# Vortex Lattice Simulations of Attached and Separated Flows around Flapping Wings

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## Abstract

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## 1. Introduction

- Birdlike flapping, which involves mainly low flapping frequencies and mainly attached flow, although flow separation can be encountered under specific circumstances;
- Insect-like flapping, which involves high flapping frequencies and separated flow [1].

## 2. Experiments

- Low-pass filter the flap, pitch, lift and drag signals with a cut-off frequency of 3 Hz (twice the highest flapping frequency).
- Identify the positive zero crossings of the flap time response and use them to define start times and end times for ${n}_{c}$ complete cycles.
- Use linear interpolation at each start and end time to determine the exact time when $\gamma =0$. Resample the time vector at 64 equal intervals between the start and end time of each cycle.
- Interpolate linearly the flap, pitch, lift and drag signals at the re-sampled time instances for each cycle.
- Calculate the mean and standard deviation values for the flap, pitch, lift and drag signals at each re-sampled time instance over the number of cycles, ${n}_{c}$. Furthermore, calculate the mean period.
- The mean values of each signal at the 64 time instances constitute the cycle averaged signal.

## 3. Vortex Lattice Model

- the velocity due to the wing’s motion ${\mathbf{U}}_{ij}^{m}$,
- the velocity induced by the vorticity in the wake ${\mathbf{U}}_{ij}^{w}$ and
- the velocity induced by only the bound chordwise vortex segments, ${\mathbf{U}}_{ij}^{bc}$

## 4. Estimation of Separated Flow Aerodynamic Loads Using the Leishman–Beddoes Model

## 5. Pitch Leading

## 6. Pure Flapping

## 7. Pitch Lagging

## 8. Comparison of Mean Aerodynamic Loads

## 9. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

VLM | Vortex Lattice Method |

CFD | Computational Fluid Dynamics |

DSV | Dynamic Stall Vortex |

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**Figure 3.**Visualization of the wing cross-section undergoing pitch lagging and pitch leading motions (pitch angle not to scale): (

**a**) pitch lag; (

**b**) pitch lead.

**Figure 4.**Wind-on and wind-off flap and pitch angle signals for the NACA 6409 wing, $U=6$ m/s, $f=1.23$ Hz.

**Figure 5.**Wind-on and wind-off cycle averaged flap and pitch angle signals for the NACA 6409 wing, $U=6$ m/s, $f=1.23$ Hz.

**Figure 7.**Lift measurements and Kirchoff theory curve fits for the NACA 6409 airfoil: (

**a**) $Re=4\times {10}^{5}$; (

**b**) $Re=5\times {10}^{6}$.

**Figure 8.**Example of the convergence with time step $\Delta t$ for the aerodynamic loads predicted by the Katz method: (

**a**) Lift coefficient; (

**b**) drag coefficient.

**Figure 9.**Pitch leading, $U=6$ m/s, $f=0.79$ Hz, ${\theta}_{min}=-{4}^{\circ}$, ${\theta}_{max}={8}^{\circ}$: (

**a**) lift coefficient; (

**b**) drag coefficient.

**Figure 10.**Pitch leading, $U=6$ m/s, $f=1.23$ Hz, ${\theta}_{min}=-{10}^{\circ}$, ${\theta}_{max}={2}^{\circ}$: (

**a**) lift coefficient; (

**b**) drag coefficient.

**Figure 11.**Pitch leading, $U=14.8$ m/s, $f=1.23$ Hz, ${\theta}_{min}=-{8}^{\circ}$, ${\theta}_{max}={4}^{\circ}$: (

**a**) lift coefficient; (

**b**) drag coefficient.

**Figure 12.**Drag coefficients for pure flapping, $U=9.4$ m/s, $f=1.5$ Hz: (

**a**) ${\theta}_{0}=-{8}^{\circ}$; (

**b**) ${\theta}_{0}=-{4}^{\circ}$; (

**c**) ${\theta}_{0}={4}^{\circ}$; (

**d**) ${\theta}_{0}={8}^{\circ}$.

**Figure 13.**VLM results, pitch lagging, $U=9.4$ m/s, $f=1.23$ Hz, ${\theta}_{min}=-{5}^{\circ}$, ${\theta}_{max}={7}^{\circ}$: (

**a**) lift coefficient; (

**b**) drag coefficient.

**Figure 14.**VLM results with separated flow contributions, pitch lagging, $U=6$ m/s, $f=1.5$ Hz, ${\theta}_{min}=-{12}^{\circ}$, ${\theta}_{max}={0}^{\circ}$: (

**a**) lift coefficient; (

**b**) drag coefficient.

**Figure 15.**VLM results with separated flow contributions, pitch lagging, $U=6$ m/s, $f=1.5$ Hz, ${\theta}_{min}={4}^{\circ}$, ${\theta}_{max}={16}^{\circ}$: (

**a**) lift coefficient; (

**b**) drag coefficient.

**Figure 16.**Mean aerodynamic load coefficients for pitch lagging cases, $V=9.4$ m/s: (

**a**) lift coefficient, $f=1.0$ Hz; (

**b**) drag coefficient, $f=1.0$ Hz; (

**c**) lift coefficient, $f=1.23$ Hz; (

**d**) drag coefficient, $f=1.23$ Hz; (

**e**) lift coefficient, $f=1.5$ Hz; (

**f**) drag coefficient, $f=1.5$ Hz.

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**MDPI and ACS Style**

Lambert, T.; Abdul Razak, N.; Dimitriadis, G.
Vortex Lattice Simulations of Attached and Separated Flows around Flapping Wings. *Aerospace* **2017**, *4*, 22.
https://doi.org/10.3390/aerospace4020022

**AMA Style**

Lambert T, Abdul Razak N, Dimitriadis G.
Vortex Lattice Simulations of Attached and Separated Flows around Flapping Wings. *Aerospace*. 2017; 4(2):22.
https://doi.org/10.3390/aerospace4020022

**Chicago/Turabian Style**

Lambert, Thomas, Norizham Abdul Razak, and Grigorios Dimitriadis.
2017. "Vortex Lattice Simulations of Attached and Separated Flows around Flapping Wings" *Aerospace* 4, no. 2: 22.
https://doi.org/10.3390/aerospace4020022