Aerodynamic efficiency explains flapping strategies used by birds

A faster cruising speed increases drag and thereby the thrust (<i>T</i>) needed to fly, while weight and lift (<i>L</i>) requirement remains constant. Birds can adjust their wingbeat in multiple ways to accommodate this change in aerodynamic force, but the relative costs of different strategies remain largely unknown. To evaluate the efficiency of several kinematic strategies, I used a robotic wing [E. Ajanic, A. Paolini, C. Coster, D. Floreano, C. Johansson, <i>Adv. Intell. Syst.</i> <b>5</b>, 2200148 (2023)] and quantitative flow measurements. I found that, among the tested strategies, changing the mean wingbeat elevation provides the most efficient solution to changing thrust-to-lift ratio (<i>T</i>/<i>L)</i>, offering insight into why birds tend to beat their wings with a greater ventral than dorsal excursion. I also found that although propulsive efficiency (η_p) may peak at a Strouhal number (<i>St</i>, measure of relative flapping speed) near 0.3, the overall efficiency of generating force decreases with <i>St</i>. This challenges the expectance of a specific optimal <i>St</i> for flapping flight and instead suggest the chosen <i>St</i> depends on <i>T</i>/<i>L</i>. This may explain variation in preferred <i>St</i> among birds and why bats prefer flying at higher <i>St</i> than birds [G. K. Taylor, R. L. Nudds, A. L. Thomas, <i>Nature</i> <b>425</b>, 707-711 (2003)], since their body shape imposes relatively higher thrust requirements [F. T. Muijres, L. C. Johansson, M. S. Bowlin, Y. Winter, A. Hedenström, <i>PLoS One</i> <b>7</b>, e37335 (2012)]. In addition to explaining flapping strategies used by birds, my results suggest alternative, efficient, flapping motions for drones to explore aiming to extend their flight range.