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Falling paper flutters and tumbles through air, whereas a paper airplane glides smoothly if its leading edge is appropriately weighted. We investigate this transformation from ‘plain paper’ to ‘paper plane’ through experiments, aerodynamic modelling and free flight simulations of thin plates with differing centre of mass (CoM) locations. Periodic modes such as fluttering, tumbling and bounding give way to steady gliding and then downward diving as the CoM is increasingly displaced towards one edge. To explain these observations, we formulate a quasi-steady aerodynamic model whose force and torque coefficients are informed by experimental measurements. The dependencies on angle of attack reflect the transition from attached to separated flow and a dynamic centre of pressure, effects that prove critical to reproducing the observed motions of paper planes in air and plates in water. Because the model successfully accounts for unsteady and steady flight modes, it may be usefully applied to further problems involving actuated motions, feedback control and interactions with ambient flows.
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This content will become publicly available on July 10, 2026
Aerodynamic equilibria and flight stability of plates at intermediate Reynolds numbers
The passive flight of a thin wing or plate is an archetypal problem in flow–structure interactions at intermediate Reynolds numbers. This seemingly simple aerodynamic system displays an impressive variety of steady and unsteady motions that are familiar from fluttering leaves, tumbling seeds and gliding paper planes. Here, we explore the space of flight behaviours using a nonlinear dynamical model rooted in a quasisteady description of the fluid forces. Efficient characterisation is achieved by identification of the key dimensionless parameters, assessment of the steady equilibrium states and linear analysis of their stability. The structure and organisation of the stable and unstable flight equilibria proves to be complex, and seemingly related factors such as mass and buoyancy-corrected weight play distinct roles in determining the eventual flight patterns. The nonlinear model successfully reproduces previously documented unsteady states such as fluttering and tumbling while also predicting new types of motions, and the linear analysis accurately accounts for the stability of steady states such as gliding and diving. While the conditions for dynamic stability seem to lack tidy formulae that apply universally, we identify relations that hold in certain regimes and which offer mechanistic interpretations. The generality of the model and the richness of its solution space suggest implications for small-scale aerodynamics and related applications in biological and robotic flight.
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- Award ID(s):
- 1847955
- PAR ID:
- 10651260
- Publisher / Repository:
- Journal of Fluid Mechanics
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 1014
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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