# Examination of bending rules in nature

Florian Bouard, Thierry Jardin, Laurent David

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Florian Bouard, Thierry Jardin, Laurent David. Examination of bending rules in nature. EU-ROMECH Colloquium 654 “Bio-Inspired Fluid-Structure Interaction”, Jul 2025, Vienna, Austria. hal-05335326

HAL Id: hal-05335326

https://isae-ensma.hal.science/hal-05335326v1

Submitted on 28 Oct 2025

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Examination of bending rules in nature

F. Bouard¹,², T. Jardin¹ and L. David²

¹ ISAE-SUPAERO, Université de Toulouse, France

² Institut PPrime, Université de Poitiers-ENSMA-CNRS, UPR 3346, France

Recent observations on a large number of flying/swimming animals (Lucas et al., 2014) have revealed common bending patterns in the animal propulsor deformation during flapping flight/swim. In particular, it has been observed that flexion location and angle typically cluster around 0.65*R* (*R* is the radius of the wing) and $\theta_0 \approx 25^\circ$ (Figure 2). However, it remains unclear whether natural species have converged towards these flexion location and angle for propulsive efficiency purposes or not. The present study focuses on the role of those parameters on aerodynamic performance of a flapping wing under hovering flight conditions.

The wing undergoes a flapping kinematic consisting in sinusoidal revolving ($\dot{\phi}$) and pitching ($\dot{\alpha}$) motions as shown on Figure 1. The wing flaps over $\phi_0 = 120^\circ$ and reaches $\alpha_0 = 45^\circ$ at mid-stroke. StarCCM+ flow solver is used to directly solve the Navier-Stokes equations using a cell-centered finite volume method around rigid and actively bending wings. Bending is here in phase with the flapping motion. Flexion location $r/R$ and angle $\theta_0$ are systematically varied along from 0 to 1 and 0° to 45°, respectively. Figure 2 shows the efficiency ($\bar{C}_L/\bar{C}_P$) contours of the different parameter combination.

<figure><img src="image_2.png" /><figcaption>Figure 1: Illustration of a flapping kinematic.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2: Bending rules parameters and efficiency contours.</figcaption></figure>

Efficiency can be improved by 4.8% for animal flexion location and angle with respect to a rigid wing (Figure 2). At the wingtip and for a relatively high $\theta_0$, the efficiency can be further increased up to a difference of 8.7% compared to a rigid wing. This study also highlighted a maximum lift production, increased by 11.2% compared to a rigid wing, (not shown here) with flexion location and angle $r/R = 0$ and $\theta_0 = 25^\circ$, respectively. The final presentation will discuss on the physics underlying these observations

# References

Lucas, K.N. et al. Bending rules for animal propulsion. Nature Communications, 3293(5), 2014.

This work was supported by the French government program Investissements d'Avenir (EUR INTREE, Reference No. ANR-18-EURE-0010).