# Three-Dimensional Hydrodynamic Analysis of a Flexible Caudal Fin

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Geometric Model

#### 2.2. Computational Domain and Boundary Conditions

^{®}Core™ i5-5257U CPU 2.70 GHz × 4 processor and it took about 22 h for the calculation of seven flapping cycles in the lower frequency range.

## 3. Results

#### 3.1. Validation and Grid Independence Study

#### 3.2. Hydrodynamic Analysis

#### 3.2.1. Thrust Production

#### 3.2.2. Pressure Distribution and Vortex Structure

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic view of a square flexible plate. The leading edge is indicated by the red line and the trailing edge is indicated by the blue line.

**Figure 2.**(

**a**) Schematic view of a computational domain. (

**b**) Grid generation. The leading edge and symmetry plane of the plate are indicated by red and yellow lines, respectively. The origin of coordinate is at the center of the leading edge of the plate.

**Figure 3.**Comparison between the simulation and experiment for different: (

**a**) Trailing edge amplitude ${A}_{t}$. (

**b**) Phase lag of the plate ${\varphi}_{lag}$.

**Figure 5.**(

**a**) Mean thrust. (

**b**) The maximum relative deformation of trailing edge ${\delta}_{max}$ for different non-dimensional frequencies ${f}^{*}$.

**Figure 6.**Displacement of the trailing edge relative to the leading edge, ${z}_{t}-{z}_{r}$, for different ${f}^{*}$ in a flapping cycle.

**Figure 9.**Pressure distributions on the spanwise symmetry plane of the plate during a half-flapping cycle for the case of ${f}^{*}$ = 0.93 (case 1).

**Figure 10.**Three-dimensional vortex structures around the plate during a half-flapping cycle for the case of ${f}^{*}$ = 0.93 (case 1).

**Figure 11.**(

**a**) Pressure distributions on the spanwise symmetry plane of the plate and (

**b**) three-dimensional vortex structures at the phase instant when a peak value of thrust appears in each case.

**Figure 12.**Pressure distributions on the; (

**a**) top and (

**b**) bottom surfaces of the plate at a phase instant when a peak value of positive thrust appears in each case (The symmetry plane and leading edge are represented by a black line and red line, respectively).

**Figure 13.**Pressure distributions on the spanwise symmetry plane of the plate at a phase instant when a peak value of negative thrust appears for the case of: (

**a**) ${f}^{*}$ = 0.93. (

**b**) ${f}^{*}$ = 1.2. (

**c**) ${f}^{*}$ = 1.47.

**Figure 14.**(

**a**) Front view and (

**b**) bottom view of three-dimensional vortex structures at the phase instant when a peak value of thrust appears for the cases of; ${f}^{*}$ = 0.93, ${f}^{*}$ = 1.20 and ${f}^{*}$ = 1.47.

${\mathit{f}}^{*}$ | Experiment | Simulation |
---|---|---|

0.93 | 6.54 mm | 6.62 mm |

1.20 | 5.88 mm | 5.96 mm |

1.47 | 4.98 mm | 5.10 mm |

${\mathit{f}}^{*}$ | Experiment | Simulation |
---|---|---|

0.93 | 0.456 × $\pi $ rad | 0.440 × $\pi $ rad |

1.20 | 0.630 × $\pi $ rad | 0.620 × $\pi $ rad |

1.47 | 0.750 × $\pi $ rad | 0.716 × $\pi $ rad |

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**MDPI and ACS Style**

Khin, M.H.W.; Obi, S.
Three-Dimensional Hydrodynamic Analysis of a Flexible Caudal Fin. *Appl. Sci.* **2022**, *12*, 12693.
https://doi.org/10.3390/app122412693

**AMA Style**

Khin MHW, Obi S.
Three-Dimensional Hydrodynamic Analysis of a Flexible Caudal Fin. *Applied Sciences*. 2022; 12(24):12693.
https://doi.org/10.3390/app122412693

**Chicago/Turabian Style**

Khin, May Hlaing Win, and Shinnosuke Obi.
2022. "Three-Dimensional Hydrodynamic Analysis of a Flexible Caudal Fin" *Applied Sciences* 12, no. 24: 12693.
https://doi.org/10.3390/app122412693