UC Irvine

More than a million species, including insects, birds and bats, rely on flapping wings to fly. Despite being one of the most common modes of transportation in the animal kingdom, flapping flight has long been an enigma for fluid dynamicists. The complex air motion around moving wings generates an unsteady flow characterized by nonlinear interactions and intricate flow structures. The complexity of these dynamics does not permit an analytical distillation of the physical processes occurring in unsteady flows and, consequently, does not provide a means to predict the flight conditions in which interesting unsteady phenomena may occur. The lack of an analytical systematic method to predict and inspect the characteristics of these higher-order effects may be attributed to a scarcity of tools capable of analyzing nonlinear systems. Geometric control theory is defined as the application of differential geometry to the study of nonlinear control systems. Its capability to analyze higher-order effects allows for the study of the nonlinear interactions between the inputs of a system, which may lead to symmetry breaking or the generation of motion in an unactuated direction. The present work introduces the potential of geometric control theory in the analysis of unsteady flows, providing a framework for the systematic discovery of nonlinear unsteady phenomena. This study focuses on the analysis of the mean aerodynamic forces on a wing performing harmonic pitching, plunging, and surging oscillations. The present research introduces a state-space formulation for the aerodynamics of a pitching-plunging-surging wing, which is sufficiently rich to capture the main physical aspects of the flow but efficient and compact to permit an analytical study using gefometric control theory. A combination of averaging and geometric control tools are applied to the dynamical model to derive analytical expressions for the mean aerodynamic forces on the oscillating wing. These average unsteady forces are then compared to their steady couterparts to identify any force generation mechanisms that may appear due to symmetry breaking. Further analysis reveals flight regimes with lift enhancement and drag reduction/thrust generation mechanisms as a product of the unsteady motion. The same problem is also pursued using a more complex, well-established reduced-order model. The Beddoes-Leishman model for dynamic stall is a condensed state-space model capable of capturing the unsteady aerodynamic forces on an oscillating airfoil, including the contributions of the leading-edge vortex dynamics. Being in a state-space form makes it suitable for a geometric control analysis. The application of the averaging theorem in a geometric control framework permits an analytical study of the averaged dynamics of the unsteady forces. Similarly, the results uncover flight regimes presenting force generation mechanisms leading to lift enhancement and drag reduction.