A universal law for predicting the thrust performance of flapping thin airfoils

Accurate prediction of thrust performance is critical for understanding the flight capabilities of biological organisms and optimizing bio-inspired aerial vehicles. For a given amplitude-based Strouhal number (St), extensive research has shown that a specific pitch amplitude maximizes thrust in two-dimensional airfoils undergoing coupled heaving-pitching motion. However, thrust variation depends not only on heaving amplitude and frequency but also on the kinematic details of pitching motion, and the explicit relationship between the optimal pitch amplitude and St remains unclear. In this study, we propose a predictive method to identify the optimal pitch amplitude ( θ opt ) through a very concise equation θ opt = arctan ( π St ) / 2 that make thrust for a flapping thin airfoil maximum under given conditions, and accurately estimates the pitch amplitude at the thrust-drag transition through θ zero = arctan ( π St ) . Furthermore, the method's applicability to different Reynolds numbers (Re) confirms its universality in low Re. These findings provide valuable insights into thrust performance prediction for bio-inspired flight systems and offer a practical tool for rapid optimization of flapping flexibility in design applications.