Optimization of Circumferentially Non-uniform Acoustic Liners in Annular Ducts

Abstract

Acoustic liners are widely used in jet engine inlet ducts, as a passive means of noise reduction. The use of liners carries a weight penalty and may also adversely aect thrust and fuel consumption. The optimization of acoustic liners gains additional freedom by the possibility of non-uniform impedance distribution. This is illustrated by considering a cylindrical o annular duct with impedance varying circumferentially. The solution of the convected wave equation, with uniform axial flow, in cylindrical coordinates, is used together with non-uniform impedance wall boundary conditions, to specify the acoustic modes. The radial eigenfunctions in this case are Bessel functions, and the method applies equally well to sheared and swirling mean flows, provided that the appropriate eigenfunctions are used. The eigenvalues or radial wavenumbers are determined by: (i) the roots of a linear combination of Bessel functions for a cylindrical nozzle with uniform wall impedance; (ii) the roots of a 2x2 determinant whose terms are linear combinations of Bessel and Neumann functions, for an annular nozzle with uniform but distinct impedances at each wall; (iii) the roots of an infinite determinant for a cylindrical nozzle with circumferentially nonuniform wall impedance; (iv) the roots of the determinant of a 2x2 block of infinite matrices for an annular nozzle with distinct, non-uniform impedance distributions at the two walls. An example of liner optimization by maximizing the decay of a particular wave mode, e. g. the slowest decaying, is presented and other possible optimization criteria are discussed.