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Experiments with paper airplanes reveal surprisingly complex aerodynamics

How these gliders keep level flight is different from the stability of airplanes.

Drop a flat piece of paper and it will flutter and tumble through the air as it falls, but a well-fashioned paper airplane will glide smoothly. Although these structures look simple, their aerodynamics are surprisingly complex. Researchers at New York University’s Courant Institute of Mathematical Sciences conducted a series of experiments involving paper airplanes to explore this transition and develop a mathematical model to predict flight stability, according to a March paper published in the Journal of Fluid Mechanics.

“The study started with simple curiosity about what makes a good paper airplane and specifically what is needed for smooth gliding," said co-author Leif Ristroph . "Answering such basic questions ended up being far from child’s play. We discovered that the aerodynamics of how paper airplanes keep level flight is really very different from the stability of conventional airplanes.”

Nobody knows who invented the first paper airplane, but China began making paper on a large scale around 500 BCE, with the emergence of origami and paper-folding as a popular art form between 460 and 390 BCE. Paper airplanes have long been studied as a means of learning more about the aerodynamics of flight. For instance, Leonardo da Vinci famously built a model plane out of parchment while dreaming up flying machines and used paper models to test his design for an ornithopter. In the 19th century, British engineer and inventor Sir George Cayley —sometimes called the "father of aviation"—studied the gliding performance of paper airplanes to design a glider capable of carrying a human.

An amusing "scientist playing with paper planes" anecdote comes from physicist Theodore von Kármán . In his 1967 memoir The Wind and Beyond , he recalled a formal 1924 banquet in Delft, The Netherlands, where fellow physicist Ludwig Prandtl constructed a paper airplane out of a menu to demonstrate the mechanics of flight to von Kármán's sister, who was seated next to him. When he threw the paper plane, "It landed on the shirtfront of the French minister of education, much to the embarrassment of my sister and others at the banquet," von Kármán wrote.

While scientists have clearly made great strides in aerodynamics—particularly about aircraft—Ristroph et al . noted that there was not a good mathematical model for predicting the simpler, subtler gliding flight of paper airplanes. It was already well-known that displacing the center of mass results in various flight trajectories, some more stable than others. “The key criterion of a successful glider is that the center of mass must be in the ‘just right’ place,” said Ristroph . “Good paper airplanes achieve this with the front edge folded over several times or by an added paper clip, which requires a little trial and error.”

He and his team verified this by test-flying various rectangular sheets of paper, changing the front weight by adding thin metallic tape to one edge. They found that an unweighted sheet tumbled end over end while descending left to right under the force of gravity. Adding a small weight to shift the center of mass slightly forward also produced a tumbling trajectory. Overall, they found that flyers with greater front-loading produced erratic trajectories full of swoops, climbs, flips, and dives.

The next step was to conduct more controlled and systematic experiments. Ristroph et al . decided to work with thin plastic plates "flying" through a large glass tank of water. The plates were laser-cut from an acrylic plastic sheet, along with two smaller "fins" embedded with lead weights to displace the center of mass, and they also serve as aerodynamic stabilizers. There were 17 plastic plates, each with a different center of mass. Each was released into the tank by sliding it down a short ramp, and the team recorded its free-flight motion through the water.

They found the same dynamics played out. If the weight was centered, or nearly so, at the center of the wing, the plate would flutter and tumble erratically. Displace the center of mass too far toward one edge, and the plate would rapidly nosedive and crash. The proverbial "sweet spot" was placing the weight between those extremes. In that case, the aerodynamic force on the plane's wing will push the wing back down if it moves upward, and push the wing back up if it moves downward. In other words, the center of pressure will vary with the angle of flight, thereby ensuring stability.

This differs substantially from conventional aircraft, which rely on airfoils—structures designed to generate lift. “The effect we found in paper airplanes does not happen for the traditional airfoils used as aircraft wings, whose center of pressure stays fixed in place across the angles that occur in flight,” said Ristroph . “The shifting of the center of pressure thus seems to be a unique property of thin, flat wings, and this ends up being the secret to the stable flight of paper airplanes. This is why airplanes need a separate tail wing as a stabilizer while a paper plane can get away with just a main wing that gives both lift and stability.”

