ON THE CORRELATION OF ACOUSTIC MODES IN A NOZZLE AND IN THE FAR-FIELD

摘要

Acoustic liners are widely used in jet engine inlet and exhaust ducts as a passive means of noise reduction, and the problem of liner optimization has received some attention recently. The use of circumferentially non-uniform liners has been shown to lead to an attenuation in the acoustic amplitude of some of the modes present in the duct. However, the sound attenuation perceived by an observer in the far-field is arguably the most important effect to be achieved. In this paper the far-field acoustic pressure field radiated from the exit of a lined cylindrical duct is considered and the case of uniform and circumferentially non-uniform liners is compared. The radiation of sound from the duct to the free space is specified by a Kirchhoff integral, the sound source distribution being specified by the pressure distribution on the duct exit plane. This pressure distribution depend on the radial, axial and circumferential modes in the cylindrical duct, which are determined by the impedance boundary conditions. When the impedance distribution varies circumferentially, the evaluation of the wavenumbers involves the determination of the roots of an infinite determinant. The evaluation of the radiation integrals show that (ⅰ) the total acoustic field consists of a spherical wave multiplying a sum of directivity factors which depend on the radial n= 1,…, ∞ and az-imuthal m= 1,…, ∞ modes; (ⅱ) each mode consists of a monopole term and a dipole term and depend on the frequency and the radial and axial wavenumbers that had been determined by the boundary condition at the duct wall. In the case of external noise of an aircraft, the observer is on the ground, at a distance much greater than the duct diameter, and the radiation integrals for an observer in the far-field can be simplified, since the dipole term is weak. The evaluation of the acoustic pressure for an observer in the far-field, shows that the directivity factors depend on the radial wavenumbers in the nozzle, which are specified by the wall boundary conditions, and thus depend on the acoustic impedance distribution. This allows comparison of hard-walled nozzles, with liners with constant impedance and non-uniform liners, the latter with impedance distribution varying circumferentially.