# Unsteady Aerodynamic Design of a Flapping Wing Combined with a Bionic Wavy Leading Edge

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Aerodynamic Characteristics Analysis of Baseline Flapping Wing

#### 2.1. Flapping Wing Motion Model

**L**and aspect ratio is $AR=\frac{L}{c}$. The set flapping wing motion law is expressed as Equation (1) [35], where, $f$ is the flapping frequency, $\psi \left(t\right)$ represents the flapping motion, ${\psi}_{1}$ is the amplitude of the flapping motion, $\alpha \left(t\right)$ represents the pitching motion, ${\alpha}_{0}$ is the pre-installation angle of the wing, ${\alpha}_{1}$ is the amplitude of the pitching motion, $\varphi $ is the phase difference, and the center of the pitching motion is the quarter chord length of each section of the wing. Figure 2 shows the time history plot corresponding to Equation (1).

#### 2.2. Calculation Method and Verification

#### 2.3. Grid and Time Step Independence Verification

#### 2.4. Flow Field Structure Characteristics of Baseline Flapping Wing

## 3. Design of Wavy Leading Edge Flapping Wing

#### 3.1. Flow Control Mechanism of Wavy Leading Edge

#### 3.2. Characteristic Parameters of Wavy Leading Edge

#### 3.3. Grid Independence Verification

## 4. Numerical Results

#### 4.1. Effects of Different Wavelengths on Aerodynamic Performance of Flapping Wing

#### 4.2. Effects of Different Amplitudes on Aerodynamic Performance of Flapping Wing

#### 4.3. Sensitivity Analysis of Design Parameters

## 5. Conclusions

- The wavy leading edge separates the airflow, which reduces the pressure on the upper surface of the flapping wing, increases the pressure difference between the upper and lower surfaces, and increases the lift. Moreover, the different velocity distributions at the peak and trough also changes the aerodynamic performance of the flapping wing. The wavy leading edge wing loses part of the trust while gaining flapping lift.
- The calculated amplitude is $\overline{A}=0.05\u20130.2$, and the wavelength is $\overline{W}=0.01\u20130.1$. At the same amplitude, the larger the wavelength of the wavy leading edge flapping wing, the smaller the time averaged lift coefficient and the larger the time averaged thrust coefficient. At the same wavelength, the larger the amplitude, the larger the time averaged lift coefficient and the smaller the time averaged thrust coefficient.
- In order to obtain the maximum lift coefficient while losing the least thrust, the wavelength and amplitude should be selected as small as possible in the design, and the wavelength has the greatest influence on the time averaged lift coefficient. In the configuration calculated in this paper, the configuration 1–4 with wavelength $\overline{W}=0.01$ and amplitude $\overline{A}=0.05$ is the best. Compared with the straight wing, this configuration can increase the average lift coefficient by 32.86% and reduce the average thrust coefficient by 14.28%.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Grid and boundary conditions. (

