Back-scattering correction and further extensions of Amiet's trailing-edge noise model. Part 1: theory

A previously published analytical formulation aimed at predicting broadband trailing-edge noise of subsonic airfoils is extended here to account for all the effects due to a limited chord length, and to infer the far-field radiation off the mid-span plane. Three-dimensional gusts are used to simulate the incident aerodynamic wall pressure that is scattered as acoustic waves. A leading-edge back-scattering correction is derived, based on the solution of an equivalent Schwarzschild problem, and added to the original formula. The full solution is found to agree very well with other analytical results based on a vanishing Mach number Green's function tailored to a finite-chord flat plate and sources close to the trailing edge. Furthermore, it is valid for any subsonic ambient mean flow velocity. The back-scattering correction is shown to have a significant effect at lower reduced frequencies, for which the airfoil chord is acoustically compact, and at the transition between supercritical and subcritical gusts. It may be important for small-size airfoils, such as automotive fan blades and similar technologies. The final far-field noise formula can be used to predict trailing-edge noise in an arbitrary configuration, provided that a minimum statistical description of the aerodynamic pressure fluctuations on the airfoil surface close to the trailing edge is available.