I will discuss two recent results from our group. First, I will show how the insect flight stability problem was perplexing to biologists and engineers for decades. We recently resolved a part of this puzzle; we discovered a fascinating stabilization mechanism that insects unconsciously exploit during flight: the natural oscillation of an insect's body around the hovering equilibrium position provides a passive/free stabilizing action through the vibrational stabilization phenomenon. This stabilization technique could not be captured using the averaging approach commonly used in literature. In contrast, it is discovered using higher-order averaging via a special type of calculus: the chronological calculus. This problem serves as a good motivation for reduced-order modeling of aerodynamics; such a discovery would hardly be made by direct simulations of the Navier-Stokes equations.
Second, driven by such a motivation, I will discuss our recently developed variational theory of aerodynamics. The Euler equation does not possess a unique solution for the flow over a two-dimensional object. This problem has serious repercussions in aerodynamics; it implies that the inviscid aero-hydrodynamic lift force over a two-dimensional object cannot be determined from first principles; a closure condition must be provided. The Kutta condition has been ubiquitously considered for such a closure in the literature, even in cases where it is not applicable (e.g. unsteady). In this talk, I will present a special variational principle that we revived from the history of analytical mechanics: Hertz' principle of least curvature. Using this principle, we developed a general (dynamical) closure condition that is, unlike the Kutta condition, derived from first principles. In contrast to the classical theory, the proposed variational theory is not confined to sharp edged airfoils; i.e., it allows, for the first time, theoretical computation of lift over arbitrarily smooth shapes, thereby generalizing the century-old lift theory of Kutta and Zhukovsky. Moreover, the new variational condition reduces to the Kutta condition in the special case of a sharp-edged airfoil, which challenges the widely accepted concept regarding the viscous nature of the Kutta condition. We also generalized this variational principle to Navier-Stokes', thereby discovering the fundamental quantity that Nature minimizes in every incompressible flow.