# Physics of optimal hovering flapping flight

Florian Bouard, Thierry Jardin, Laurent David

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Florian Bouard, Thierry Jardin, Laurent David. Physics of optimal hovering flapping flight. 4th Colloquium on Vortex Dominated Flows – DisCoVor 2025, Jun 2025, Bozeman, United States. hal-05335335

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# Physics of optimal hovering flapping flight

Florian BOUARD<sup>a,b*</sup>, Thierry JARDIN<sup>a</sup> and Laurent DAVID<sup>b</sup>

<sup>a</sup> ISAE-SUPAERO, Université de Toulouse, France

<sup>b</sup> Institut PPrime, Université de Poitiers-ENSMA-CNRS, UPR 3346, France<sup>1,2</sup>

*Correspondence: florian.bouard@univ-poitiers.fr

## 1 Introduction

These past decades, flapping wings have fostered considerable attention by both zoologists and aerospace engineers with the goal of unraveling the physics behind bird and insect flight and develop insect-scale bio-inspired flying robots. Extensive research has revealed the role of kinematic parameters (e.g. angle-of-attack, flapping amplitude, phase between pitching and flapping motions, etc) on the aerodynamic performance of rigid wings. The main conclusion was that most natural flyers present large flapping amplitude (≈ 120°) and moderate (≈ 35°) angle-of-attack. Yet, recent observations on mosquitoes [1,2] suggest that smaller amplitude (≈ 45°) and larger angle-of-attack present large lift production despite the relative absence of a strong LEV (Leading Edge Vortex), commonly admitted as the principal reason allowing flapping flight. The LEV is a coherent structure developing on the wing's upper surface during a stroke of a flapping cycle which creates a low-pressure region, thus contributing to generating lift. For hovering flight with small stroke amplitude, lift generation is not yet fully understood. Specifically, there is no consensus on the underlying mechanism(s) responsible for the lift peak observed around mid-stroke for these kinds of kinematics. [1] attributed it to LEV attachment while [2] proposed the explanation of "fast-pitching-up rotation". In other words, the wing pitches up rapidly while moving forward generating opposite-sign vorticity at wing's trailing and leading-edges which results in large aerodynamic force [2]. Hence, numerical and experimental investigations are conducted on large and low flapping amplitude motions to characterize flow topology of both configurations. A Force Partitioning Method (FPM) is then used on experimental and numerical results to correlate flow topology (and/or added-mass effects) and lift production.

## 2 Methodology

The wing is an untwisted AR = 4 NACA0012 profile extruded in the radial direction with a chord length c = 0.01m. The flapping wing kinematic is composed of revolving sinusoidal and pitching Fourier series-based motions (Figure

<figure><img src="image_2.png" /><figcaption>**Figure 1:** Efficient (left) and lift-producing (right) kinematics illustration during a downstroke phase.</figcaption></figure>

1, top) with respective angular speeds $\dot{\phi}$ and $\dot{\alpha}$. The kinematics studied here are those introduced by [3] and presented at DisCoVor 2023. These kinematics have been optimized (in terms of mean lift coefficients $\bar{C}_L$ (Figure 1, bottom right) and efficiency $\bar{C}_L/\bar{C}_P$ (Figure 1, bottom left), $\bar{C}_P$ being the mean power coefficient) using a combination of neural networks and genetic algorithm. The efficient motion is relatively classic with typical flapping amplitude of 110° and moderate angle-of-attack during the stroke. The lift-producing motion consists of relatively small flapping amplitude (20°), large pitch angle with fast pitching variations during the majority of the stroke and advanced pitch motion with respect to the revolving one.

The numerical simulations are carried out through directly solving the 3D Navier-Strokes equations. The setup is similar to that used in previous studies from our group [3,4].

The experimental setup consists in time-resolved scanning tomography PIV measurements where the test volume (limited to about one third of the outboard region) is illuminated with consecutive laser sheets. It is similar to that used in previous flapping flight investigations [5].

A Force Partitioning Method (FPM) is employed. This method allows to decompose various contributions of flapping flight physics induced loads. Specifically, added-mass and vortex-induced loads can be identified separately.