The team also developed a mathematical model as a "flight simulator" to reproduce those motions. Ristroph et al. think their findings will prove useful in small-scale flight applications like drones or flying robots, which often require a more minimal design with no need for many extra flight surfaces, sensors, and controllers. The authors also note that the same strategy might be at work in winged plant seeds, some of which also exhibit stable gliding, with the seed serving as the payload to displace the center of mass. In fact, a 1987 study of the flying seeds of the gourd Alsomitra macrocarpa  showed a center of mass and glide ratios consistent with the Ristroph group's optimal gliding requirements.

DOI: Journal of Fluid Mechanics, 2022. 10.1017/jfm.2022.89  ( About DOIs ).

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Cool Science: Paper airplanes, insect wings and the future of tiny drones

Conduct your own aerodynamics experiment at home

For nearly 200 years, in the great pursuit of knowledge about aerodynamics, researchers have been fascinated by paper airplanes. Even the simplest design – a plain sheet of paper with no wings or fins, nothing resembling a modern airplane – can provide insights into complex aerodynamic principles.

A rectangular sheet of paper will tumble to the ground like confetti, but adding just the right amount of weight, in just the right location, affects how the paper flies.

Leif Ristroph, an associate professor at New York University and self-described paper airplane fanatic, said the idea to study the aerodynamics of a falling sheet of paper began with curiosity about the movement.

“The magic that makes a paper airplane glide and not flutter is how you weight it. You don’t need all the other aspects,” Ristroph said. “It feels like someone must have observed this before because it seems so basic but I couldn’t find it. It’s worth understanding, then.”

And while designing a passenger airplane that resembles a sheet of paper is a horrible idea, he said, the research could contribute to new ways of thinking about the design of simple aircraft such as tiny drones.

Observations in air and water

Researchers added strips of metallic tape to one of the long edges of a rectangular sheet to create varying degrees of front weight. When dropped, each sheet moves in a unique way. “There is kind of a magic spot for the balance point where you can get the best glide. If the balance point is too close to the middle, you will see fluttering and tumbling motions. If the balance point is too far forward, it will nose-dive and crash. The best point is about halfway between the middle of the sheet and the front edge. Then you will see a nice glider,” Ristroph said.

Part of Ristroph’s work for his doctorate in physics was studying insect flight and how flapping wings affect flight control and stability. The paper airplane study was not part of his planned research under a U.S. National Science Foundation grant , but the relationship between insect wings and falling paper stood out, and the results contribute to a more general understanding of animal flight and swimming dynamics.

Plus, sheets of paper are easier to maintain in the laboratory than insects, he said.

Ristroph also made observations using a water tank and thin plates of plastic, whose “flight” through water (video above) are governed by the same physics that determines the motions of paper falling through the air. In both instances, the rate and control of the fall was affected by the location of the weight and the impact it had on the center of mass. There is a “sweet spot” for the center of mass that produces a stable glide path.

Based on the results, Ristroph and his team created a mathematical model that serves as the basis of a “flight simulator,” a computer program that reproduces the different flight motions. No such model had previously existed for predicting gliding flight.

“The simulator can produce the full range of motions accurately. It describes the forces acting on the plate and finds out how the flyers should move based on the forces used in the equation. I think for the scientific community, that will translate well into something useful for everyone involved,” he said.

The findings could also contribute to new design methods for drones. “There are a lot of people working on making very small-scale flying robots. There are even some designs that are passive flyers – not powered – that are modeled on flying plant seeds that twirl like helicopters when they fall.”

These passive drones can be used for projects like air quality monitoring.

“There is an initiative to make small flyers, and at very small scales, simpler is better,” Ristroph said. “When you think about paper airplanes, it doesn’t get much simpler than that. I’m not an engineer who can make a useful flying robot, but I hope our findings will inspire others who can.”

“Studying toys is serious fun. Whether it’s a paper airplane or a spinning top, there is something scientifically valuable in finding out how they work.”

Experiments with aerodynamics at home

Ristroph’s research includes a component dedicated to training STEM students in aspects of modern applied mathematics, emphasizing the connections among modeling, simulation and experimental observation.

And his experiments are simple enough that they can be conducted anywhere. “I like to think of the whole world as a laboratory. There are simple experiments in front of you. The things you see around you are fascinating if you look at them in the right way,” he said.

Want to replicate the experiment at home? The basic materials are a notepad or copy paper about 6 inches long and 2 inches wide that can be folded and tape or paper clips that can be used to add weight.

