Acoustic Diffraction by a Finite Airfoil in Uniform Flow

Abstract

Diffraction by a flat airfoil in uniform flow is analytically examined, focusing on the acquisition of an accurate series solution for both low- and high-frequency incident waves. Formulation of integral equations is based on the use of the Wiener-Hopf technique in the complex domain. As the kernels of the integral equations are multivalued functions having a branch cut in the complex domain, the unknown in the integral operator is assumed to be a constant Therefore, the solution is a zeroth-order approximate solution adequate for a high-frequency problem. In this study, the unknown is expanded by a Taylor series of an arbitrary order in the analytic region, and the solution is obtained in series form involving a special function called a generalized gamma function Γm (u, z). As the generalized gamma functions occurring in finite diffraction theory have the specific argument u as nonnegative integer + ½ the authors used their previously determined exact and closed-form formulas of this special function to obtain the complete series solution. The present series solution exhibits faster convergence at a high frequency compared to a low frequency, whereas the Mathieu series solution in the elliptic coordinates converges faster at a low frequency relative to a higher frequency. Through exact and asymptotic evaluations of inverse Fourier transforms, the scattered and total acoustic fields are visualized in a physical domain and each term of the solution is physically interpreted as 1) semi-infinite leading-edge scattering, 2) trailing-edge correction, and 3) interaction between leading and trailing edges, respectively.