Theory and Low-Order Modeling of Unsteady Airfoil Flows

Unsteady flow phenomena are prevalent in a wide range of problems in nature and engineering. These include, but are not limited to, aerodynamics of insect flight, dynamic stall in rotorcraft and wind turbines, leading-edge vortices in delta wings, micro-air vehicle (MAV) design, gust handling and flow control. The most significant characteristics of unsteady flows are rapid changes in the circulation of the airfoil, apparent-mass effects, flow separation and the leading-edge vortex (LEV) phenomenon. Although experimental techniques and computational fluid dynamics (CFD) methods have enabled the detailed study of unsteady flows and their underlying features, a reliable and inexpensive loworder method for fast prediction and for use in control and design is still required. In this research, a low-order methodology based on physical principles rather than empirical fitting is proposed. The objective of such an approach is to enable insights into unsteady phenomena while developing approaches to model them. The basis of the low-order model developed here is unsteady thin-airfoil theory. A time-stepping approach is used to solve for the vorticity on an airfoil camberline, allowing for large amplitudes and nonplanar wakes. On comparing lift coefficients from this method against data from CFD and experiments for some unsteady test cases, it is seen that the method predicts well so long as LEV formation does not occur and flow over the airfoil is attached. The formation of leading-edge vortices (LEVs) in unsteady flows is initiated by flow separation and the formation of a shear layer at the airfoil's leading edge. This phenomenon has been observed to have both detrimental (dynamic stall in helicopters) and beneficial (high-lift flight in insects) effects. To predict the formation of LEVs in unsteady flows, a Leading Edge Suction Parameter (LESP) is proposed. This parameter is calculated from inviscid theory and is a measure of the suction at the airfoil's leading edge. It is hypothesized, and verified with experimental and computational data, that LEV formation always occurs at the same critical value of LESP irrespective of motion kinematics. Further, the applicability of the LESP criterion in influencing the occurrence of LEV formation is demonstrated. To model the growth and convection of leading-edge vortices, the unsteady thin-airfoil theory is augmented with discrete-vortex shedding from the leading edge. The LESP criterion is used to predict and modulate the shedding of leading-edge vorticity. Comparisons with experiments and CFD for test-cases with different airfoils, Reynolds numbers and motion kinematics, show that the method performs remarkably well in predicting force coefficients and flowfields for unsteady flows. The use of a single empirical parameter - the critical LESP value, allows the determination of onset, growth and termination of leading-edge vortex shedding. In the final part of the research, the discrete-vortex model is extended to flows where the freestream velocity is varying or small in comparison with motion velocity. With this extension, the method is made applicable to a larger set of 2D flows such as perching and hovering maneuvers, gusts, and sinusoidally varying freestream. Abstractions of perching and hovering are designed as test cases and used to validate the low-order model's performance in highly-unsteady, vortex-dominated flows. Alongside development of the low-order methodology, several features of unsteady flows are studied and analyzed with the aid of CFD and experiments. While remaining computationally inexpensive and retaining the essential flow-physics, the method is seen to be successful in prediction of both force coefficients and flow histories.