Effects of Plunging Acceleration on the Passive Morphing of Avian-Inspired Flexible Foils

This study utilizes two-way coupled simulations to demonstrate that the aerodynamic performance of passively morphing foils under accelerated plunging is critically dependent on the interplay between wing geometry, bending rigidity, and the extent of trailing-edge flexibility, revealing that bio-inspired cambered shapes offer superior stability and performance compared to symmetric foils in unsteady environments.

The Gist

Imagine you are holding a stiff plastic ruler and a flexible rubber spatula. Now, imagine someone suddenly yanks both of them upward through the air.

The plastic ruler stays straight, fighting the air with a rigid snap. The rubber spatula, however, bends, wiggles, and twists. It doesn't just move up; it changes its shape as it moves.

This paper is a deep dive into exactly what happens when those "rubber spatulas" are actually bird wings, and the "yank" is a sudden, violent gust of wind. The researchers wanted to know: Does bending help a wing fly better in a storm, or does it make things worse?

Here is the story of their discovery, broken down into simple concepts.

1. The Experiment: The "Wind Tunnel" and the "Birds"

The scientists built a digital wind tunnel. Inside, they tested three types of wings:

• The Standard Wing: A classic, symmetrical airplane wing shape (NACA0012). Think of this as a stiff, uniform ruler.
• The Falcon Wing: A wing shape inspired by a Peregrine Falcon (built for speed and diving).
• The Owl Wing: A wing shape inspired by a Barn Owl (built for silent, gliding flight).

They didn't just move these wings up and down; they accelerated them. Imagine a car that doesn't just drive fast, but slams on the gas pedal instantly. This mimics a bird hitting a sudden gust of wind or diving for prey.

They also made the back half of these wings "flexible" (like the rubber spatula) to see how much bending was good.

2. The Big Discovery: "Goldilocks" Stiffness

The researchers found that there is no single "perfect" amount of flexibility. It depends entirely on the shape of the wing.

• Too Stiff: The wing acts like a brick. It fights the wind, but it doesn't generate extra lift.
• Too Flexible: The wing flaps around like a wet noodle. It creates chaos, shaking the lift up and down wildly.
• Just Right (The Sweet Spot):
  • For the Owl and the Standard wing, the perfect stiffness was a specific "medium" level.
  • For the Falcon wing, the perfect stiffness was slightly more flexible.

The Analogy: Think of a trampoline. If it's too hard (concrete), you don't bounce. If it's too soft (mud), you sink and get stuck. But if it's just right, the bounce propels you higher than you could jump on your own. The Falcon wing needed a slightly "muddier" trampoline to get the best bounce than the Owl wing did.

3. The Secret Weapon: The "Vortex Dance"

Why does bending help? It's all about the invisible whirlpools of air (called vortices) that form around the wing.

• The Rigid Wing: When it moves up fast, it creates a whirlpool that quickly falls apart. It's like a sandcastle wave that crashes immediately.
• The Flexible Wing: As the wing bends, it actually holds onto that whirlpool longer. The bending shape acts like a cradle, keeping the air swirling tightly against the wing. This creates a suction effect that pulls the wing upward with more force.

The Creative Metaphor: Imagine you are trying to scoop up a scoop of ice cream with a spoon.

• A rigid spoon just pushes the ice cream aside.
• A flexible spoon bends slightly, cupping the ice cream perfectly and lifting it up without spilling.
  The flexible wing "cups" the air, creating a stronger lift force than a rigid wing ever could.

4. The Danger Zone: The "Wet Noodle" Effect

The study found a warning sign. If you make the flexible part of the wing too long (extending 75% of the way from the back), things get messy.

• For the Standard Wing: It turned into a "wet noodle." The lift started shaking violently (up and down), which is bad for a drone or plane trying to stay stable.
• For the Bird Wings (Falcon & Owl): They were surprisingly resilient! Even when very flexible, their unique curved shapes (camber) acted like a stabilizer. They kept the air flowing smoothly, preventing the violent shaking.

The Lesson: Nature has designed bird wings with a specific curve that acts as a shock absorber. A simple flat wing doesn't have this protection; if you make it too flexible, it goes haywire.

5. The "Gust" Factor

The researchers also tested how sudden the wind hit.