**a**) Background grid; (

**b**) Component grid; (

**c**) Boundary layer and surface mesh.

**Figure 7.**The comparison of time history of lift and thrust coefficient. (

**a**) Lift coefficient; (

**b**) Trust coefficient.

**Figure 11.**Flipper’s front structure of humpback whale [41].

**Figure 14.**Construction method of wavy leading edge wing. (

**a**) Wing spanwise section of wavy leading edge wing; (

**b**) Airfoil of wavy leading edge wing.

**Figure 15.**Wavy leading edge flapping wing configuration. (

**a**) Configuration 1-1; (

**b**) Configuration 1-2; (

**c**) Configuration 1-3; (

**d**) Configuration 1-4; (

**e**) Configuration 2-1; (

**f**) Configuration 2-4; (

**g**) Configuration 1-3; (

**h**) Configuration 2-4; (

**i**) Configuration 3-1; (

**j**) Configuration 3-2; (

**k**) Configuration 3-3; (

**l**) Configuration 3-4.

**Figure 16.**The time history of lift and thrust coefficient (Configuration 11). (

**a**) Lift coefficient; (

**b**) Trust coefficient.

**Figure 17.**The time history of lift and thrust coefficient (Configuration 3-4). (

**a**) Lift coefficient; (

**b**) Trust coefficient.

**Figure 18.**Time history of lift coefficient in a flapping cycle. (

**a**) $\overline{A}=0.05$; (

**b**) $\overline{A}=0.1$; (

**c**) $\overline{A}=0.2$.

**Figure 19.**Time history of trust coefficient in a flapping cycle. (

**a**) $\overline{A}=0.05$; (

**b**) $\overline{A}=0.1$; (

**c**) $\overline{A}=0.2$.

**Figure 20.**Comparison of average lift coefficient and thrust coefficient. (

**a**) Time averaged lift coefficient; (

**b**) Time averaged trust coefficient.

**Figure 21.**Comparison of average lift coefficient of downstroke and upstroke. (

**a**) Time averaged lift coefficient of downstroke; (

**b**) Time averaged lift coefficient of upstroke.

**Figure 22.**Comparison of average thrust coefficient of downstroke and upstroke. (

**a**) Time averaged trust coefficient of downstroke; (

**b**) Time averaged trust coefficient of upstroke.

**Figure 23.**Pressure contours of the upper surface of flapping wing (Pa). (

**a**) t/T = 0; (

**b**) t/T = 0.2; (

**c**) t/T = 0.5; (

**d**) t/T = 0.8.

**Figure 24.**Pressure coefficient of 80% wingspan section. (

**a**) t/T = 1; (

**b**) t/T = 0.2; (

**c**) t/T = 0.5; (

**d**) t/T = 0.8.

**Figure 25.**Pressure contours and streamlines of 50% spanwise section of flapping wing at t/T = 0. (

**a**)Straight Wing; (

**b**) $\overline{W}=0.1$; (

**c**) $\overline{W}=0.05$; (

**d**) $\overline{W}=0.02$; (

**e**) $\overline{W}=0.01$.

**Figure 26.**Time history of lift coefficient in a flapping cycle. (

**a**) $\overline{W}=0.1$; (

**b**) $\overline{W}=0.05$; (

**c**) $\overline{W}=0.02$; (

**d**) $\overline{W}=0.01$.

**Figure 27.**Time history of trust coefficient in a flapping cycle. (

**a**) $\overline{W}=0.1$; (

**b**) $\overline{W}=0.05$; (

**c**) $\overline{W}=0.02$; (

**d**) $\overline{W}=0.01$.

**Figure 28.**Comparison of average lift coefficient and thrust coefficient. (

**a**) Time averaged lift coefficient; (

**b**) Time averaged trust coefficient.

**Figure 29.**Comparison of average lift coefficient of downstroke and upstroke. (

**a**)Time averaged lift coefficient of downstroke; (

**b**) Time averaged lift coefficient of upstroke.

**Figure 30.**Comparison of average thrust coefficient of downstroke and upstroke. (

**a**) Time averaged trust coefficient of downstroke; (

**b**) Time averaged trust coefficient of upstroke.

**Figure 31.**Pressure contours of the upper surface of flapping wing. (

**a**) t/T=0; (

**b**) t/T = 0.2; (

**c**) t/T = 0.5; (

**d**) t/T = 0.8.

**Figure 32.**Mach number contours of 50% spanwise section of flapping wing at t/T = 0. (

**a**) Straight Wing; (

**b**) $\overline{A}=0.05$; (

**c**) $\overline{A}=0.1$; (

**d**) $\overline{A}=0.2$.

Author | Object | Method | Reynolds Number | Aerodynamic Performance Change |
---|---|---|---|---|

Fish et al. [11] | NACA63 | Experiment | $6\times {10}^{6}$ | The stall angle of attack is delayed by more than 5°. |

Sudhakar et al. [12] | A high aspect-ratio UAV | Experiment | $1.8\times {10}^{5}$ $2.7\times {10}^{5}$ | The lift-drag ratio is increased by up to 25%. |

Sudhakar et al. [13] | NACA 4415 | Experiment | $1.2\times {10}^{5}$ | The separation bubble on the upper wing surface is obviously reduced. |

Wei et al. [14] | Swept conical SD7032 wing | Experiment | $2.2\times {10}^{5}$ | The stall attack angle is delayed by 2–4 degrees. |

Abdelrahman et al. [15] | S1223 | Simulation | $1\times {10}^{5}$ $3\times {10}^{5}$ $1.5\times {10}^{6}$ | No considerable difference occurs in lift and drag before the stall. |

Seyhan et al. [16] | NACA 0015 | Experiment | $6.3\times {10}^{4}$ | Lift coefficients have increased by at least 26.2% after stall angles. |

Gopinathan et al. [17] | NACA 0015NACA 4415 | Experiment/Simulation | $1.83\times {10}^{5}$ | The t/c ratio of the HW flipper is strategically reduced to 0.15. |

Torró et al. [18] | NACA0021 | Simulation | $1.2\times {10}^{5}$ | The lift coefficient increased by 0.064 and the drag coefficient decreased by 0.045. |

Ikeda et al. [19] | Serrated single-feather wing | Experiment | $5.9\times {10}^{3}$ $9.8\times {10}^{3}$ | The lift and lift-to-drag ratio decrease at AoAs < 15°, and the aerodynamic performance of the two is basically the same at AoAs > 15°. |

Ramachandiran et al. [20] | NACA 0015NACA 4415 | Simulation | $1.83\times {10}^{5}$ | The lift increases at the angle of attack of 11–15 degrees, and the stall angle of attack is delayed by about three degrees. |

Rostamzadeh et al. [30] | NACA 0021 | Simulation | $1.2\times {10}^{5}$ $1.5\times {10}^{6}$ | The lift loss is delayed at 18–25° angle of attack. |

Anwar et al. [33] | insect-like flapping wing | Simulation | 400 | The lift-drag ratio is reduced by 0.34%. |

Parameter | Value | Parameter | Value |
---|---|---|---|

Reynolds number Re | $1.2\times {10}^{5}$ | $\mathrm{Plunging}\mathrm{amplitude}{\psi}_{1}(\xb0)$ | 15 |

