Unsteady Aerodynamic Modeling with Time-Varying Free-Stream Mach Numbers

The development of a reduced-order unsteady airfoil theory is described for application to subsonic compressible flows with variable free-stream Mach number. The airfoil theory, which is suitable for application to most types of blade element based comprehensive helicopter rotor analyses, is developed for arbitrary, time-dependent combined variations in angle of attack and Mach number. The approach is validated using computational fluid dynamics (CFD) solutions based on the Euler equations. The results show that existing blade element unsteady aerodynamic theories can give good results only if the local Mach number variations are small in amplitude and vary slowly in time (i.e., a nearly quasi-steady assumption). However, helicopter rotor problems involve significant nonsteady variations in local Mach number at the blade element level and so violate the quasi-steady Mach number assumptions that are inherent in most existing unsteady aerodynamic models. The new model is developed using the indicial theory as a basis, and shows excellent agreement with direct CFD solutions for a wide range of practical flows relevant to helicopter rotor applications. For supercritical flows, nonlinearities associated with the formation and movement of shock waves are observed in the CFD solutions, which are responsible for differences in the predicted results. The estimation of the pitching moment is found to be particularly challenging because it is sensitive to these nonlinear phenomena. Overall, this study shows that the new approach provides significant improvements in sectional airloads predictions over existing methods that might be used in various types of comprehensive helicopter rotor analyses.