A Nonlinear Indicial Method for the Calculation of Unsteady Airloads

The development of a nonlinear, indicial based unsteady airfoil theory is discussed. The theory is numerically efficient, and is suitable for use in routine helicopter aero-acoustics and rotor performance predictions. A principal aim of the study is to explore the limits of linear unsteady aerodynamic theory, and to extend the basis of the indicial method to the treatment of pertinent nonlinear unsteady aerodynamic phenomena. The main advantage of the method is that when the indicial response to a particular forcing mode is determined, the cumulative response to an arbitrary forcing in that mode can be obtained by means of Duhamel superposition. This technique leads to nearly four orders of magnitude reductions in computational time to find the unsteady airloads. The primary expense with the indicial method, however, is in the proper identification and functional representation of the indicial response functions themselves. In the present work, the indicial responses (both linear and nonlinear) have been extracted from computational fluid dynamics (CFD) calculations based on an Euler method. The various responses were then represented, albeit approximately, in exponential functional form. For oscillating airfoil problems, the study shows that the linear indicial theory gives good agreement with results from direct CFD calculations, as long as any nonlinearities in the flow are weak, i.e., at low mean angle of attack and at sub-critical free-stream Mach numbers. The nonlinear indicial method is shown to provide better predictions of the unsteady airloads when significant aerodynamic nonlinearities exist, such as those associated with supercritical flows and shock wave movement.