# Performance of Semi-Active Flapping Hydrofoil with Arc Trajectory

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## Abstract

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## 1. Introduction

## 2. Description and Definition of Flapping Foil Propulsion

#### 2.1. Geometric Structure and Motion

_{z}denotes the fluid moment imposed on foil, N.m; and $\theta $ is the pitching angle, radian.

_{rel,}

_{0}), shown in Figure 1, the AoA of flapping foil at different times can be expressed as Equation (3). The maximum AoA in a period of motion can be expressed as ${\alpha}_{max}=max\left[\left|\alpha \left(t\right)\right|\right]$.

_{rel,}

_{0}), which can be obtained according to the movement of the swinging arm; ${V}_{A}$ is the speed of the hull, which is a constant value under a certain working condition.

_{A}used in this study is 0.2~4 m/s. The advance coefficient J of the semi-active flapping foil is about 0.67 to 15, and the corresponding Re is about 2 × 10

^{5}~9 × 10

^{5}. The rotational inertia ($I$) around the axis is related to the frequency ratio of the pitching motion of the flapping foil. In our previous studies [28], the effect of the resonance mechanism on the semi-active flapping foil performance was studied. The analysis of varying frequency ratio showed that the system resonance makes the foil deviate from the ideal angle of attack and the propulsion performance of the system declines or even loses propulsion entirely. When the frequency ratio is small, that is, the foil has a small moment of inertia, the self-pitching flapping foil can work well. A more suitable frequency ratio could be selected below 0.5. Considering that the effect on the flapping foil performance is very small when the frequency ratio is small [28], we set rotational inertia I = 0.001 Kg$\xb7$m

^{2}in all the examples described in this paper. Additionally, it has been verified that the performance of the flapping foil is almost not affected by rotational inertia near this value.

#### 2.2. Nondimensional Propulsive Indicators

_{A}) and the average thrust ($\overline{{F}_{x}}$). Therefore, the efficiency of the flapping foil is defined as the ratio of output power (${P}_{out}$) to input power (${P}_{in}$), which is expressed as Equation (9)

## 3. Computational Method and Validation

#### 3.1. Governing Equations

_{j}is the components of area vector $\overrightarrow{s}$; and $\rho $ is the density of water.

#### 3.2. Mesh and Method

#### 3.3. Validation

_{0}/c) is 0.75, and the pitching axis is one-third chord from the leading edge of the foil. The experimental and the calculated results for f = 1.2 Hz and a

_{max}= 20° are compared and shown in Figure 5. The comparison presents excellent agreement between the two results in terms of the propulsive efficiency (η) and the thrust coefficient (c

_{T}). The definition of thrust coefficient given by Read et al. [32] and Schouveiler et al. [33] is slightly different from that presented in this paper, and the specific definition can be referred to the description in the literature.

## 4. Results and Analysis

#### 4.1. Propulsive Efficiency and the Thrust Coefficient

#### 4.2. Angle of Attack Analysis

_{max}) of each working condition is analyzed in this part.

_{max}decreases with the advance coefficient, and with the increase in spring stiffness, the α

_{max}under the same working condition gradually increases. This is also easy to understand, because with the increase in the spring stiffness, the hydrodynamic moment that the spring can resist is increasing, and the AoA increases with the increase in the hydrodynamic moment. As a result, the AoA of the flapping foil system is increasing. At the same time, the increase in swing arm length also increases the α

_{max}under the same working condition. In order to analyze the relationship between α

_{max}and KT, Figure 9 also depicts the α

_{max}trajectory corresponding to the maximum thrust coefficient, as shown by the black solid line. It is obtained by comparing with the results of thrust coefficient in Figure 8. From the trajectory of this black line, it can be found that at a low advance coefficient (J = 0.67), the α

_{max}corresponding to the maximum thrust coefficient is the smallest in all the different spring stiffness conditions, about 0.75 rad (the point A in the black solid line). That is, at a lower advance coefficient, the thrust generated by the flapping foil with smaller AoA is higher. However, at a higher advance coefficient (J ≈ 6), the highest thrust coefficient point has the maximum α

