Editor's Note: This article originally appeared in SIAM News on December 05, 2022 (https://sinews.siam.org/Details-Page/the-unconventional-aerodynamics-of-paper-airplanes).
Paper airplanes and other flying toys were serious sources of inspiration in the early days of aviation. Although researchers abandoned efforts to understand paper flyers after the emergence of modern aeronautics in the last century, new compelling biological and robotics applications have reawakened interest in the physics of low-speed and small-scale flight. The relevant aerodynamic regime is challenging, but recent progress is coming from mathematical models that explain the intricate motions of falling and flying paper. The latest models and simulations reveal the secrets of a good paper glider, which is less about the design and more about the unusual aerodynamics of thin wings.
Paper is a fascinating flight system. Its aerial repertoire is impressively diverse, from the chaotic flutters and tumbles of confetti to the smooth gliding of a well-crafted paper airplane. The ability of an unassuming object to display such complicated behaviors has attracted the attention of many physicists, mathematicians, and engineers [1, 3, 5-7]. Part of its simple charm comes from its status as a passive flight system that is solely powered by the downward pull of gravity. The surrounding fluid is the only other contributing influence, meaning that the motion’s complexity reflects the complexity of the aerodynamics.
Prior research has primarily focused on the simplest flight system: a rigid, thin plate or sheet. This is the mathematical plane that lends its name to airplane. As such a plate falls through a fluid, diverse aerodynamic conditions lead to a spectrum of interesting motions. This scenario connects to the two-dimensional problem of a thin body moving within a planar domain of fluid, which is the context wherein most modeling advances have occurred [1, 3, 7].
But a simple plate or slip of paper is no glider. Depending on the conditions, a plate may exhibit back-and-forth fluttering, end-over-end tumbling, or chaotic combinations of both motions [1, 3, 5, 7]. These modes and their determining factors have motivated much of the prior research on inherently unsteady or time-varying aerodynamic mechanisms [10]. We recently built upon previous studies to investigate how the unsteady flutters and tumbles of plain paper transform into the steady gliding of a paper plane [4]. A paper airplane’s capability to remain level and fly smoothly is a question of gliding stability. Many factors affect aircraft stability, but the center of mass (CoM)—or the weight’s effective point of action—is one of the most critical [2]. Our investigations indicate that the CoM position is the only essential ingredient in paper airplane stability, since the unusual aerodynamics of thin plates handle everything else [4].
Figure 1. Folding and flying paper airplanes and airfoils. 1a. Instructions for making a thin planar sheet of rectangular planform. Adding copper tape displaces the center of mass (CoM) towards the leading edge, which is folded over and affixed with double-sided tape. Side fins may be useful to prevent lateral motion. 1b. Instructions for making a paper airfoil. 1c. Examples of finished products that are weighted such that the CoM or balance point is near the quarter-chord point. 1d-1g. Flight modes for paper planes with different CoM values; the images are overlaid frames from videos. Tumbling, bounding, swooping, and diving are displayed by planes and foils, whereas gliding is achieved only by planes. Figures 1a-1c courtesy of the authors, and Figures 1d-1g adapted from [4].
Our story begins with experiments in the Applied Mathematics Laboratory at New York University’s Courant Institute of Mathematical Sciences — though anyone can conduct these tests at home. First, we confirmed that a simple rectangular sheet is capable of gliding flight if the CoM is appropriately located in the fore-aft direction [4]. Our design consists of a rectangle that is cut from standard copy paper, with a thin strip of heavy metallic tape added along the leading edge to shift the CoM forward (see Figure 1a); we are happy to provide interested readers with template files, further instructions, and information about suppliers of the necessary materials. We determined the CoM as the balance point (see Figure 1c) and performed test flights that reveal the aerial tendencies for different CoM positions. Adjustable tabs on the sides of the design help suppress lateral or sideways motion.
Changing the degree of weighting and hence the CoM yields the motions in Figure 1d-1f. This simple flyer can achieve smooth gliding, but only if the CoM is near the quarter-chord point (i.e., halfway between the middle of the plate and its leading edge). A CoM that is too close to the middle leads to tumbling, bounding, and swooping, and one that is too far forward causes nose-down diving.
The “just right” CoM location at the quarter chord may be familiar for some readers based on the theory of thin airfoils, which predicts this same point as the center of pressure or effective location of the aerodynamic forces for thin airfoils at low angles of attack (AoA) [2]. We were therefore surprised to discover that an airfoil cannot glide. Figure 1b depicts the creation of a foil-shaped paper flyer by looping over the front edge. The addition of metallic tape again yields different CoM locations, but test flights resulted in frustration; varied CoM values trigger tumbles, swoops, and dives, but never smooth gliding.
We turn to mathematical modeling to solve the mystery of paper airplane stability. The end goal is a dynamical system that accounts for the evolution of the body’s rotations and translations during free flight. Such a treatment of the aerodynamics is an approximation at best, since the model does not explicitly include the state of the surrounding fluid [1, 7]. Instead, we assume that all aerodynamic forces and torques are expressible as physically informed laws that involve the body’s dynamical state.
