# Quasi-Steady versus Navier–Stokes Solutions of Flapping Wing Aerodynamics

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2})–O(10

^{4}), as opposed to O(10

^{6}) or higher for more conventional aircraft. Operating in this regime introduces new challenges associated with aerodynamic efficiency and lift generation, while their small size makes them particularly susceptible to environmental factors leading to challenges associated with stability and control [3,4,5].

## 2. Materials and Methods

#### 2.1. Case Setup and Wing Kinematics

_{2}[31], i.e., U = 2πfZ${\hat{r}}_{2}$R, where f is the stroke frequency, Z is the half peak-to-peak stroke amplitude, and ${\hat{r}}_{2}$ = 0.57 is the spanwise location of the center of the second moment of wing area, normalized by the wing length R. The aspect ratio AR = R/c = 3.02 was determined by using an actual fruit fly wing to create a 3D model as shown in Section 2.2.2. With the Reynolds number held constant at 100, the stroke frequency f and amplitude Z are dependent on each other. The reduced frequency is defined as k = 2πfc/(2U) = (2Z${\hat{r}}_{2}$AR)

^{−1}. For this study, the stroke amplitude is kept constant at Z = 49.3°, so that the reduced frequency is applicable to a fruit fly flight k = 0.33 [3].

_{d}and α

_{u}are the forward and return pitch amplitudes, respectively. In this study the pitch amplitude is A = α

_{d}= α

_{u}. The parameter Δτ

_{r}is a non-dimensional number that determines the duration of the pitch rotation in terms of the period. When the pitching duration is small the pitching motion is rapid and is confined to the end of each stroke. When the pitching duration is Δτ

_{r}= 0.5, the pitching motion is as slow as it can be because it lasts the entire stroke. Depending on Δτ

_{r}, the duration of the pitch the pitching waveform can vary from sinusoidal (Δτ

_{r}= 0.5) to a square wave when Δτ

_{r}approaches 0. The parameters τ

_{1}, τ

_{2}, and τ

_{3}determine the timing of the rotation. Since only symmetric rotation is considered in this study

_{,}τ

_{1}= Δτ

_{r}/2, τ

_{2}= τ

_{1}+ Δτ

_{r}/2, and τ

_{3}= 0.5 − Δτ

_{r}/2. The pitching motion is imposed on the pitch axis location x

_{pa}, which is normalized by c and measured from the leading edge. When x

_{pa}= 0.0, the pitch axis is at the leading edge and when x

_{pa}= 1 the pitch axis is at the trailing edge. Figure 1b shows a diagram of the pitching motion for a half-stroke for A = 45°, x

_{pa}= 0.5, and Δτ

_{r}= 0.2.

_{pa}= 0, 0.25, 0.5, 0.75, 1, and the duration of the pitch Δτ

_{r}= 0.2, 0.3, 0.5. The study focuses on the pitching kinematics because the angle of attack has been shown to greatly affect the aerodynamic performance [4]. There are 45 combinations for these values, which are simulated for each of the aerodynamic models. A case by case breakdown for all 45 cases is shown in Figure 1d and Table A1 in Appendix A.

#### 2.2. Aerodynamic Models

- A 3D Navier–Stokes equations solution of the rigid wing motion.

_{L}, drag C

_{D}, and moment about the pitching axis (C

_{M}), defined as

#### 2.2.1. Quasi-Steady Aerodynamic Model

_{trans}, rotational lift from circulation F

_{rot}, and added mass force F

_{a}, to the aerodynamic force on the wing. Additional forces generated via wake capture are omitted from the model.

_{t}is the wing’s instantaneous velocity.

_{rot}is C

_{rot}= C

_{rot,theo}= π [18]. However, Sane and Dickinson [18] show that C

_{rot}depends on the nondimensional rotational velocity $\widehat{\mathsf{\omega}}$ = ωc/U

_{tip}, where ω is the absolute angular velocity of the wing about its pitch axis and U

_{tip}is the reference velocity at the wing tip. Based on their experiments, an empirical fit was proposed as C

_{rot}= C

_{rot,exp}= (−11.77$\widehat{\mathsf{\omega}}$ + 0.8152)(0.75 − x

_{pa}). In this study, we use C

_{rot}

_{,exp}to calculate the rotational force and quantify its benefit by comparing the resulting force against the values obtained with C

_{rot}

_{,theo}in Section 3.

