Geometric control analysis of the unsteady aerodynamics of a pitching–plunging airfoil in dynamic stall

Abstract

Geometric control theory is the application of differential geometry to the study of nonlinear dynamical systems. This control theory permits an analytical study of nonlinear interactions between control inputs, such as symmetry breaking or force and motion generation in unactuated directions. This paper studies the unsteady aerodynamics of a harmonically pitching–plunging airfoil in a geometric control framework. The problem is formulated using the Beddoes–Leishman model, a semi-empirical state space model that characterizes the unsteady lift and drag forces of a two-dimensional airfoil. In combination with the averaging theorem, the application of a geometric control formulation to the problem enables an analytical study of the nonlinear dynamics behind the unsteady aerodynamic forces. The results show lift enhancement when oscillating near stall and thrust generation in the post-stall flight regime, with the magnitude of these force generation mechanisms depending on the parameters of motion. These findings demonstrate the potential of geometric control theory as a heuristic tool for the identification and discovery of unconventional phenomena in unsteady flows.