Unsteady Aerodynamics

1.3 Preliminaries

A lifting surface embedded in a disturbed flow experiences time vari­ations of both lift and drag forces. These variations are sources of sound according to the acoustic analogy. For airfoil-like designed bodies with at­tached flows considered later on, lift variations are much larger than drag forces, therefore the latter are neglected and the unsteady problem is ad­dressed based on non-viscous flow arguments. Random incident velocity disturbances cause time variations of both the magnitude and the angle of attack of the relative velocity vector experienced by the airfoil. These vari­ations induce a total instantaneous force F(t), or at a more precise level the corresponding local instantaneous lift forces distributed over the surface, noted £(t). When calculating the noise from the airfoil according to Ffowcs Williams & Hawkings’ equation, the major difficulty to deal with is the evaluation of this force field with enough accuracy. In the fluid-dynamics community, the mean value of F(t) or £(t), referred to as steady loading, is primarily addressed because only the steady-state is directly related to the aerodynamic efficiency the surface must ensure. Typically the airfoil is designed for a desired value of the mean lift coefficient CL, as a function of the angle of attack a. The fluctuations of F(t) or £(t) around the mean, essentially responsible for the noise, are much more difficult to quantify, and depend on external conditions not intrinsic to the surface design. The fluctuations bring no benefit to the aerodynamic efficiency and are only an undesirable source of noise (or vibration, even though vibrations are not addressed here). So any reduction of the unsteadiness, if possible, is a good deal from the acoustical point of view and does not essentially suffer from efficiency constraints.