_{trans}and F

_{rot}, are applied at the quarter chord, although these assumptions may be over-simplified as recent studies [27,33] have shed more light on the location of the center of pressure in flapping wing motions. There is a residual pitching moment that arises due to wing rotation given by Leishman [34], which is

#### 2.2.2. Navier–Stokes Equation Model

^{2}), similar to the present study. Additionally, Lian et al. [38] and Kamakoti and Shyy [39] used it for Re = O(10

^{4}). Appendix B demonstrates low Re motion being simulated with sufficient accuracy.

#### 2.2.3. Spatial and Temporal Sensitivity Study

_{a}= 1.5, Δτ

_{r}= 0.5, x

_{pa}= 0.5, α

_{d}= α

_{u}= 45° (case 26 in Table A1). The medium grid (23 × 46 × 92) with 631,800 number of cells and 480 time steps per period yields sufficiently converged solution for lift as shown in Figure 3 and Table 1.

## 3. Results and Discussion

#### 3.1. Aerodynamic Response under the Three-Dimensional Pitch-Flap Motion

_{pa}, A, and Δτ

_{r}affect the resulting C

_{L}, C

_{D}, and C

_{M}under the 3D pitch-flap motion given by Equations (1) and (2).

_{L}in the design space of x

_{pa}, A, and Δτ

_{r}. The middle pitching amplitude α = 45° has the largest C

_{L}at x

_{pa}= 0 and Δτ

_{r}= 0.2 (rapid rotation). For x

_{pa}> 0.5, larger lift is seen at lower A and, therefore, at higher AoAs (AoA is roughly 90°–α). When the pitching axis is close to the leading edge, a longer pitching duration reduces lift. On the other hand, when x

_{pa}> 0.5, the lift is nearly insensitive to Δτ

_{r}. The effects of the pitching duration are the most noticeable when the pitch axis is near the trailing edge at A = 60°. The lowest lift is found in this region when A = 60°, x

_{pa}= 1, and Δτ

_{r}= 0.2.

_{r}< 0.3, and the pitching axis is at the leading or trailing edge have the most effect on the lift coefficient. On the other hand, the lift response is mild when the pitch axis is at the midchord x

_{pa}= 0.5 and the duration is relatively long Δτ

_{r}≥ 0.3. To analyze the flapping wing physics further, we consider three parametric studies where a single design parameter is changed at extreme and mild motions as summarized in Table 2.

#### 3.1.1. Pitching Amplitude Trends

_{r}= 0.3 and x

_{pa}= 0.5. Figure 5b shows a case with extreme pitching motion with kinematics Δτ

_{r}= 0.2 and x

_{pa}= 1.0.

_{r}and pitch axis x

_{pa}at the trailing edge) show the lift coefficient has much larger peaks during the ends of the strokes than seen in the less extreme case in Figure 5a. The difference in the lift during the midstroke is much smaller than during the ends of the strokes.

#### 3.1.2. Pitching Duration Trends

_{r}= 0.5 implies that the pitching takes place over the entire stroke. As the pitching duration shortens, i.e., as the wing pitches faster, the lift coefficient at the beginning and end of the stroke shows larger peaks in Figure 8a. Then at the middle of the stroke (t/T = 0.25) the lift coefficient has similar magnitudes. This trend is magnified for the rapid pitch extreme motions in Figure 8b with much larger lift peaks. The drag and moment coefficients show similar trends as the lift coefficient.

_{r}= 0.2 shows slightly larger iso-Q-surface. The larger vortical structures at the stroke ends due to quicker rotation are consistent with the higher force magnitudes in Figure 8.