• Slow Wind: The wing just bends a little.
• Sudden, Violent Gust: The wing bends a lot, but because of the "Goldilocks" stiffness, it actually generates a massive burst of lift to counter the wind.

The faster the wind hits, the more the wing bends, and the stronger the "whirlpool" becomes. It's a self-correcting system: The harder the wind hits, the more the wing bends to catch it.

Why Does This Matter? (The Real-World Impact)

This isn't just about understanding birds. It's about building better drones and UAVs (Unmanned Aerial Vehicles).

Current drones have rigid wings. If they hit a sudden gust, they crash or lose control.

• The Future: Imagine a drone with wings made of smart, flexible materials that mimic a falcon or an owl.
• The Benefit: When a sudden gust hits, the wing would automatically bend, "cup" the air, and generate extra lift to stay stable. It would be like a surfer who automatically adjusts their board to a wave, rather than a rigid boat that gets tossed around.

Summary

• Birds are smart engineers: Their wings aren't just stiff bones; they are flexible, curved structures designed to handle sudden changes in wind.
• Flexibility is a tool, not a weakness: If tuned correctly, bending wings can create more lift than stiff wings.
• One size does not fit all: The "perfect" flexibility depends on the shape of the wing. You can't just copy a falcon wing and make it out of rubber; you have to tune the rubber to match the shape.
• Nature's shock absorbers: The curved shape of bird wings helps them stay stable even when they are very flexible, something simple flat wings struggle to do.

In short, the next generation of flying machines might not look like rigid metal planes, but like flexible, bending birds that dance with the wind rather than fighting it.

Technical Summary

1. Problem Statement

The study addresses a critical gap in the understanding of Fluid-Structure Interaction (FSI) for flexible lifting surfaces under transient, accelerated kinematics. While previous research has extensively covered flexible foils under harmonic (periodic) motion or static conditions, realistic flight maneuvers and gust responses involve rapid acceleration.

• The Gap: It is unclear whether the force augmentation benefits observed in periodic flexible wings persist under impulsive acceleration, or if the altered vortex dynamics (specifically Leading-Edge Vortex or LEV formation) decouple structural deformation from force generation.
• The Biological Context: Real avian wings possess complex, non-uniform stiffness distributions (stiff leading edge, compliant trailing edge) and specific camber/thickness profiles. Most existing studies simplify these to symmetric NACA airfoils or flat plates with uniform stiffness, failing to capture the structural reality of biological flyers.
• Objective: To systematically investigate how plunging acceleration, wing geometry (NACA0012 vs. bio-inspired falcon/owl), bending rigidity, and the extent of the flexible trailing edge interact to influence aerodynamic performance and vortex dynamics.

2. Methodology

The authors employed a high-fidelity two-way coupled FSI framework using a partitioned strong-coupling approach.

• Geometries: Three wing sections were simulated:
  1. NACA0012: A conventional symmetric foil.
  2. Peregrine Falcon: Bio-inspired profile with physiologically motivated camber and thickness.
  3. Barn Owl: Bio-inspired profile with distinct camber and thickness distributions.
  • Note: The bio-inspired geometries were derived from experimental measurements of primary feather cross-sections.
• Kinematics: The wings underwent accelerated plunging to mimic a transverse discrete gust. The motion was defined by a smooth ramp-up and ramp-down profile controlled by a transition speed parameter ($a_s$), ranging from 3 (gradual gust) to 11 (severe, impulsive gust).
• Structural Modeling:
  • Modeled as a Neo-Hookean hyperelastic material to handle large deformations.
  • The trailing edge (25%, 50%, or 75% of the chord) was flexible, while the leading edge remained rigid.
  • Non-dimensional bending rigidity ($K_B$) was varied from 5 to 100.
• Numerical Setup:
  • Fluid: Incompressible Navier-Stokes equations solved via OpenFOAM (pimpleFoam solver) at $Re = 10^3$. No turbulence modeling (laminar flow).
  • Structure: Solved via CalculiX (Finite Element Method).
  • Coupling: preCICE library facilitated data exchange (forces/displacements) using Radial Basis Function (RBF) interpolation. Strong implicit coupling with IQN-ILS acceleration was used for stability.
• Validation: The framework was validated against experimental data for rigid wings (Jackson et al., 2025) and flexible trailing-edge static cases (Boughou et al., 2024), showing close agreement in vortex formation angles and displacement magnitudes.