_{max}in all different spring stiffness conditions, about 0.3 rad (the point B in the black solid line). That is, under the condition of high advance coefficient, the thrust generated by the semi-active flapping foil with larger AoA is higher. For the intermediate advance coefficient conditions (points between point A and point B), the α

_{max}corresponding to the maximum thrust transits from a smaller value to a larger value step by step, which indicates that the α

_{max}corresponding to the maximum thrust coefficient is almost centered. That is, when the advance coefficient is at the middle value, α Too large or too small α

_{max}is not conducive to the generation of thrust. It can be understood that when the α

_{max}is too large, there is obvious separation phenomenon on the flapping foil surface, which will not produce effective lift, thus affecting the generation of thrust. When the α

_{max}is too small, the AoA decreases, the lift coefficient of the airfoil decreases, leading to reduced thrust. In addition, it can be observed that the jumping of the black solid line (α

_{max}trajectory corresponding to the maximum thrust coefficient) between different spring stiffness roughly occurs in the range ${\alpha}_{max}=0.3~0.4\mathrm{rad}$, which indicates that the maximum thrust tends to occur in the range of this AoA.

_{max}of $K=1.0\mathrm{N}.\mathrm{m}/\mathrm{rad}$ is greater than that of $K=2.0\mathrm{N}.\mathrm{m}/\mathrm{rad}$. In the last figure of Figure 9, the α

_{max}of $K=1.0\mathrm{N}.\mathrm{m}/\mathrm{rad}$ changes to the smallest one. It was found through inspection that this is due to a sudden increase in the AoA time-history curves at the low speed point under these conditions. Taking the working condition $L/c=5.0$ as an example, Figure 10 shows the time-history curve of AoA at low advance coefficient point ($J=0.67)$. It can be seen that, unlike the approximate sinusoidal AoA time-history curve of the conventional fully active flapping foil, the AoA time-history curve of this semi-active flapping foil driven by the swing arm is trapezoidal. This kind of trajectory has little effect on the maximum efficiency value of flapping foil, while it can maintain high efficiency with a relatively wider working range [16]. With the increase in spring stiffness, the AoA of the foil increases gradually. Whereas, in the process of direction changing of AoA, a convex peak appears on the curve of $K=1.0\mathrm{N}.\mathrm{m}/\mathrm{rad}$, which leads to the change in the maximum AoA, as shown in Figure 9. We speculate that the reason for the AoA jump-like phenomenon may be that when the advance speed V

_{A}is too small, the relative velocity angle of the foil changes abruptly.

#### 4.3. Analysis of Vortex Structure

## 5. Conclusions

- The influence of arm length and spring stiffness on the performance of the semi-active flapping foil is very clear. Increasing the length of the swing arm is beneficial to improving the peak efficiency of this semi-active flapping foil with circular-arc trajectory. At the same swing arm length, reducing the spring stiffness is also conducive to improving the peak efficiency of the flapping foil. The analysis of the maximum angle of attack shows that there is a definite corresponding relationship between the maximum angle of attack and the peak efficiency. For the flapping foil with the small aspect ratio NACA0012 airfoil structure, its peak efficiency is usually concentrated near ${\alpha}_{max}=0.2\mathrm{rad}$, and it can maintain high efficiency within a certain range of ${\alpha}_{max}$.
- The influencing factors of the thrust coefficient of the semi-active flapping foil propulsion are complex. The length of the swinging arm, the spring stiffness, and the advance coefficient can all have a significant impact on the thrust coefficient of the flapping foil. On the whole, compared with the conventional semi-active foil, the elliptical trajectory system also has a large thrust coefficient at low advance coefficient. The curve of the thrust coefficient decreases monotonically with the increase in advance coefficient when the spring stiffness is small. Under the condition of high spring stiffness, there is a peak value and a valley value of the thrust coefficient. According to the analysis of the flow field, the reason for the thrust valley may be the combined effect of the swing angle and the angle of attack. At the valley point of the thrust coefficient, the angle of attack of the flapping foil is large, so the vortex separation of foil is significant and the lift value is low. At the same time, the swing angle is small, so the contribution of lift to the thrust is low, which leads to the appearance of the thrust valley. In addition, by comparing the thrust coefficient and the maximum angle of attack ${\alpha}_{max}$, it is found that too large or too small ${\alpha}_{max}$ is unfavorable to the thrust under the working condition of the intermediate advance stage, and the peak thrust tends to appear in the range ${\alpha}_{max}=0.3~0.4\mathrm{rad}$.
- The flow-field analysis of the low aspect ratio airfoil shows that the vortex rings are interlocked in the wake field of the flapping foil at a low advance speed. With the increase in advance coefficient, the vortex rings are gradually lengthened first, and then separated from each other. When the advance coefficient is further increased, the vortex ring is split into a tip vortex and separated vortex on the airfoil surface. From the reverse analysis, the vortex ring is the result of the tip vortex and the separated vortex on the airfoil surface sticking together, while the vortex ring interlocking is formed by the compression of the vortex ring in space.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 5.**Comparisons of the propulsive efficiency η and the thrust coefficient c