## 3 Results

**Efficient kinematic:** Figure 2 (a) shows the instantaneous lift coefficient $C_L$ for the efficient motion computed by integration of wing surface pressure and FPM. We note that both computations give similar results and thus we leverage the

<footnote n="1">¹This work was supported by the French government program "Investissements d'Avenir" (EUR INTREE, Reference No. ANR-18-EURE-0010).</footnote>

<footnote n="2">²This work was supported by the French government program "France 2030" (LABEX INTERACTIFS, reference ANR-11-LABX-0017-01).</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 2:** Lift coefficient $C_L$ for both efficient (a) and lift-producing (b) motions from wing surface integration and FPM. FPM contributions are also shown.</figcaption></figure>

use of FPM to decompose the lift in different contributions, i.e. added-mass ($C_{L,\kappa}$), vortex-induced lift ($C_{L,\omega}$). Moreover we note that $C_{L,\omega}$ dominates lift variations and thus we propose to correlate flow topology and instantaneous lift variations in Figure 3 where numerical (left) and experimental (right) Q-criterion isosurfaces colored by vortex-induced lift are shown. First, it is observed that wing-wake interactions are identified as negative lift source (blue isosurfaces) at $t/T = 2.07$. Further in the stroke, at $t/T = 2.43$, the lift is almost exclusively driven by $C_{L,\omega}$ and the LEV is identified as the main source of lift production as shown by the red isosurfaces. Right column shows experimental flow results on which a good agreement can be found with the numerical flow. The final presentation will propose FPM applied to both numerical and experimental results.

**Lift-producing kinematic:** Figure 2 (b) shows the instantaneous lift coefficient $C_L$ for the lift-producing motion computed by wing surface integration and FPM. Again, results from both approaches are similar and we use FPM to decompose lift. $C_L$ variations are here driven by both $C_{L,\kappa}$ and $C_{L,\omega}$. In other words, added-mass effects are here more important for lift production when compared with the efficient motion. However, fluctuations in lift are closely associated with variations in $C_{L,\omega}$, highlighting the significant role of vortex dynamics in the physics of low flapping flight amplitude. For example, at $t/T = 2.25$, where a significant lift peak occurs, $C_{L,\omega}$ also peaks, indicating a strong influence of vortex dynamics. As for the efficient motion, we propose

<figure><img src="image_4.png" /><figcaption>**Figure 3:** Q isosurfaces colored by vortex-induced lift ($C_{L,\omega}$) for the efficient motion (cf Figure 2 (a)). Numerical (left) and experimental (right) results.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 4:** Q isosurfaces colored by vortex-induced lift ($C_{L,\omega}$) for the lift-producing motion (cf Figure 2 (b)).</figcaption></figure>

$Q$ isosurfaces colored by vortex-induced lift on Figure 4. It can be noted that vortical structures on the wing upper surface, found at both trailing and leading edges, are associated with lift production (red) while those on wing lower surface are found to be responsible for negative lift (blue). The final presentation will detail vortex dynamics (and induced-loads) associated with this kinematics.

## 4 Conclusion

Numerical simulations and experiments are conducted on large and low hovering flapping amplitude motions. While the large amplitude motion presents well-known flapping flight physics (strong LEV attachment), observed on both experimental and numerical results, the low amplitude motion exhibits complex vortex dynamics and lift production revealed through FPM.

## References

[1] Bomphrey, R. et al. Smart wing rotation and trailing-edge vortices enable high frequency mosquito flight. *Nature*, 2017.

[2] Liu, L.-G. et al. Aerodynamic-force production mechanisms in hovering mosquitoes. *Journal of Fluids Mechanics*, 2020.

[3] Corban, B. et al. Discovering optimal flapping wing kinematics using active deep learning. *Journal of Fluid Mechanics*, 2023.

[4] Bouard, F. et al. Hovering flapping wings with dynamic twist. *Physics of Fluids*, 2024.

[5] David, L. et al. Time-resolved scanning tomography PIV measurements around a flapping wing. *Experiments in Fluids*, 2012.