Reynold number scaling of circulation growth on flapping wings.

This article focuses on developing and validating a scaling relationship for the thrust coefficient (\(C_T\)) of insect-scale flapping wings based on wing kinematics, specifically reduced frequency (\(k\)) and Strouhal number (\(St\)). Using a model grounded in von Kármán and Sears' potential flow theory and incorporating a Blasius-like stream function to represent circulation transport in the leading-edge vortex (LEV) shear layer, the study compares two cases: one with Reynolds number (\(Re\)) scaling and one without. Analysis of 168 flights from 38 live mountain pine beetles shows that both models predict thrust similarly well, but only the Reynolds number–scaled model accurately predicts circulation in the wake, highlighting the Reynolds number's critical role in modeling flow field dynamics despite the inviscid assumptions of classical theory. The findings suggest that while force scaling can be approximated without explicit Reynolds number dependence, incorporating it is essential for accurate flow field representation, with implications for understanding insect flight aerodynamics and informing engineered flapping-wing designs.