Influence of camber on sound generation by airfoils interacting with high-frequency gusts

Abstract

A theoretical model is developed for the sound generated when a convected disturbance encounters a cambered airfoil at non-zero angle of attack. The model is a generalization of a previous theory for a flat-plate airfoil, and is based on a linearization of the Euler equations about the steady, subsonic flow past the airfoil. High-frequency gusts, whose wavelengths are short compared to the airfoil chord, are considered. The airfoil camber and incidence angle are restricted so that the mean flow past the airfoil is a small perturbation to a uniform flow. The singular perturbation analysis retains the asymptotic regions present in the case of a flat-plate airfoil: local regions, which scale on the gust wavelength, at the airfoil leading and trailing edges; a ‘transition’ region behind the airfoil which is similar to the transition zone between illuminated and shadow regions in optical problems; and an outer region, far away from the airfoil edges and wake, in which the solution has a geometric-acoustics form. For the cambered airfoil, an additional asymptotic region in the form of an acoustic boundary layer adjacent to the airfoil surface is required in order to account for surface curvature effects. Parametric calculations are presented which illustrate that, like incidence angle, moderate amounts of airfoil camber can significantly affect the sound field produced by airfoil–gust interactions. Most importantly, the amount of radiated sound power is found to correlate very well with a single aerodynamic loading parameter, αeff, which is an effective mean-flow incidence angle for the airfoil leading edge.