Turbulence-based parametrization of animal flight

Abstract

Animals capable of powered flight range in wingspan from a few hundred microns to a few meters. The inertial turbulence to which these animals are exposed features vortices ranging from a few hundred micrometers to hundreds of kilometers in size. Yet, the impact of ambient turbulence on animal flight is virtually uncharted and most studies on animal flight are conducted in still air or under laminar conditions. Here, we propose a novel parameterization that links animal flight with turbulence, through a proxy for the energy injected into the atmosphere, $E_{sp}=b^3 f^2$, with $f$ the animal's flapping frequency and $b$ the wingspan. We model this parameter using a scaling relation in the shape of a power law $E_{sp} \propto k^α$, with $k=1/b$ the wavenumber corresponding to the animal inverse wingspan. Literature provides four theoretical predictions on the exponent $α$: two connected to aerodynamic and energetic aspects of flight, $α_{aero}=-2$ and $α_{power}=-5/3$, and two linked to physiological limits. Drawing from experimental data of over 400 species spanning 13 insect orders and two vertebrate classes, we recover $α_{power}=-5/3$ as the best scaling relation across the animal kingdom. Grouping per animal clade however reveals a secondary power law with $α=-5/2$ exponent for invertebrate orders, with a family-dependent coefficient. This new scaling relation suggests a yet-unknown universal physical mechanism in insect flight, likely depending on wing morphology and mechanical properties.