#### 3.1.3. Pitching Axis Trends

_{pa}= 0.0, to the trailing edge, x

_{pa}= 1.0. Figure 10a shows a case with a relatively small pitching motion with A = 30° and Δτ

_{r}= 0.3. Figure 10b shows a case with large, rapid pitching motion with A = 60° and Δτ

_{r}= 0.2.

_{pa}= 0.5, the leading-edge vortex is much smaller and can be found only near the wing tip (Figure 11b). Also, as the pitching axis moves from the leading edge to trailing edge, the wingtip vortex becomes stronger. The reverse is true for the wing root vortex, which is larger when pitching axis is at the leading edge.

#### 3.2. Assessment of the QS Model

_{pa}is away from 0.5, either the leading edge (LE) or trailing edge (TE) will experience the largest velocity due to rotation. For symmetric rotations, which we employ in this study, the LE will experience a greater relative flow velocity when x

_{pa}= 1 then vice versa, because the LE rotates into the flow, whereas the TE rotates away from the flow.

_{pa}= 1.0, with the smallest typically being at x

_{pa}= 0.5 or x

_{pa}= 0.25. Secondly, as the pitching amplitude increases we see an increase in differences in cycle-averaged lift coefficient. Thirdly, a longer pitching duration leads to a smaller difference in cycle-averaged lift coefficient.

#### 3.2.1. Largest Difference Motion: Case 39

_{pa}= 1.0, A = 60°, and Δτ

_{r}= 0.2. The NSe model predicts a lift coefficient of −0.59, whereas the QS model overpredicts the lift coefficient as 1.11.

_{rot}

_{,theo}= π worsens the accuracy significantly during this portion of the stroke. This significant inaccuracy is likely due to the fact that the empirical terms in the Sane and Dickenson QS model [18] are obtained for x

_{pa}≤ 0.66, where the present study considers x

_{pa}≤ 1. In addition to this assumption, the range of dimensionless rotational velocity ($\hat{\mathsf{\omega}}$ in C

_{rot}of Equation (5)) ranges from 0.166 to 0.374 in Sane and Dickinson’s QS formulation [18]. However, the present study includes a range of $\hat{\mathsf{\omega}}$ between 0.5 and 1.75 (which is close to the range typical of insects [16]), resulting in values completely outside that used by Sane and Dickinson [18]. Considering the fact that the rotational velocity scales the rotational forces quadratically, it is unsurprising that this term creates large discrepancies between models, especially at such high values. Additionally, a more accurate rotational drag term could be used in the model, such as the formulations presented in recent studies [27,28]. Since the rotational drag term is highly dependent on the rotational axis, it is expected that improved accuracy would rectify the differences between the QS and NSe solutions even at more extreme pitch axis locations.

_{pa}= 1.0 results in a large drag force being produced, overpredicting the NSe solutions significantly. At the beginning of the forward stroke, the drag is almost equally composed of the translational and rotational components. Based on the lift comparison, we argue that the inaccuracy of the rotational lift causes the large difference observed in this interval. As the pitching ends, the pitching angle is constant and the drag is made up entirely of the translational component. A closer agreement exists in this phase.

#### 3.2.2. Smallest Difference Motion: Case 22

_{pa}= 0.5, A = 30°, and Δτ

_{r}= 0.3. This motion shows a difference in the cycle-averaged lift of only 0.01 between QS and NSe.

_{r}= 0.3, the wing does not pitch as quickly. This motion is directly seen in the QS lift coefficient which is mostly sinusoidal. The QS shows a peak at the middle of the forward stroke and low values at stroke transition. The NSe model also predicts lowest values at the stroke transitions. However, the lift and drag time histories consist of multiple peaks. The first peak in the NSe lift is associated with wake capture, which a QS model cannot capture, and the second with the delayed stall due to the LEVs [40].

_{rot,exp}= 3.14, which is close to the theoretical value of C

_{rot,theo}= π [18]. It should also be noted that the added mass contribution in both Figure 13 and Figure 16 are not symmetric between the upstroke and downstroke. It is, however, expected that the added mass contribution would be symmetric between the two strokes when symmetric kinematics are implemented. It appears that this asymmetry is a result of the QS model’s formulation of the added mass force (Equation (7)) which is dependent on the sign convention used in defining the wing motion. This is demonstrated by Sane and Dickinson’s plots of lift due to added mass, which show slight asymmetries as well [18].

