By Matthew R. Francis
Many readers have likely heard the common trope that bumblebees should not be able to fly. This is obviously a misleading statement—bumblebees can fly, so their ability to do so is self-evident—but it highlights some of the complications that researchers face in their quest to understand insect flight. Due to their small body size and relatively low flight speeds, insects occupy a middle region of aerodynamics that lies between the macroscopic regime of airplanes and the microscopic world of bacteria; in this intermediate realm, the rules that govern each domain do not fully apply.
But insect flight is not simply about aerodynamics. It also pertains to evolution and the biological workings inside insect bodies. “I’m interested in insects as a living system,” Jane Wang of Cornell University said. “The balancing act of the insect is very interesting, because it depends on the physics of flight as well as their evolved neurocircuitry. My hunch has been that part of the neural behavior can be understood best if we start from the physical principles of flight.”
During her invited presentation at the 2020 SIAM Conference on the Life Sciences, which took place virtually this June, Wang described the wide range of theoretical models and laboratory experiments that are required to ask and answer questions pertaining to insect flight. “It’s such a rich problem and contains many elements that we do not normally think about in non-living systems,” she said.
Wang and her colleagues examined several species of insect to understand the mechanism that is needed to achieve controlled flight. They measured reaction times, quantified wing and body kinematics, computed flight maneuvers, and even found a potential link between a specific muscle and flight stability in flies.
Insects are the most diverse group of animals on the planet, which complicates matters. Biologists have identified over 900,000 species, and as many as 30 million are still undescribed. From the tiniest fairyflies (wasps that are barely visible without a microscope) to the heaviest Goliath beetles, many insects fly and possess distinct flight mechanics. Bumblebees, dragonflies, and locusts—just to name a few—fly in different ways that reflect their various strategies for survival and feeding. These behavioral aspects accompany more straightforward issues of body size, weight, and shape while tying into the insects’ sensory organs and nervous systems.
Flight on the Edge of Instability
However, a few commonalities do exist across species. Despite insects’ relatively wide range of sizes, their flight is intrinsically unstable — they must therefore exert effort to maintain their orientation and avoid falling. Wang likens insect flight strategy at the border of stability and instability to our own human bipedal stance. Because we stand upright on two legs without additional balancing organs, we are always very close to falling over (anyone who has ever experienced equilibrium issues knows this well). Even standing in place requires constant tiny adjustments in posture. However, bipedalism means that we can translate our state of perpetual almost-falling into the efficient forward motion known as walking.
Wang postulates that the boundary between stable and unstable flight may provide similar advantages for flying insects [1]. Just as we exploit our unstable human uprightness to walk forward, insects might use flight instabilities to stay aloft and preserve maneuverability.
The starting point when modeling flapping insect wings is the Navier-Stokes equations for incompressible fluids:
\[\frac{\partial \vec{u}}{\partial t}+ \vec{u} \cdot \triangledown \vec{u}=-\triangledown p + \frac{1}{\textrm{Re}}\triangledown\vec{u}\]
\[\triangledown \cdot \vec{u}=0.\]
Here, all physical quantities are expressed as dimensionless quantities and scaled by a characteristic velocity \(U\) and length scale \(L\).
This dimensionless form of the Navier-Stokes equation reveals a vital parameter: the Reynolds number \(\textrm{Re}\). \(\textrm{Re}=LU/\nu\), where \(\nu\) is the kinematic viscosity of the fluid. \(\textrm{Re}\) is approximately 150 for a fruit fly and around 3,000 for a dragonfly. Since the transition of smooth fluid flow to unsteady flows occurs roughly at \(\textrm{Re}=1,000\) and the flapping wings move at high angles of attack, many insects literally fly among vortices of turbulent air. Airplanes have Reynolds numbers that are several orders of magnitude higher, while bacteria live in a world of very tiny \(\textrm{Re}\) values (because of their small size).
“At this intermediate range of Reynolds numbers, you cannot ignore either the viscous force or the fluid inertia,” Wang said. “In the case of flapping flight, the wings reverse periodically. Each time they do this they generate vortices. These vortices interact, so the flow is complex and hard to wrap our heads around.”
In fact, Wang believes that it is possible for the vortices to provide some of the extra lift forces that insects need to stay aloft. To test this, she computed the flows and forces by solving the Navier-Stokes equations and studied plates falling through air and water, which have a similar range of Reynolds numbers. This allowed Wang and her students to recreate the types of vortices that are shed by flapping wings and extract theoretical models for the aerodynamic forces exerted on these wings.
Hovering and Maneuvering
Figure 1. The phase diagram for flight control. The colors represent the root-mean-square pitch angle of the insect body in degrees. Smaller values (blue) indicate well-controlled flight and higher values indicate insect tumbling. Figure courtesy of [1].
The mechanics and aerodynamics comprise only part of the puzzle that Wang wishes to solve. Flight is dynamic; just as humans must actively maintain balance (a mostly reflexive and subconscious process), insects need to constantly react to their surroundings to control their body orientation and keep from falling.
Another important aspect of flight is hovering, which many flying insects can do (even if only for a short time). “This ability to pause has an advantage for maneuvering,” Wang said. “Like driving a car, you can change your orientation very effectively if you stop instead of turning in a big loop. In order to hover, insects need neurofeedback algorithms so that they maintain their upright posture.”
Wang’s control algorithm for her insect model has two timescales: the sensing rate \(T_s\) and the reaction delay time \(T_d\). Both govern the way in which the simulated insect adjusts to its body state. In particular, the phase diagram that spans these two timescales (see Figure 1) revealed the presence of a maximum reaction time that enables controlled flight, and demonstrated that effective control occurs when the sensing rate is in half-integer multiples of the rate at which the insects’ wings beat.
For fruit flies, the most robust control transpires at the rate of the wing beat frequency (roughly four milliseconds), which is shorter than the flies’ visual response time. They thus presumably receive sensory information from balancing organs called halteres, which scientists first identified in the early 18th century as essential for flight in flies. Subsequent research revealed that halteres—which are likely the evolutionary remnants of the second set of wings that many other insect groups possess—act much like gyroscopes and allow insects to detect their bodies’ three-dimensional rotational rate. In addition, Wang speculates that a muscle called b1, which fires with every wing beat, is essential for flight stability.
These final points indicate the power of interactions between mathematics and biological studies. Physical analyses inspired the connection to insect neuroanatomy, which in turn informed the experiments’ design and helped refine the theoretical models of flight control. Recognizing that insect flight involves a reflexive set of actions and exists on the boundary between stability and instability proved evolutionarily successful. We can observe the flight of the bumblebee, after all; understanding its flight, however, requires the right balance of seemingly disparate fields.
This article is based on Jane Wang’s invited talk at the 2020 SIAM Conference on the Life Sciences, which occurred virtually earlier this year. Wang’s presentation is available on SIAM’s YouTube Channel.
References
[1] Wang, Z.J. (2016). Insect flight: from Newton’s laws to neurons. Ann. Rev. Cond. Matt. Phys., 7, 281.
Matthew R. Francis is a physicist, science writer, public speaker, educator, and frequent wearer of jaunty hats. His website is BowlerHatScience.org.