Single Mode Theory for Impedance Eduction in Large-Scale Ducts with Grazing Flow

Abstract

An impedance eduction theory for a rigid wall duct containing an acoustic liner with an unknown impedance and uniform grazing flow is presented. The unique features of the theory are: 1) non-planar waves propagate in the hard wall sections of the duct, 2) input data consist solely of complex acoustic pressures acquired on a wall adjacent to the liner, and 3) multiple higher-order modes may exist in the direction perpendicular to the liner and the opposite rigid wall. The approach is to first measure the axial propagation constant of a dominant higher-order mode in the liner sample section. This axial propagation constant is then used in conjunction with a closed-form solution to a reduced form of the convected Helmholtz equation and the wall impedance boundary condition to educe the liner impedance. The theory is validated on a conventional liner whose impedance spectrum is educed in two flow ducts with different cross sections. For the frequencies and Mach numbers of interest, no higher-order modes propagate in the hard wall sections of the smaller duct. A benchmark method is used to educe the impedance spectrum in this duct. A dominant higher-order vertical mode propagates in the larger duct for similar test conditions, and the current theory is applied to educe the impedance spectrum. Results show that when the theory is applied to data acquired in the larger duct with a dominant higher-order vertical mode, the same impedance spectra is educed as that obtained in the small duct where only the plane wave mode is present and the benchmark method is used. This result holds for each higher-order vertical mode that is considered.