Our model consists of the Newton-Euler equations for a rigid body with forcing terms that account for weight and aerodynamic effects like added mass, lift, drag, and their associated torques [4]. It takes the form of a system of coupled nonlinear ordinary differential equations for the translational velocity \((v_{x^\prime},v_{y^\prime})\) in the frame of the plate with orientation angle \(\theta\) and angular velocity \(\omega=\dot \theta\):
\[(m+m_{11}) \dot v_{x^\prime} = (m+m_{22}) \omega v_{y^\prime} - m_{22} \omega^2 \ell_{CM} + L_{x^\prime} + D_{x^\prime} - m ^\prime g \sin \theta,\tag1\]
\[(m+m_{22}) \dot v_{y^\prime} = -(m+m_{11}) \omega v_{x^\prime} + m_{22} \dot \omega \ell_{CM} + L_{y^\prime} + D_{y^\prime} - m^\prime g \cos \theta,\tag2\]
\[(I+I_a) \dot \omega = \tau_T + \tau_R + \tau_B.\tag3\]
The aerodynamic characteristics for a given wing shape are specified within the various terms, e.g., the lift and drag dependencies on speed, angle, and so forth. We solved the model equations numerically via a “flight simulator” code, the output data of which we can reanimate and further analyze.
Figures 2a and 2b compare the simulated flight trajectories for plates or planes versus foils with systematically varied CoMs. The results for a thin plate impressively match all of the behaviors that we observed in experiments on paper planes, thus validating the model [4]. Furthermore, the results for a foil confirm the nonexistence of gliding. In fact, gliding stands out as the unique flight mode for a plane or plate, but not an airfoil.
What is the secret to the plate’s stability? Our investigations point to the translational torque \(\tau_T\) in \((3)\), which corresponds to the pitching moment that is associated with the fluid forces acting at a location that is different from the CoM [4]. Expressed in terms of the center of pressure (CoP), \(\tau_T\) is the pressure force multiplied by the CoP-to-CoM distance. The subtlety lies in the CoP’s dependence on the AoA. Experimental measurements indicate that a thin plate has a CoP profile that decreases with AoA (see the magenta curve in Figure 2c), meaning that the forces are concentrated closer to the front at low angles and towards the middle at higher angles. This relationship implies that perturbations to the angle, whether increasing or decreasing, face appropriate restoring torques as the CoP moves to the fore or aft of the CoM. But the same is not true for an airfoil, whose flat CoP profile (see the cyan curve in Figure 2c) indicates a lack of response to changes in AoA [2].
Figure 2. Flight simulations of planes and foils. 2a. Representative trajectories of thin plates that are falling in air, as computed from our model [4]. The colors correspond to CoM locations that vary from the center of the plate (red) to near the leading edge (purple). Gliding (blue) occurs when the CoM is near the quarter-chord point. 2b. Simulated trajectories of foils that fail to glide for any CoM. 2c. Center of pressure dependence on angle of attack (AoA) for a plate (magenta) and foil (cyan). 2d. The flow over the plate separates from the leading edge and reattaches to form a “bubble” whose size varies with AoA. 2e. Flow over the foil remains smoothly attached. Figure courtesy of the authors.
This explanation evokes the question of why the different CoP profiles occur. The value of \(\mathrm{CoP} = 0.25\) and its invariance with AoA are results of the Kutta-Joukowski theorem for airfoils, whose more famous prediction applies to lift [2]. The theory assumes that the flow over the leading edge is smoothly attached (see Figure 2e), which typically holds if the AoA is small and the foil is thin. Yet this logic does not apply if the foil is too thin—i.e., paper thin—since a thin plate or sheet behaves altogether differently than a foil. Its sharp leading edge triggers flow separation even at small angles [8], and reattachment further downstream traps a vortex or so-called “separation bubble” on the upper side of the plate (see Figure 2d). The separation region has low pressure and a size that varies with AoA, allowing for dynamical modifications of the pressure distribution and hence the CoP.
The intriguing picture that emerges is that a paper airplane hang-glides under a “bubble” that sits atop its leading edge and inflates and contracts in a way that ensures a smooth ride. This mechanism of dynamic stabilization is wholly different from that of a conventional airplane, which requires a tail because the airfoils of the main wing lack any intrinsic responsiveness.
These findings highlight the many twists, turns, and unexpected directions that result from seemingly simple questions about falling paper and flying paper planes. These systems are much more than curiosities — they drive advances in the modeling of flight dynamics, with applications to wing shapes, speeds, scales, and flow states that are outside the domain of standard aerodynamics. Once we better understand the relevant effects, we can exploit them to design small-scale flying robots that are inspired by the flapping-wing flight of insects or the various falling styles of plant seeds [9, 10]. We therefore anticipate many future applications for models and simulations that can accurately capture such unconventional aerodynamics.
References
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[8] Smith, J.A., Pisetta, G., & Viola, I.M. (2021). The scales of the leading-edge separation bubble. Phys. Fluids, 33(4), 045101.
[9] Viola, I.M., & Nakayama, N. (2022). Flying seeds. Curr. Biol., 32(5), R204-R205.
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Huilin Li is a Ph.D. student in mathematics at New York University Shanghai and a member of the Applied Mathematics Laboratory (AML) at New York University’s (NYU) Courant Institute of Mathematical Sciences. Leif Ristroph is an applied mathematician and experimental physicist who directs the AML at NYU’s Courant Institute of Mathematical Sciences.