On the roles of chord-wise flexibility in a flapping wing with hovering kinematics

The aerodynamic performance of a flapping two-dimensional wing section with simplified chord-wise flexibility is studied computationally. Bending stiffness is modelled by a torsion spring connecting two or three rigid components. The leading portion of the wing is prescribed with kinematics that are characteristic of biological hovering, and the aft portion responds passively. Coupled simulations of the Navier–Stokes equations and the wing dynamics are conducted for a wide variety of spring stiffnesses and kinematic parameters. Performance is assessed by comparison of the mean lift, power consumption and lift per unit power, with those from an equivalent rigid wing, and two cases are explored in greater detail through force histories and vorticity snapshots. From the parametric survey, four notable mechanisms are identified through which flexible wings behave differently from rigid counterparts. Rigid wings consistently require more power than their flexible counterparts to generate the same kinematics, as passive deflection leads to smaller drag and torque penalties. Aerodynamic performance is degraded in very flexible wings undergoing large heaving excursions, caused by a premature detachment of the leading-edge vortex. However, a mildly flexible wing has consistently good performance over a wide range of phase differences between pitching and heaving – in contrast to the relative sensitivity of a rigid wing to this parameter – due to better accommodation of the shed leading-edge vortex into the wake during the return stroke, and less tendency to interact with previously shed trailing-edge vortices. Furthermore, a flexible wing permits lift generation even when the leading portion remains nearly vertical, as the wing passively deflects to create an effectively smaller angle of attack, similar to the passive pitching mechanism recently identified for rigid wings. It is found that an effective pitch angle can be defined that accounts for wing deflection to align the results with those of the equivalent rigid wing.