_{T}with previous experimental results for a

_{max}= 20°.

**Figure 9.**Maximum AoA with different spring stiffness and swing arm length. (black solid line is α

_{max}trajectory corresponding to the maximum thrust coefficient).

**Figure 10.**AoA time-history curve of the flapping foil with different spring stiffness for $L/c=5.0,J=0.67$.

**Figure 13.**The flow field and vortex structure of a flapping foil with different advance coefficients ($\frac{L}{c}=5.0,K=10.0\mathrm{N}.\mathrm{m}/\mathrm{rad}$).

Symbol | Units | Definition |
---|---|---|

V_{A} | m/s | speed of the hull |

T | s | period of the flapping hydrofoil |

L/c | m | arm length |

c | m | chord length of the flapping hydrofoil |

H | m | spanwise of the flapping hydrofoil |

R | m | radius of round corners designed at both ends of the span direction of the flapping hydrofoil |

β | radian | swing angle of the swing arm |

f | Hz | swing frequency of the swing arm |

θ | radian | pitching angle of the flapping hydrofoil |

V_{rel,}_{0} | m/s | pitching center speed of the foil relative to the hull |

V_{x}_{0},V_{y}_{0} | m/s | velocity components of V_{rel,0} |

F_{x}_{0},F_{y}_{0} | m | X-direction force, Y-direction force exerted by the swing arm on the pitching center of flapping foil, respectively |

M_{z}_{0} | N m | torque exerted by the swing arm on the pitching center of flapping foil |

M_{z} | N m | fluid moment imposed on foil |

K | Nm/rad | torsion spring stiffness |

I | Kg m^{2} | rotational inertia about the axis of the flapping hydrofoil considering the attached water |

$\overline{{F}_{x}}$ | N | average thrust in the forward direction |

α (AoA) | radian | angle of attack of flapping hydrofoil |

St | St number of flapping hydrofoil | |

J | advance coefficient | |

KT | thrust coefficient | |

η | propulsive efficiency of flapping hydrofoil |

Direction | Hydrofoil Surface Region | Refinement Area |
---|---|---|

spanwise/Z | $H/128$ | $c/64$ |

chordwise/X | $c/128$ | |

normal direction/Y | $c/128$ |

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**MDPI and ACS Style**

Zhou, J.; Yan, W.; Mei, L.; Shi, W. Performance of Semi-Active Flapping Hydrofoil with Arc Trajectory. *Water* **2023**, *15*, 269.
https://doi.org/10.3390/w15020269

**AMA Style**

Zhou J, Yan W, Mei L, Shi W. Performance of Semi-Active Flapping Hydrofoil with Arc Trajectory. *Water*. 2023; 15(2):269.
https://doi.org/10.3390/w15020269

**Chicago/Turabian Style**

Zhou, Junwei, Wenhui Yan, Lei Mei, and Weichao Shi. 2023. "Performance of Semi-Active Flapping Hydrofoil with Arc Trajectory" *Water* 15, no. 2: 269.
https://doi.org/10.3390/w15020269