From fluttering and tumbling flight to stable gliding and periodic soaring

In this piece Timmy Shum takes us on a journey of mathematical reflecting during Covid.

Illustration by Nino Mekanarishvili
Something peculiar that I like to do, as a distraction from the Covid-19 pandemic, is to imagine the neighborhood that I live in as a natural “research lab for fluid dynamics”, with fundamental “experiments” occurring everywhere around me.  Viewing my surroundings through the lens of applied mathematics, curious motions are thrusted to the foreground of my view: a leaf fluttering around in a soft wind; a dirty napkin tumbling and drifting away; a small bird gliding through the air; and an innocent butterfly in hovering flight.  In a moment of social withdrawal, and a quiet rebellion against the status quo, I indulge in the secret patterns and dynamics that no one else seems to notice.

And in my ghostly presence, with no closure and searching for meaning in a horrific era, I find comfort in drifting back to a time when I worked with Professor Leif Ristroph on the mathematical modeling and simulation of freely-falling wings — following the dreamy past work of Professor Jane Wang and her team at Cornell.  Mathematics had come to life, using differential equations to elucidate various flight patterns, and I swing into the present moment, watching a falling leaf flutter in the soft wind outside of Mud Cafe, in the East Village neighborhood, and I wonder what the motion looks like initially — when it falls from the tree and instantly feels the aerodynamic forces, lift and drag, acting on its body, and with gravity pulling it to the ground.  I wonder whether its initial flight behavior, if captured by a high-speed camera, would look something like this numerical solution:
Figure 1

And if so, then after this erratic, transient behavior dissipates, would the falling leaf settle into its elegant, fluttery flight, oscillating from side to side while descending, much like in this simulation:

Figure 2

Drifting back into the present, I get an iced chai latte at Mud Cafe and sit outside on their rusty metal bench, since there’s no indoor dining allowed.  The neighborhood feels like a ghost town, and a war zone.  My face is draped in a colorful Batik face covering, bought from a local store which sells, exclusively, Indian cloth, made by a fourth-generation family owner, who studied engineering at MIT.  Moments later, a thin, dirty napkin flies into the wind, in a tumbling motion, rotating and drifting away while descending, similar to this numerical solution:

Figure 3

I sip on my refreshing iced chai latte, as I wonder whether the barber shops will ever reopen.  And as I ponder about my long, disheveled hairstyle, a small bird comes to view, gliding through the air, and it evokes thoughts of the moment when Prof. Ristroph and I first simulated a freely-falling wing that can glide stably in air, rather than flutter or tumble to the ground:

Figure 4

The intrinsic stability seemed interesting, with the wing’s glide angle remaining roughly constant, after initial, transient behavior:

Figure 5
Drifting back into the present, it is 7 pm, and I hear clanging pots and pans, and bittersweet cheering and clapping for the first responders of Covid-19 patients.  I finish my iced chai latte at Mud, and head back home to use my bathroom.  My refrigerator is empty.  There’s no food at home.

Laying down for the evening, I tinker with the aerodynamic model which now has elegant gliding flight solutions, and I revisit the idea of adding a “shear flow” to the equations — whose numerical solution seems rather striking:
Figure 6
Hmm, the wing seems to be settling into a longtime periodic flight behavior, using the shear flow to fly in orbits.  In fact, the solution appears periodic in all variables but x, with the lack of periodicity in x giving rise to passive-dynamic locomotion, powered by a wind shear — a phenomenon beautifully described to me by Professor Andy Ruina.

I then wanted to visualize the net fluid forces (in magenta) acting on the wing, as well as the wing’s instantaneous velocity (in green), while the wing “dynamically soars” in the shear flow (in cyan):
Figure 7
Since dynamic soaring behaviors have long been observed in the flight of the albatross, near sea level, my natural “research lab” must now extend from Manhattan to the oceans, where the albatross uses nothing more than the fast-changing speeds of the horizontal winds to achieve stable flight, without flapping its wings to generate lift.

I’m probably being grandiose with my thoughts again.  Somebody please reign in my thoughts, so that they do not roam wild.  Is this math model physically reasonable?  Does it have any predictive capacity?  Could it someday inform the development of passive-dynamic or minimally-controlled robotics?  Could I derive a PDE, whose solution describes the evolution of a simple shear flow, subject to a moving boundary (the soaring wing)?  I shouldn’t be so grandiose with my thoughts.  Perhaps someday, when I work in an actual research lab again, in a post-Covid-19 era, I’ll resume comparing math models and simulations with physical experiments, making sharper estimates.  But for now, in the tragic world we’re living in, the people around me arguing over politics and conspiracy theories, the local coffee shops that remain open as “essential businesses”, and the ambulances hurrying to the hospital with their unsettling sirens form the backdrop of my daily life, as I contemplate on the fluid dynamics of the gliding bird, the tumbling napkin, and the fluttering leaf in a still wind.

Timmy Shum

NY, NY, USA

Illustration by Liani Malherbe

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