No abstract

In the current study, we experimentally investigated the flexibility effects on the aerodynamic performance of flapping wings and the correlation with aspect ratio at angle of attack α  =  45°. The Reynolds number based on the chord length and the wing tip velocity is maintained at Re  =  5.3  ×  10. Our result for compliant wings with an aspect ratio of 4 shows that wing flexibility can offer improved aerodynamic performance compared to that of a rigid wing. Flexible wings are found to offer higher lift-to-drag ratios; in particular, there is significant reduction in drag with little compromise in lift. The mechanism of the flexibility effects on the aerodynamic performance is addressed by quantifying the aerodynamic lift and drag forces, the transverse displacement on the wings and the flow field around the wings. The regime of the effective stiffness that offers improved aerodynamic performance is quantified in a range of about 0.5-10 and it matches the stiffness of insect wings with similar aspect ratios. Furthermore, we find that the aspect ratio of the wing is the predominant parameter determining the flexibility effects of compliant wings. Compliant wings with an aspect ratio of two do not demonstrate improved performance compared to their rigid counterparts throughout the entire stiffness regime investigated. The correlation between wing flexibility effects and the aspect ratio is supported by the stiffness of real insect wings.

Abstract. Tukey depth function is one of the most famous multivariate tools serving robust purposes. It is also very well known for its computability problems in dimensions p ≥ 3. In this paper, we address this computing issue by presenting two combinatorial algorithms. The first is naive and calculates the Tukey depth of a single point with complexity O n p−1 log(n) , while the second further utilizes the quasiconcave of the Tukey depth function and hence is more efficient than the first. Both require very minimal memory and run much faster than the existing ones. All experiments indicate that they compute the exact Tukey depth.