## 4. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Case | α (°) | Δτ_{r} | x_{pa} | Case | α (°) | Δτ_{r} | x_{pa} | Case | α (°) | Δτ_{r} | x_{pa} |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 30 | 0.2 | 0 | 16 | 30 | 0.5 | 0.25 | 31 | 30 | 0.3 | 0.75 |

2 | 45 | 0.2 | 0 | 17 | 45 | 0.5 | 0.25 | 32 | 45 | 0.3 | 0.75 |

3 | 60 | 0.2 | 0 | 18 | 60 | 0.5 | 0.25 | 33 | 60 | 0.3 | 0.75 |

4 | 30 | 0.3 | 0 | 19 | 30 | 0.2 | 0.5 | 34 | 30 | 0.5 | 0.75 |

5 | 45 | 0.3 | 0 | 20 | 45 | 0.2 | 0.5 | 35 | 45 | 0.5 | 0.75 |

6 | 60 | 0.3 | 0 | 21 | 60 | 0.2 | 0.5 | 36 | 60 | 0.5 | 0.75 |

7 | 30 | 0.5 | 0 | 22 | 30 | 0.3 | 0.5 | 37 | 30 | 0.2 | 1 |

8 | 45 | 0.5 | 0 | 23 | 45 | 0.3 | 0.5 | 38 | 45 | 0.2 | 1 |

9 | 60 | 0.5 | 0 | 24 | 60 | 0.3 | 0.5 | 39 | 60 | 0.2 | 1 |

10 | 30 | 0.2 | 0.25 | 25 | 30 | 0.5 | 0.5 | 40 | 30 | 0.3 | 1 |

11 | 45 | 0.2 | 0.25 | 26 | 45 | 0.5 | 0.5 | 41 | 45 | 0.3 | 1 |

12 | 60 | 0.2 | 0.25 | 27 | 60 | 0.5 | 0.5 | 42 | 60 | 0.3 | 1 |

13 | 30 | 0.3 | 0.25 | 28 | 30 | 0.2 | 0.75 | 43 | 30 | 0.5 | 1 |

14 | 45 | 0.3 | 0.25 | 29 | 45 | 0.2 | 0.75 | 44 | 45 | 0.5 | 1 |

15 | 60 | 0.3 | 0.25 | 30 | 60 | 0.2 | 0.75 | 45 | 60 | 0.5 | 1 |

## Appendix B

_{L}> of 0.64, which is in sufficient agreement with Fry, who reports a <C

_{L}> of 1.06, and with Aono et al. [44] whose NSe simulation result in a <C

_{L}> of 0.58. For a more detailed description of the kinematics and computational setup, refer to Aono et al. [44].

**Figure A1.**Comparison of fruit fly (

**a**) lift coefficient C

_{L}and (

**b**) drag coefficient C

_{D}between Fry et al. [43], Aono et al. [44], and the present NSe solver. Note that the gray shaded region around the gray trace bounds the upper and lower force values for the experimental results of Fry et al. [43].

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**Figure 1.**Schematic of the (

**a**) wing geometry, (

**b**) orthographic view of the flapping wing motion, (

**c**) 2D view of the pitching motion for Δτ

_{r}= 0.2, and (

**d**) 3D design space with the design variables, pitch amplitude A, pitching duration Δτ

_{r}and pitching axis location x

_{pa}along the chord.

**Figure 3.**Time history of the lift coefficient for the (

**a**) spatial and (

**b**) time sensitivity studies during the third flapping period. Note, all NSe results reported are from the third flapping period.

**Figure 4.**NSe solutions of the cycle-averaged lift coefficient in the design space. (

**a**) Iso-surfaces of <C

_{L}> in the design space. (

**b**) Slices of <C

_{L}> in the design space at various x

_{pa}values.

