Flapping dynamics of a compliant membrane in a uniform incoming flow

We numerically investigate the effects of the stretching coefficient, K_S, the flapping frequency, St, and the pitching amplitude on the propulsive performance of a compliant membrane undergoing combined heaving and pitching in uniform flow. Distinct optimal values of K_S are identified that respectively maximize thrust and efficiency: thrust can be increased by 200%, and efficiency by 100%, compared to the rigid case. Interestingly, these optima do not occur at resonance but at frequency ratios (flapping to natural) below unity . Using a force decomposition based on the second invariant of the velocity gradient tensor Q, we show that thrust primarily arises from Q-induced and body-acceleration forces. The concave membrane surface can trap the leading-edge vortex (LEV) from the previous half-stroke, generating detrimental Q-induced drag. However, moderate concave membrane deformation weakens this LEV and enhances body-acceleration-induced thrust. Thus, the optimal K_S for maximum thrust occurs below resonance, balancing beneficial deformation against excessive drag. Furthermore, by introducing the membrane's deformation into the leading-edge tangential angle and substituting it into an existing scaling law developed for rigid plates, we obtain predictive estimates for the thrust and power coefficients of the membrane. The good agreement confirms the validity of this approach and offers insights for performance prediction.