#
Aerodynamics of a Wing with a Wingtip Flapper^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Model and Methods

^{2}) bonded to the spar with a rubber cement. The total wing model span is 320 mm and the total area is S = 24,225 mm

^{2}. The flapper area is 1/5th of the total wing area.

_{2}-axis is parallel to the x-axis and passes through the point 3 in the middle of the leading edge spar. Point 4 belongs to the trailing edge. Video images were digitized by employing a direct linear transformation method and software [13]. This procedure provides all three coordinates of marked points of the flapper.

_{2}-axis.

## 3. Results and Discussions

#### 3.1. Kinematic Parameters and Aerodynamic Coefficients of Wing Model

_{L}and C

_{D}are shown in Figure 6 and Figure 7, respectively. In order to illustrate variations in the aerodynamic coefficients, error bars were plotted in the figures based on standard deviation at each flapping frequency.

_{L}/C

_{D}, is an important performance measure, specifically, of a flight range of an aircraft. The results given in Table 1 show that the lift-to-drag ratio increases with a flapping frequency increase. In the present work, a conventional wing case was obtained by fixing a flapper horizontally. The lift-to-drag ratio for the conventional wing is denoted in the table by the zero frequency case. The wing with the wingtip flapper surpasses the conventional wing on average by 60% or within a margin of 30–80% depending on frequency.

#### 3.2. Smoke-Wire Flow Visualization

#### 3.3. Near-field PIV Measurements

_{0}), at the time t

_{0}, when the vortex system left the flapper. Time histories of vortex center motion for all vortices are presented in Figure 11.

_{0}, when vortices shed from the flapper at x/c = 0 and the moment, t, when they reached the PIV measurements plane x/c. In the present study, the value Δt = t − t

_{0}is determined by using the PIV technique. Four small reflective markers were attached to the base and the tip of the flapper, and along the leading edge of the rigid part of the model. The markers located at x/c = 0 have been observed as bright spots on PIV images taken at a given station x/c. Tracking them and digitizing their positions allowed the determination of the time variation of the flapping angle of the leading edge spar, ${\phi}_{r}$(t). The obtained function is similar to the one presented in Figure 5. Simultaneously, the displacement of the center of the vortex FO in the vertical direction, z(t), was determined. For a given measurement plane, the time difference between time instants, corresponding to maxima of z(t) and of ${\phi}_{r}$(t), was calculated, which is approximately equal to the time delay Δt. Then, the flapping angle of the leading edge spar can be found as $\phi $ = $\phi $(t

_{0}) = ${\phi}_{r}$(t − Δt).

_{n}and v

_{s}, respectively. On the first half of the downstroke from φ = 44° DN to 9° DN, the normal component, v

_{n}, increases up to 7.2 m/s (Figure 14a,b). The maximum of v

_{n}reaches 6.1 m/s during the first half of the upstroke from φ = −34° UP to −17° UP (Figure 14c,d). In plots in Figure 14b,d it is seen that v

_{n}maxima are closer to FO and RO, respectively, where higher velocity magnitudes and gradients are observed. For all the four phases, the magnitude of v

_{s}is much smaller than that of v

_{n}, indicating the jet-like airflow.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

C_{L} | lift coefficient $[{C}_{L}=L/(0.5\rho {V}^{2}S)]$ |

C_{D} | drag coefficient $[{C}_{D}=D/(0.5\rho {V}^{2}S)]$ |

c | chord |

c_{m} | mean chord length |

D | drag force |

FI | secondary undulating vortex |

FO | primary undulating vortex |

f | flapping frequency, Hz |

k | reduced frequency, $k=\pi f{c}_{m}/V$ |

L | lift force |

m | tip of membrane flapper |

R_{c} | vortex core size, mm |

r | tip of rigid wing |

RI | inner part of vortex ring |

RO | outer part of vortex ring |

S | total wing area, including both fixed and flapping parts, mm^{2} |

t | time, s |

T | stroke period [T = 1/f], s |

UP | upstroke |

DN | downstroke |

V | tunnel velocity, m/s |

v | velocity component along y-axis, m/s |

v_{n} | velocity component of jet flow along n-axis, m/s |

v_{s} | velocity component of jet flow along s-axis, m/s |

w | velocity component along z-axis, m/s |

xyz | model-fixed frame of reference |

G | vortex circulation, m^{2}/s |

a | angle of attack, deg |

b | pitching angle, deg |

f | flapping angle of the leading edge spar, deg |

r | air density, kg/m^{3} |

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**Figure 4.**High-speed video and particle image velocimetry (PIV) setup with model, laser, and cameras.

**Figure 9.**Flow patterns in horizontal plane z = 0 for three time instants at flapping frequency of 20 Hz.

**Figure 13.**Average velocity fields at x/c = 2 (rigid part of the wing and the area swept by the leading edge spar are shown by dashed lines).

Frequency | 0 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{L}/C_{D} | 2.379 | 3.414 | 3.613 | 3.896 | 3.872 | 3.708 | 3.062 | 3.461 | 3.998 | 4.421 | 3.878 | 4.315 |

x/c | 2 | 3 | 5 | ||||||
---|---|---|---|---|---|---|---|---|---|

min | max | ave | min | max | ave | min | max | ave | |

FO | 0.051 | 0.210 | 0.113 | 0.039 | 0.217 | 0.116 | 0.011 | 0.142 | 0.066 |

FI | −0.023 | −0.100 | −0.050 | −0.006 | −0.086 | −0.047 | −0.002 | −0.046 | −0.012 |

RO | −0.020 | −0.176 | −0.068 | −0.006 | −0.163 | −0.063 | - | - | - |

RI | 0.016 | 0.094 | 0.049 | 0.010 | 0.053 | 0.028 | - | - | - |

x/c | 2 | 3 | 5 | ||||||
---|---|---|---|---|---|---|---|---|---|

min | max | ave | min | max | ave | min | max | ave | |

FO | 3.82 | 11.3 | 8.56 | 5.39 | 13.0 | 9.36 | 4.29 | 13.2 | 8.68 |

FI | 4.83 | 8.54 | 6.40 | 4.05 | 9.78 | 7.32 | 1.95 | 7.9 | 4.32 |

RO | 4.52 | 9.97 | 7.43 | 2.77 | 11.9 | 7.53 | - | - | - |

RI | 2.92 | 8.86 | 6.23 | 4.16 | 8.38 | 5.90 | - | - | - |

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**MDPI and ACS Style**

Zhao, L.; Shkarayev, S.; Su, E.
Aerodynamics of a Wing with a Wingtip Flapper. *Fluids* **2018**, *3*, 29.
https://doi.org/10.3390/fluids3020029

**AMA Style**

Zhao L, Shkarayev S, Su E.
Aerodynamics of a Wing with a Wingtip Flapper. *Fluids*. 2018; 3(2):29.
https://doi.org/10.3390/fluids3020029

**Chicago/Turabian Style**

Zhao, Longfei, Sergey Shkarayev, and Erlong Su.
2018. "Aerodynamics of a Wing with a Wingtip Flapper" *Fluids* 3, no. 2: 29.
https://doi.org/10.3390/fluids3020029