“Studying toys is serious fun,” Ristroph said. “Whether it’s a paper airplane or a spinning top, there is something scientifically valuable in finding out how they work.”  

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For new insights into aerodynamics, scientists turn to paper airplanes

Findings unveil mechanisms that explain flight stability.

A series of experiments using paper airplanes reveals new aerodynamic effects, a team of scientists has discovered. Its findings enhance our understanding of flight stability and could inspire new types of flying robots and small drones.

"The study started with simple curiosity about what makes a good paper airplane and specifically what is needed for smooth gliding," explains Leif Ristroph, an associate professor at New York University's Courant Institute of Mathematical Sciences and an author of the study, which appears in the Journal of Fluid Mechanics . "Answering such basic questions ended up being far from child's play. We discovered that the aerodynamics of how paper airplanes keep level flight is really very different from the stability of conventional airplanes."

"Birds glide and soar in an effortless way, and paper airplanes, when tuned properly, can also glide for long distances," adds author Jane Wang, a professor of engineering and physics at Cornell University. "Surprisingly, there has been no good mathematical model for predicting this seemingly simple but subtle gliding flight."

Since we can make complicated modern airplanes fly, the researchers say, one might think we know all there is to know about the simplest flying machines.

"But paper airplanes, while simple to make, involve surprisingly complex aerodynamics," notes Ristroph.

The paper's authors began their study by considering what is needed for a plane to glide smoothly. Since paper airplanes have no engine and rely on gravity and proper design for their movement, they are good candidates for exploring factors behind flight stability.

To investigate this phenomenon, the researchers conducted lab experiments by launching paper airplanes with differing centers of mass through the air. The results, along with those from studying plates falling in a water tank, allowed the team to devise a new aerodynamic model and also a "flight simulator" capable of predicting the motions.

To find the best design, the researchers placed different amounts of thin copper tape on the front part of the paper planes, giving them varied center of mass locations. Lead weights added to the plates in water served the same purpose.

"The key criterion of a successful glider is that the center of mass must be in the 'just right' place," Ristroph explains. "Good paper airplanes achieve this with the front edge folded over several times or by an added paper clip, which requires a little trial and error."

In the experiments, the researchers found that the flight motions depended sensitively on the center of mass location. Specifically, if the weight was at the center of the wing or only displaced somewhat from the middle, it underwent wild motions, such as fluttering or tumbling. If the weight was displaced too far toward one edge, then the flier quickly dove downwards and crashed. In between, however, there was a "sweet spot" for the center of mass that gave stable gliding.

The researchers coupled the experimental work with a mathematical model that served as the basis of a "flight simulator," a computer program that successfully reproduced the different flight motions. It also helped explain why a paper airplane is stable in its glide. When the center of mass is in the "sweet spot," the aerodynamic force on the plane's wing pushes the wing back down if the plane moves upward and back up if it moves downward.

"The location of the aerodynamic force or center of pressure varies with the angle of flight in such a way to ensure stability," explains Ristroph.

He notes that this dynamic does not occur with conventional aircraft wings, which are airfoils -- structures whose shapes work to generate lift.

"The effect we found in paper airplanes does not happen for the traditional airfoils used as aircraft wings, whose center of pressure stays fixed in place across the angles that occur in flight," Ristroph says. "The shifting of the center of pressure thus seems to be a unique property of thin, flat wings, and this ends up being the secret to the stable flight of paper airplanes."

"This is why airplanes need a separate tail wing as a stabilizer while a paper plane can get away with just a main wing that gives both lift and stability," he concludes. "We hope that our findings will be useful in small-scale flight applications, where you may want a minimal design that does not require a lot of extra flight surfaces, sensors, and controllers."

The paper's other authors were Huilin Li, a doctoral candidate at NYU Shanghai, and Tristan Goodwill, a doctoral candidate at the Courant Institute's Department of Mathematics.

The work was supported by grants from the National Science Foundation (DMS-1847955, DMS-1646339).

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Story Source:

Materials provided by New York University . Note: Content may be edited for style and length.

Journal Reference :

• Huilin Li, Tristan Goodwill, Z. Jane Wang, Leif Ristroph. Centre of mass location, flight modes, stability and dynamic modelling of gliders . Journal of Fluid Mechanics , 2022; 937 DOI: 10.1017/jfm.2022.89

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Paper Airplane Study Finds New Aerodynamic Effects

A team of researchers analyzing the flight characteristics of paper airplanes have found a whole new set of unexpected aerodynamic effects. Initially designed to understand the basic mechanics behind what makes a good paper airplane fly, the novel aerodynamic effects found by researchers may lead to the development of a whole new class of flying robots or small drones that function with very few moving parts.

