Numerical simulation and reduced-order modeling of a flapping airfoil

Recent advances in many fields have made the design of micro-aerial vehicles that implement flapping wings a possibility. However, there are many outstanding problems that must be solved before flapping flight can be implemented as a practical means of propulsion. This dissertation focuses on two important aspects of flapping flight: the physics of the flow of a fluid around a heaving airfoil and the development of a reduced-order model for the control of a flapping airfoil.; To study the physics of the flow, a numerical model for two-dimensional flow around an airfoil undergoing prescribed oscillatory motions in a viscous flow is developed. The model is used to examine the flow characteristics and power coefficients of a symmetric airfoil heaving sinusoidally over a range of frequencies and amplitudes. Both periodic and aperiodic solutions are found. Additionally, some flows are asymmetric in that the up-stroke is not a mirror image of the down-stroke.; For a given Strouhal number---defined as the product of dimensionless frequency and heave amplitude---the maximum efficiency occurs at an intermediate heaving frequency. This is in contrast to ideal flow models, in which efficiency increases monotonically as frequency decreases. Below a threshold frequency, the separation of the leading edge vortices early in each stroke reduces the force on the airfoil and leads to diminished thrust and efficiency. Above the optimum frequency, the efficiency decreases similarly to inviscid theory. For most cases, the efficiency can be correlated to interactions between leading and trailing edge vortices, with positive reinforcement leading to relatively high efficiency, and negative reinforcement leading to relatively low efficiency. Additionally, the efficiency is related to the proximity of the heaving frequency to the frequency of the most spatially unstable mode of the average velocity profile of the wake; the greatest efficiency occurs when the two frequencies are nearly identical. The importance of viscous effects for low Reynolds number flapping flight is discussed.; The computational model is used as the basis for developing a reduced-order model for active control of a flapping wing. Using proper orthogonal decomposition (POD), sets of orthogonal basis functions are generated for simulating flows at various heaving and pitching parameters. With POD, most of the energy in the flow is concentrated in just a few basis functions. These functions are used for the projection of the Navier-Stokes equations using a Galerkin projection, reducing them to a small set of coupled, non-linear ordinary differential equations. The Galerkin projection is used to simulate oscillatory motions that are both similar to, and different from, the motion used to generate the POD modes; however; errors are introduced into the model from several sources. The focus of the current work is on the causes and effects of errors in the model on important aspects of the flow, chiefly input and output power and efficiency. The suitability of this approach for controlling a flapping wing over a broad range of parameters is discussed.