On the optimization of non-uniform acoustic liners on annular nozzles

The solution of the convected wave equation, with uniform axial flow, in cylindrical co-ordinates, is used together with non-uniform impedance wall boundary conditions, to specify the acoustic modes in a cylindrical or annular nozzle. 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 2×2 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 non-uniform wall impedance; (iv) the roots of the determinant of a 2×2 block of infinite matrices for an annular nozzle with distinct, non-uniform impedance distributions at the two walls. The case of circumferentially non-uniform but axially uniform wall impedance, allows the existence of an axial wavenumber for each frequency and each eigenvalue or radial wavenumber. The acoustic liner may be optimized by maximizing the decay of a particular wave mode, e.g., the slowest decaying, or a combination of them, e.g., the total acoustic energy.