$\mathrm{Wing}\mathrm{chord}\mathrm{length}c(\mathrm{m})$ | 0.12 | $\mathrm{Plunging}\mathrm{motion}\mathrm{equation}(\xb0)$ | $\psi (t)=15\mathrm{cos}(16\pi t)$ |

Wing aspect ratio AR | 8 | $\mathrm{Pitching}\mathrm{amplitude}{\alpha}_{1}(\xb0)$ | 15 |

$\mathrm{Approaching}\mathrm{velocity}{U}_{\infty}(\mathrm{m}/\mathrm{s})$ | 15 | $\mathrm{Setting}\mathrm{angle}\mathrm{of}\mathrm{wing}{\alpha}_{0}(\xb0)$ | 0 |

Mach number Ma | 0.0435 | $\mathrm{Pitching}\mathrm{axis}\mathrm{location}(m)$ | 0.25c |

$\mathrm{Flapping}\mathrm{frequency}f(\mathrm{Hz})$ | 8 | $\mathrm{Phase}\mathrm{difference}\varphi (\xb0)$ | 90 |

Reduced frequency k | 0.2 | $\mathrm{Pitching}\mathrm{motion}\mathrm{equation}(\xb0)$ | $\alpha \left(t\right)=15\mathrm{cos}\left(16\pi t+90\right)$ |

Case | Number of Background Grid (w) | Number of Component Grids (w) | Total Number of Grids (w) | Total Faces (w) | Total Nodes (w) |
---|---|---|---|---|---|

Case1 | 70 | 562 | 632 | 3612 | 2565 |

Case2 | 135 | 706 | 841 | 4554 | 3150 |

Case3 | 289 | 836 | 1125 | 6908 | 4816 |

Configuration | $\overline{\mathit{A}}$ | $\overline{\mathit{W}}$ | Configuration | $\overline{\mathit{A}}$ | $\overline{\mathit{W}}$ | Configuration | $\overline{\mathit{A}}$ | $\overline{\mathit{W}}$ |
---|---|---|---|---|---|---|---|---|

1-1 | 0.05 | 0.1 | 2-1 | 0.1 | 0.1 | 3-1 | 0.2 | 0.1 |

1-2 | 0.05 | 0.05 | 2-2 | 0.1 | 0.05 | 3-2 | 0.2 | 0.05 |

1-3 | 0.05 | 0.02 | 2-3 | 0.1 | 0.02 | 3-3 | 0.2 | 0.02 |

1-4 | 0.05 | 0.01 | 2-4 | 0.1 | 0.01 | 3-4 | 0.2 | 0.01 |

$\overline{\mathit{W}}$ | 0.1 | 0.05 | 0.02 | 0.01 |
---|---|---|---|---|

$\mathrm{Time}\mathrm{averaged}\mathrm{lift}\mathrm{coefficient}\overline{{C}_{L}}$ | 16.01%↑ | 28.89%↑ | 34.32%↑ | 41.04%↑ |

$\mathrm{Time}\mathrm{averaged}\mathrm{trust}\mathrm{coefficient}\overline{{C}_{T}}$ | 11.66%↓ | 14.04%↓ | 20.71%↓ | 25.97%↓ |

$\overline{\mathit{A}}$ | 0.05 | 0.1 | 0.2 |
---|---|---|---|

$\mathrm{Time}\mathrm{averaged}\mathrm{lift}\mathrm{coefficient}\overline{{C}_{L}}$ | 24.34%↑ | 29.33%↑ | 36.52%↑ |

$\mathrm{Time}\mathrm{averaged}\mathrm{trust}\mathrm{coefficient}\overline{{C}_{T}}$ | 8.75%↓ | 15.53%↓ | 29.03%↓ |

Wavelength $\overline{\mathit{W}}$ | Amplitude $\overline{\mathit{A}}$ | |
---|---|---|

$\mathrm{Time}\mathrm{averaged}\mathrm{lift}\mathrm{coefficient}\overline{{C}_{L}}$ | −77.4% | 22.6% |

$\mathrm{Time}\mathrm{averaged}\mathrm{trust}\mathrm{coefficient}\overline{{C}_{T}}$ | 52.9% | −47.1% |

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## Share and Cite

**MDPI and ACS Style**

Bai, X.; Zhan, H.; Mi, B. Unsteady Aerodynamic Design of a Flapping Wing Combined with a Bionic Wavy Leading Edge. *Appl. Sci.* **2023**, *13*, 1519.
https://doi.org/10.3390/app13031519

**AMA Style**

Bai X, Zhan H, Mi B. Unsteady Aerodynamic Design of a Flapping Wing Combined with a Bionic Wavy Leading Edge. *Applied Sciences*. 2023; 13(3):1519.
https://doi.org/10.3390/app13031519

**Chicago/Turabian Style**

Bai, Xuan, Hao Zhan, and Baigang Mi. 2023. "Unsteady Aerodynamic Design of a Flapping Wing Combined with a Bionic Wavy Leading Edge" *Applied Sciences* 13, no. 3: 1519.
https://doi.org/10.3390/app13031519