Falling leaves

Autumn is the “season of mists and mellow fruitfulness,” as John Keats described it in his poem “To Autumn”. It’s an idyllic season anticipated by the serotinal period where you experience a pleasant nostalgia about summer and you remember how happy you were going back to school, at this time, with good intentions for the new academic year.

During this time of the year, if you meander in a leafy area, you can gaze in wonder at how the natural scenery has dressed in yellowish and reddish splotched clothes. As you step on the mushy foliage that wallpapers the humus, you can start meditating and mooning about, breathing the breezy and crispy air that wafts across the bucolic landscape. Waldeinsamkeit, as Germans say, if you are walking alone in a forest.

However, if your stream of thoughts zooms in towards the falling leaves, you may want to “return to your science,” as Virgil once instructed his pupil Dante. Or, like Rick exhorted Morty, "focus on science!" Leaves don’t just simply fall. In fact, Emily Brontë in her poem describes the leaves as “fluttering from the autumn tree.”

Fall, leaves, fall; die, flowers, away;

Lengthen night and shorten day;

Every leaf speaks bliss to me

Fluttering from the autumn tree.

I shall smile when wreaths of snow

Blossom where the rose should grow;

I shall sing when night's decay

Ushers in a drearier day.

— Emily Brontë, "Fall, Leaves, Fall"

Your physics teacher has probably told you that even when you leave the classroom, physics is always around you. As a matter of fact, the term physics comes from the Ancient Greek word “physis,” which means nature. 

If you carefully observe falling leaves, you will notice that they may follow different types of motion, fluttering, if they oscillate from side to side, tumbling, if they rotate and drift sideways, and gyrating, if they spiral. This dance is governed by the combination of aerodynamic and gravitational forces. When these forces do not act at the same point in the falling object, a torque is induced.

Since leaves fall in a fluid, i.e. the air, they are subjected to the gravitational force as well as air resistance, or aerodynamic drag. The drag force can be divided into inertial drag and viscous drag. The inertial drag comes from the fact that the moving object must push away the fluid, while the viscous drag comes from the friction between the fluid and the surface of the object. The ratio of inertial drag to viscous drag is called Reynolds number (Re). 

Some of the biggest minds of the 19th-century, such as James Maxwell, Lord Kelvin, and Gustav Kirchhoff, studied the motion of falling strips in a fluid. Unfortunately, they couldn’t travel to the future to get access to a supercomputer to run simulations. With the advent of chaos theory and the increasing computational power, physicists are investigating this phenomenon with new eyes. “Everything depends on the color of the crystal through which one sees it,” as Pedro Calderón de la Barca once stated. Watch carefully! The motion of falling leaves can appear very complex and hard to predict.

Dmitry Kolomenskiy and Kai Schneider have studied the influence of the Reynolds number on falling leaves by solving the Navier–Stokes equations with a numerical approach [1]. They found out that at a low Re, leaves fall steadily. At Re = 100, the motion is oscillatory while at Re = 1000, the motion appears to be chaotic due to the presence of vortices.

These simulations are consistent with the experimental observations of the dynamics of disks falling in water/glycerol mixtures conducted by Stuart B. Field et al. [2]. At lower Re, they observed a steady-falling regime in which the disks fall with horizontal orientation. At higher Re and low moments of inertia, a disk is subjected to a periodic-oscillating motion. A chaotic motion appears when both Re and the moment of inertia are high. In this regime a disk oscillates with larger and larger amplitude until it flips over and starts tumbling until it goes back to the oscillating regime. At very large moments of inertia they observe tumbling motion.

A. Belmonte et al. found that at high Re, the dynamics of falling strips in water or water/glycerol mixture is governed by the Froude number (Fr) [3]. This number is commonly defined in many applications as the ratio of inertial and gravitational forces. According to their study, the transition from fluttering to tumbling motion occurs at Fr = 0.67 ± 0.05. If a fluttering strip has enough angular momentum at the overturning point, it will rotate completely. In their model, the Fr is proportional to the square root of mass and inversely proportional to the length of the strip. Therefore, long strips will flutter while shorter strips will tumble. Since the length of the trajectories was insufficient, the authors of that study couldn't determine if chaotic trajectories were also present.

These models can be used to elucidate the physics behind falling leaves. However, it is safe to say that the shape of a leaf is far more complex compared to the shape of a disk or a strip. Small perturbations will have a big effect on its aerodynamics and the motion will appear chaotic.

Next time you err in the woods, look at the falling leaves and pay attention to their motion. Will you observe a fluttering motion? A tumbling motion? Or a chaotic trajectory?

 In addition, since you will be ambling in the woods, it is worth mentioning that your Fr will be below the subcritical value of 0.5. If you start walking faster, your Fr will increase. Once it reaches the subcritical value, you will spontaneously change your motion and you will start running. The transition occurs at a speed of 2 m/s. It turns out that at this speed you may still be able to walk with some effort. However, at speeds greater than 3 m/s, corresponding to a Fr greater than 1, the critical value, you will not be able to walk anymore and you will be forced to run. Try it!

 References

[1] D. Kolomenskiy, K. Schneider Numerical simulations of falling leaves using a pseudo-spectral method with volume penalization. Theor. Comput. Fluid Dyn. 24, 169–173 (2010)

[2] S. B. Field, M. Klaus, M. G. Moore, F. Nori Chaotic dynamics of falling disks. Nature 388 (1997)

[3] A. Belmonte, H. Eisenberg,  E. Moses From Flutter to Tumble: Inertial Drag and Froude Similarity in Falling Paper. Phys. Rev. Lett. 81, 345 (1998)