Reduced-Order Modeling of Unsteady Aerodynamics: Theoretical and Computational Approaches
Unsteady aerodynamics emerges in several applications in mechanical and aerospace engineering such as the flutter problem, dynamic stall, bio-inspired flying robots, helicopters, and wind turbines. If a leap were to happen in the design of these machines, it would have to happen in the preliminary design phase where millions of design alternatives are investigated, which cannot and should not be performed with high-fidelity simulations. Therefore, the development of reduced-order models for unsteady aerodynamic loads will be of paramount importance for advancing these applications, which is the main focus of this Dissertation.
The common (perhaps only) reduced-order models of unsteady aerodynamics in literature are based on potential flow theory, which is neither complete (invokes a closure condition) nor descriptive of viscous effects. While the Kutta condition is quite successful in small-angle of attack and large Reynolds number situations, it was originally devised for steady flows. Hence, its illegitimate use in unsteady environments may only be successful at low-frequencies. Therefore, there are numerous research reports that invoked another closure condition for potential flow in the unsteady case.
Three main contributions are achieved in this Dissertation. First, realizing that the lift development and vorticity production are essential viscous processes, we develop a viscous extension of the classical theory of unsteady aerodynamics by matching the potential flow theory with a special boundary layer theory that pays close attention to the flow details in the vicinity of the trailing edge: the triple deck boundary layer theory. Based on this extension, we develop for the first time a Reynolds-number-dependent lift frequency response. The theory is validated against high-fidelity simulations of the Navier-Stokes equations showing a remarkable matching. It is found that viscosity induces more phase lag at high reduced frequencies and low Reynolds numbers where the viscous effects are more pronounced.
Second, to extend the applicability of this viscous unsteady theory to account for arbitrary airfoil shapes, arbitrary kinematics, and wake deformation, a numerical method is developed. The developed numerical technique represents a viscous extension of the classical unsteady vortex lattice method (UVLM). In such an extension, we exploit Hilbert matrix algebra to show the relationship between the Kutta condition and the location of the control and collocation points on the panel. As such, we relax (correct for) the implicit Kutta condition in the UVLM by updating the locations of these points within the panel at each time based on the viscous correction coming from the triple deck boundary layer theory, which is based on the instantaneous airfoil motion.
Finally, due to the fact that many aerial vehicles operate at moderate Reynolds number, which is prone to laminar-to-turbulent transition and theoretical approaches are not expected to perform efficiently in those conditions, the transition effects on the lift and circulation dynamics on a pitching airfoil are investigated numerically. It is found that transition induces significant nonlinearities in the lift and circulation dynamics even at very small amplitudes down to half a degree. It is shown that this nonlinearity in the lift dynamics is attributed to the violation of the Ku0tta condition in this regime. Based on this connection, we delineate how the potential flow theory can be extended with the aid of high-fidelity simulation data to capture the nonlinear transition effects.