Effects of flexibility on the aerodynamic performance of flapping wings

Abstract

Effects of chordwise, spanwise, and isotropic flexibility on the force generation and propulsive efficiency of flapping wings are elucidated. For a moving body immersed in viscous fluid, different types of forces, as a function of the Reynolds number, reduced frequency (k), and Strouhal number (St), acting on the moving body are identified based on a scaling argument. In particular, at the Reynolds number regime of $O(1{0}^{3} \ensuremath{-} 1{0}^{4} )$ and the reduced frequency of $O(1)$, the added mass force, related to the acceleration of the wing, is important. Based on the order of magnitude and energy balance arguments, a relationship between the propulsive force and the maximum relative wing-tip deformation parameter ($\gamma $) is established. The parameter depends on the density ratio, St, k, natural and flapping frequency ratio, and flapping amplitude. The lift generation, and the propulsive efficiency can be deduced by the same scaling procedures. It seems that the maximum propulsive force is obtained when flapping near the resonance, whereas the optimal propulsive efficiency is reached when flapping at about half of the natural frequency; both are supported by the reported studies. The established scaling relationships can offer direct guidance for micro air vehicle design and performance analysis.

On the diverse roles of fluid dynamic drag in animal swimming and flying

“…Of course, 'insect-inspired' is a rather broad label, considering the wide variety of insect wing morphologies and kinematics, and one can reasonably expect different force balances in, say, tiny insects performing clap-fling [100,101] or larger insects in cruising flight [102] or taking-off [103]. Wing compliance has been identified as one of the key points that determine the performance of flapping wings [82,83,[104][105][106]. More precisely, it has been observed that during a stroke cycle, the trailing edge response of the wing was characterized by a strong lag with respect to the imposed motion of the leading edge.…”

Aerodynamics, sensing and control of insect-scale flapping-wing flight

“…Wing flexibility can enhance propulsive force generation, while reducing the power consumption (e.g. [9,[123][124][125][126]). Also, optimal efficiency are observed for an effective Strouhal number based on the deformed wing motion between 0.25 and 0.35 [127], similar to pitching and plunging rigid wings [15].…”

“…[9,124]) by capturing the first-order, important mechanism in the fluidstructure interaction. These relationships provide time-averaged estimates for the lift, power input and efficiency [9,124,127] (figure 7), consistent with relationships based on experiments [124,127]. In general, the fluid dynamic force acting on a moving wing can be decomposed into two terms: (i) the acceleration-reaction force, often linearized by the added mass [128], and (ii) the force induced by the vorticity in the flow field [9,129], or the aerodynamic damping term.…”

“…For example, the rigid wing models by Theodorsen [67] forward flight and Dickinson et al [26] for hover flight also include both forces. For fast flapping motions, the wing acceleration and hence the acceleration-reaction forces are greater than the vorticity-induced force [9]. Under these circumstances, the fluid dynamic force can be approximated by the added mass and scaling relationships can be established for time-averaged aerodynamic quantities [9,124,130].…”

“…which is a combination of the dimensionless parameters introduced in tables 1 and 3 [9,74]. Then the time-averaged lift coefficientC L correlate to γ as…”