**Figure 5.**Plots of the predicted lift, drag, and moment coefficient for cases with changing pitching amplitude, A = 30° (black), A = 45° (red) and A = 60° (blue). (

**a**) Mild motions with Δτ

_{r}= 0.3 and x

_{pa}= 0.5; (

**b**) Extreme motions with Δτ

_{r}= 0.2 and x

_{pa}= 1.0.

**Figure 6.**Contour plots of the NSe iso-surfaces of Q-criterion at Q = 0.75 for motions with Δτ

_{r}= 0.3, x

_{pa}= 0.5 and (

**a**) A = 30°, (

**b**) A = 45°, and (

**c**) A = 60°.

**Figure 7.**Contour plots of the NSe coefficient of pressure on the top and bottom of the wing at the middle of the forward stroke (t/T = 0.25) for Δτ

_{r}= 0.3 and x

_{pa}= 0.5; (

**a**) A = 30°, (

**b**) A = 45°, and (

**c**) A = 60°.

**Figure 8.**Plots of the predicted lift, drag, and moment coefficient for cases with changing pitching duration, Δτ

_{r}= 0.2 (black), Δτ

_{r}= 0.3 (red), and Δτ

_{r}= 0.5 (blue). (

**a**) Mild motion with A = 30° and x

_{pa}= 0.5; (

**b**) Extreme motion with A = 60° and x

_{pa}= 1.0.

**Figure 9.**Contour plots of the NSe iso-surfaces of Q-criterion at Q = 0.75 at t/T = 0.25 (

**top**) and t/T = 0.5 (

**bottom**) for motions with pitching duration, (

**a**) Δτ

_{r}= 0.2, (

**b**) Δτ

_{r}= 0.3, and (

**c**) Δτ

_{r}= 0.5; Mild motions with A = 30° and x

_{pa}= 0.5.

**Figure 10.**Plots of the NSe predicted lift, drag, and moment coefficient for cases with changing pitching axis, x

_{pa}= 0.0 (black), x

_{pa}= 0.25 (red), x

_{pa}= 0.5 (blue), x

_{pa}= 0.75 (purple), and Δτ

_{r}= 0.5 (green) (

**a**) cases with A = 30° and Δτ

_{r}= 0.3 (

**b**) cases with A = 60° and Δτ

_{r}= 0.2.

**Figure 11.**NSe iso-surfaces of Q-criterion at Q = 0.75 and colored with Y- vorticity for the following motions: (

**a**) x

_{pa}= 0, A = 30°, and Δτ

_{r}= 0.3 at t/T = 0.0, (

**b**) x

_{pa}= 0.5, A = 30°, and Δτ

_{r}= 0.3 at t/T = 0.0, and (

**c**) x

_{pa}= 1.0, A = 30°, and Δτ

_{r}= 0.3 at t/T = 0.0.

**Figure 12.**Contour plots of the differences in cycle-averaged lift coefficient between QS and NSe (<C

_{L}

_{,QS}> − <C

_{L}

_{,NSe}>) in the design space of the pitching axis (x

_{pa}), pitching duration (Δτ

_{r}), and pitching amplitude (A). (

**a**) Iso-surfaces of <C

_{L}

_{,QS}> − <C

_{L}

_{,NSe}> in the design space. (

**b**) Slices of <C

_{L}

_{,QS}> − <C

_{L}

_{,NSe}> in the design space at various x

_{pa}.

**Figure 13.**Plots of wing pitching and flapping for the smallest difference motion versus the largest difference motion. The largest difference motion is characterized by A = 60°, Z = 49.3°, Δτ

_{r}= 0.2, and x

_{pa}= 1.0. The smallest difference motion is characterized by A = 30°, Z = 49.3°, Δτ

_{r}= 0.3, and x

_{pa}= 0.5.

**Figure 14.**Plots of coefficients of lift (

**top**), drag (

**middle**), and moment (

**bottom**) over one stroke for QS (red) and NSe (blue) predictions for the largest difference motion. In the figure the cycle-averaged lift coefficient for the QS is 1.11 and the NSe is −0.59. A = 60°, Δτ

_{r}= 0.2 and x

_{pa}= 1.0.

