Effect of airfoil thickness on high-frequency gust interaction noise.

Abstract

A theory is developed for the influence of airfoil thickness on sound generated by the interaction of high-frequency convected disturbances with a symmetric airfoil at zero angle of attack. Both vortical and entropic convected disturbances are considered. The analysis is based on a linearization of the inviscid equations of motion about the non-uniform mean flow past the airfoil. The mean flow is assumed to be two-dimensional and subsonic. The Mach number is assumed to be O(1), although the small Mach number limit is also investigated. The analysis utilizes singular perturbation methods, and involves two asymptotic parameters: the airfoil thickness to chord ratio is assumed small (ε ≪ 1), and the aerodynamic reduced frequency is assumed large (k = ω b/U(∞) ≫ 1), with ε k=O(1). The singular perturbation analysis shows that the primary sound generation is concentrated in a local region surrounding the airfoil leading edge. The size of this local region scales on the disturbance wavelength. The solution in the local leading-edge region reveals several sound-generating mechanisms which involve mean flow gradients. The interaction of the convected disturbance with the trailing edge does not generate sound. However, the trailing edge is a scatterer of the sound field generated in the leading-edge region. Away from the airfoil edges, the propagation of the primary sound field from the leading edge and the secondary, scattered sound field from the trailing edge is described by geometric acoustics, with the amplitude varying on the scale of the airfoil chord and the phase varying on the much smaller scale of the disturbance wavelength. In addition, diffraction-type transition regions exist along the airfoil surfaces and downstream of the airfoil. The influence of airfoil thickness on the total sound power is found to be controlled primarily by the Strouhal number St = ω r(n)/U(∞), where r(n) is the nose radius of the airfoil. Small values of St reduce the sound power relative to the level for St = 0, but with further increases of St the noise level rises sharply. The increases in sound power level due to leading-edge thickness are more substantial at higher Mach numbers. Airfoil thickness produces dramatic changes in the far-field directivity. The sound power and directivity are strong functions of the gust characteristics.