi A, Moored K.
High-Efficiency Can Be Achieved for Non-Uniformly Flexible Pitching Hydrofoils via Tailored Collective Interactions. *Fluids*. 2021; 6(7):233.
https://doi.org/10.3390/fluids6070233

**Chicago/Turabian Style**

Kurt, Melike, Amin Mivehchi, and Keith Moored.
2021. "High-Efficiency Can Be Achieved for Non-Uniformly Flexible Pitching Hydrofoils via Tailored Collective Interactions" *Fluids* 6, no. 7: 233.
https://doi.org/10.3390/fluids6070233