**Figure 15.**Plots of the translational (black), added mass (red), rotational (blue) and theoretical rotational (blue dashed), added mass and rotational lift forces in the QS model for the largest difference motion. A = 60°, Δτ

_{r}= 0.2, and x

_{pa}= 1.0.

**Figure 16.**Plots of coefficients of lift (

**top**), drag (

**middle**), and moment (

**bottom**) over one period for QS (red) and NSe (blue) predictions for the small difference motion. In the figure the cycle-averaged lift coefficient for the QS is 0.83 and the NSe is 0.82. A = 30°, Δτ

_{r}= 0.3, and x

_{pa}= 0.5.

**Figure 17.**Plots of the translational (black), added mass (red), rotational (blue) and theoretical rotational (blue dashed) lift forces in the QS model for the smallest difference motion. A = 30°, Δτ

_{r}= 0.3, and x

_{pa}= 0.5.

**Table 1.**Spatial and temporal sensitivity for 3D grids. Five different meshes and five timestep sizes were included in the spatial and temporal sensitivity study. The medium grid (23 × 46 × 92) with 631,800 cells was chosen with 480 timesteps/period for the NSe simulations.

Cells | Timesteps/Period | Total Cells | <C_{L}> | L1-Norm | L2-Norm | |
---|---|---|---|---|---|---|

spatial | 10 × 20 × 40 | 480 | 46,930 | 0.7820 | 0.0518 | 0.0571 |

15 × 30 × 60 | 480 | 168,200 | 0.7667 | 0.0220 | 0.0257 | |

23 × 46 × 92 | 480 | 631,800 | 0.7543 | 0.0113 | 0.0152 | |

34 × 68 × 136 | 480 | 2,091,874 | 0.7543 | 0.0075 | 0.0094 | |

51 × 102 × 204 | 480 | 7,181,504 | 0.7574 | --- | --- | |

temporal | 23 × 46 × 92 | 60 | 631,800 | 1.0504 | 0.1243 | 0.2990 |

23 × 46 × 92 | 120 | 631,800 | 0.8705 | 0.1071 | 0.1830 | |

23 × 46 × 92 | 240 | 631,800 | 0.7874 | 0.0848 | 0.1032 | |

23 × 46 × 92 | 480 | 631,800 | 0.7543 | 0.0519 | 0.0448 | |

23 × 46 × 92 | 960 | 631,800 | 0.7336 | --- | --- |

A | Δτ_{r} | x_{pa} | Pitching Motion |
---|---|---|---|

30°, 45°, 60° | 0.3 | 0.5 | Mild |

30°, 45°, 60° | 0.2 | 1.0 | Extreme |

30° | 0.2, 0.3, 0.5 | 0.5 | Mild |

60° | 0.2, 0.3, 0.5 | 1.0 | Extreme |

30° | 0.3 | 0.0, 0.25, 0.5, 0.75, 1.0 | Mild |

60° | 0.2 | 0.0, 0.25, 0.5, 0.75, 1.0 | Extreme |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pohly, J.A.; Salmon, J.L.; Bluman, J.E.; Nedunchezian, K.; Kang, C.-k. Quasi-Steady versus Navier–Stokes Solutions of Flapping Wing Aerodynamics. *Fluids* **2018**, *3*, 81.
https://doi.org/10.3390/fluids3040081

**AMA Style**

Pohly JA, Salmon JL, Bluman JE, Nedunchezian K, Kang C-k. Quasi-Steady versus Navier–Stokes Solutions of Flapping Wing Aerodynamics. *Fluids*. 2018; 3(4):81.
https://doi.org/10.3390/fluids3040081

**Chicago/Turabian Style**

Pohly, Jeremy A., James L. Salmon, James E. Bluman, Kabilan Nedunchezian, and Chang-kwon Kang. 2018. "Quasi-Steady versus Navier–Stokes Solutions of Flapping Wing Aerodynamics" *Fluids* 3, no. 4: 81.
https://doi.org/10.3390/fluids3040081