Geometric-control formulation and averaging analysis of the unsteady aerodynamics of a wing with oscillatory controls

Abstract

Differential-geometric-control theory represents a mathematically elegant combination of differential geometry and control theory. Practically, it allows exploitation of nonlinear interactions between various inputs for the generation of forces in non-intuitive directions. Since its early developments in the 1970s, the geometric-control theory has not been duly exploited in the area of fluid mechanics. In this paper, we show the potential of geometric-control theory in the analysis of fluid flows, exemplifying it as a heuristic analysis tool for discovery of symmetry-breaking and unconventional force-generation mechanisms. In particular, we formulate the wing unsteady aerodynamics problem in a geometric-control framework. To achieve this goal, we develop a reduced-order model for the unsteady flow over a pitching–plunging wing that is (i) rich enough to capture the main physical aspects (e.g. nonlinearity of the flow dynamics at large angles of attack and high frequencies) and (ii) efficient and compact enough to be amenable to the analytic tools of geometric nonlinear control theory. We then combine tools from geometric-control theory and averaging to analyse the developed reduced-order dynamical model, which reveals regimes for lift and thrust enhancement mechanisms. The unsteady Reynolds-averaged Navier–Stokes equations are simulated to validate the theoretical findings and scrutinize the underlying physics behind these enhancement mechanisms.