Edge-source acoustic green's function for an airfoil of arbitrary chord, with application to trailing-edge noise

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Abstract
Approximations are derived for the three-dimensional, time-harmonic acoustic Green’s function whose normal derivative vanishes on the surface of an airfoil of finite thickness and chord ℓfor source locations in the neighbourhood of either the leading or trailing edge. The acoustic wavelength is assumed to be large relative to the airfoil thickness, but no restriction is placed on its magnitude relative to ℓ. A multiple scattering calculation is performed for high frequencies that involves an expansion in terms of the successive scattering of waves from the leading and trailing edges of the airfoil. The ‘principal subseries’ of the expansion is summed and shown to provide an excellent approximation for the Green’s function when κ0ℓ ≥ 1, where κ0is the acoustic wavenumber. The solution is extended down to κ0ℓ = 0by interpolation with the corresponding Green’s function for an airfoil of acoustically compact chord. The results extend the single scattering approximation introduced by R. K. Amiet (AIAAJ.121970), and are illustrated by application to the problem of trailing-edge noise generated by nominally steady, low Mach number flow past the airfoil. Experiments and numerical simulations of such flows often include acoustic frequencies that are sufficiently small that the usual assumption of trailing-edge noise theory, that the airfoil is semi-infinite, is not valid.

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