3. Key Contributions

1. Acceleration as a Governing Parameter: The study establishes that plunging acceleration is a primary independent variable governing FSI in flexible wings, distinct from frequency-based parameters used in harmonic studies.
2. Geometry-Specific Optimal Stiffness: It demonstrates that the "optimal" bending rigidity for maximum lift is not universal but depends heavily on the wing's geometric profile.
3. Stabilizing Role of Bio-Inspired Geometry: It reveals that bio-inspired camber and thickness distributions inherently stabilize unsteady forces, decoupling large structural deformations from detrimental lift fluctuations—a phenomenon not observed in symmetric NACA foils.
4. Vortex-Structure Coupling Mechanisms: The paper provides mechanistic insights into how acceleration drives LEV/TEV (Trailing-Edge Vortex) interactions and how wing geometry modulates these interactions to either enhance or suppress force generation.

4. Key Results

A. Effect of Bending Rigidity ($K_B$)

• Optimal Stiffness: There is a non-monotonic relationship between flexibility and lift.
  • NACA0012 & Owl: Peak lift ($C_{l,max}$) occurs at $K_B = 10$.
  • Falcon: Peak lift occurs at $K_B = 7.5$.
  • Insight: Excessive flexibility ($K_B < 7.5$) degrades performance by disrupting the phase synchronization between deformation and vortex growth.
• Deformation vs. Lift: While deformation increases monotonically as stiffness decreases, lift does not. The bio-inspired wings (especially the Owl) generate significantly higher lift and deformation than the NACA0012 at optimal stiffness.

B. Effect of Flexible Segment Extent (25%, 50%, 75%)

• 25% Flexibility: Behaves almost identically to a rigid wing; negligible aerodynamic benefit.
• 75% Flexibility:
  • NACA0012: Exhibits a sharp, detrimental increase in lift fluctuations ($C_{l,RMS}$), indicating unsteady, chaotic vortex shedding.
  • Bio-inspired (Falcon/Owl): Despite large deformations, these geometries show a reduction or stabilization in $C_{l,RMS}$. The cambered geometry acts as a flow control mechanism, maintaining coherent vortex structures even under high flexibility.

C. Effect of Transition Speed ($a_s$)

• Acceleration Scaling: Increasing $a_s$ (from 3 to 11) monotonically increases:
  • Trailing-edge deflection amplitude.
  • Peak lift ($C_{l,max}$).
  • Vortex strength (circulation $\Gamma$) and coherence.
• Drag Reduction: At high acceleration ($a_s=11$), the bio-inspired wings (particularly the Falcon) transition to generating net thrust (negative drag), a result of acceleration-driven wake momentum redistribution enhanced by their specific geometry.

D. Vortex Dynamics

• LEV/TEV Coupling: Acceleration triggers rapid LEV formation. Bio-inspired wings sustain a stronger, more coherent LEV for longer durations compared to the NACA0012.
• Wake Topology: The bio-inspired geometries promote a stable vortex couple (LEV-TEV) that convects downstream with less distortion. In contrast, the highly flexible NACA0012 (75% case) suffers from vortex distortion and loss of coherence, leading to erratic lift.

5. Significance and Implications

• Design of UAVs and Drones: The findings suggest that for vehicles operating in gusty, unsteady environments, simply adding flexibility is insufficient. Designers must tailor material stiffness to the specific wing geometry.
• Passive Gust Mitigation: Bio-inspired geometries (like the Owl wing) offer a passive mechanism to suppress lift fluctuations ($C_{l,RMS}$) even when undergoing large deformations, enhancing stability without active control systems.
• Thrust Generation: The ability of flexible, bio-inspired wings to generate net thrust under high acceleration suggests new strategies for agile maneuvering and energy-efficient flight in turbulent conditions.
• Future Directions: The study highlights the need for 3D simulations and spatially graded stiffness models (smooth transitions from rigid to flexible) to fully replicate biological flight mechanics.

In conclusion, the paper demonstrates that plunging acceleration fundamentally alters the FSI landscape, where the interplay between geometry-specific compliance and vortex dynamics determines whether passive morphing enhances or degrades aerodynamic performance. The bio-inspired wings consistently outperform conventional symmetric foils by leveraging their shape to stabilize unsteady flows.