On Sound Radiation from an Open-Ended Non-Uniformly Lined Cylindrical Nozzle

Abstract

Acoustic liners are widely used in jet engine inlet and exhaust ducts, as a passive means of noise reduction. One possible way of liner optimization is the use of a non-uniform acoustic impedance distribution. It has been shown that such a liner can 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. These issues may be illustrated by considering the radiation of sound from a cylindrical duct with impedance varying circumferentially, axially or in both directions and the acoustic pressure at the far-field in each situation. The radiation of sound from an open pipe is represented by a pressure distribution on a disk, viz. the exit plane. The radiation of sound in free space, i.e. without obstacles, is specified by a Kirchhof integral. The source distribution on the duct exit plane, which allows the evaluation of radiation integrals, is specified by the radial, axial, and circumferential modes, in the cylindrical nozzle. The evaluation of the radiation integrals show that (i) the total acoustic field consists of a spherical wave multiplying a sum of radial modes n = 1, . . . ,∞ and azimuthal modes of odd order only m = 1,3,5, . . . ; (ii) each mode consists of a monopole term and a dipole term and depend on the frequency and the radial wavenumbers, determined by the boundary condition at the duct wall. When the impedance distribution varies circumferentially or axially, the evaluation of the wavenumbers involves the determination of the roots of an infinite determinant, while when the impedance aries both axially and circumferentially, the roots of a doubly infinite determinant have to be calculated. 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 it depends 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, axially or in both directions. The far-field acoustic pressure is specified by a directivity factor that is calculated for several values of the parameters for each of the cases mentioned above.