WHAT MAKES A GOOD PAPER AIRPLANE FLY?

What is it that makes a good paper airplane glide across a lengthy high school cafeteria but dooms other similar-appearing craft to dive straight into the floor? It’s a question that has flummoxed more than one would-be paper airline launcher and one that motivated a team of researchers from New York University’s (NYU) Courant Institute of Mathematical Sciences and Cornell University to search for an answer.

“The study started with simple curiosity about what makes a good paper airplane and specifically what is needed for smooth gliding,” explains NYU associate professor and study co-author Leif Ristroph in a press release announcing the research. “Answering such basic questions ended up being far from child’s play. We discovered that the aerodynamics of how paper airplanes keep level flight is really very different from the stability of conventional airplanes.”

CENTER OF MASS IS THE KEY

To begin their study, the research team evaluated the mathematics behind paper airplane flight. For instance, a paper airplane has no motor but instead relies on gravity for its gliding movement, much like a gliding bird.

“Birds glide and soar in an effortless way, and paper airplanes, when tuned properly, can also glide for long distances,” said study co-author Jane Wang, a professor of engineering and physics at Cornell University. “Surprisingly, there has been no good mathematical model for predicting this seemingly simple but subtle gliding flight.”

It soon became apparent that absent previous mathematical modeling, uncovering the secrets of paper airplane flight would require some real-world testing. First, the team used a basic triangle design for their paper airplane, which is considered the simplest and most efficient form. Next, the researchers placed thin pieces of copper tape on the front part of the paper airplanes, much like the paper clip used in many homemade models. The copper tape segments were placed in different locations on each plane, creating a differing center of mass.

“The key criterion of a successful glider is that the center of mass must be in the ‘just right’ place,” Ristroph explains. “Good paper airplanes achieve this with the front edge folded over several times or by an added paper clip, which requires a little trial and error.”

After numerous paper airplane launches in a laboratory setting, the team found the best weight location to maximize flight distances.

“If the weight was at the center of the wing or only displaced somewhat from the middle, it underwent wild motions, such as fluttering or tumbling,” the press release explains. “If the weight was displaced too far toward one edge, then the flier quickly dove downwards and crashed. In between, however, there was a “sweet spot” for the center of mass that gave stable gliding.”

The team mirrored these test flights by sinking a set of plates in a water tank and observing their “gliding” motion on the way to the bottom. Lead weights were placed at different locations on the plates to change their center of mass, much like the copper tape on the paper airplane.

The researchers combined all of this accumulated data to create customized “flight simulator” software that allowed them to analyze and simulate paper airplane flight. And to their surprise, they not only found out how to make a good airplane, but they also learned that the aerodynamic principles of a gliding paper airplane (or a sinking plate) were completely different from that of a conventional aircraft.

Will Future Spacecraft Use ‘Jaws’ Bacteria to Make Rocket Fuel?

“The effect we found in paper airplanes does not happen for the traditional airfoils used as aircraft wings, whose center of pressure stays fixed in place across the angles that occur in flight,” said Ristroph. “The shifting of the center of pressure thus seems to be a unique property of thin, flat wings, and this ends up being the secret to the stable flight of paper airplanes.”

“This is why airplanes need a separate tail wing as a stabilizer,” he added, “while a paper plane can get away with just a main wing that gives both lift and stability.”

PAPER AIRPLANE AERODYNAMICS MAY LEAD TO INNOVATIVE AIRCRAFT

Published in the Journal of Fluid Mechanics , the study offers a unique insight into a child’s toy that nearly everyone has built. More importantly, it also found a whole new set of aerodynamic principles that the researchers believe could lead to the development of new types of hyper-efficient, specialized aircraft.

“We hope that our findings will be useful in small-scale flight applications,” concluded Ristroph, “where you may want a minimal design that does not require a lot of extra flight surfaces, sensors, and controllers.”

Unfortunately, the study did not indicate where you should write the “Do you like me? Mark Yes or No” message. SMH.

Follow and connect with author Christopher Plain on Twitter:   @plain_fiction