Validation of an Inverse Semi-Analytical Technique to Educe Liner Impedance

Abstract

In this paper, the acoustic impedance of a liner is educed by a novel semi-analytical inverse technique. The liner sample is placed flush with the solid walls in a rectangular duct with grazing flow. The technique uses complex acoustic pressure measured at four positions at the wall of the duct, upstream and downstream of the lined section, and educes the impedance with a mode-matching method. Previous studies neglected grazing flow nonuniformity and the pressure discontinuity that appears at the liner-wall boundary caused by the discontinuity of the acoustic particle velocity into the wall. In the present paper, the mode-matching formulation is rederived in terms of pressure instead of velocity potential which is found to be more numerically stable. Moreover, the proposed methodology is validated with benchmark data from an experiment performed by NASA. First, the ability of the code to reproduce the pressure field for a given impedance is tested. Second, the ability to educe the correct impedance for a given pressure distribution is tested. The results of the mode-matching code are in very good agreement with the experimental data. The effect of shear flow is investigated and it can be concluded that the assumption of uniform flow is appropriate for the chosen liner, duct size, and frequency range of interest. Nomenclature A = admittance of the liner, 1=rayl a = half the width of the duct, m a q , a q = amplitudes of the incident and reflected qth mode in the inlet hard duct, Pa b q , b q = amplitudes of the incident and reflected qth mode in the lined duct, Pa c q , c q = amplitudes of the incident and reflected qth mode in the outlet hard duct, Pa j = complex unity k = normalized wave number !=c a k q m , k q n = normalized transversal wave number in the x and y directions, respectively k q z = normalized axial wave number of the qth mode L = normalized length of the lined duct M = flow Mach number p = temporal Fourier transform of the pressure, Pa q = mode number u, w = unsteady velocity field in the x and y directions, respectively, m=s x, y, z = normalized Cartesian coordinates x : x=a; z coordinate in the direction of the duct axis q = two-dimensional mode shape of the qth mode after separating the z dependence

A systematic uncertainty analysis for liner impedance eduction technology

“…Both systematic acoustic prediction and liner optimization necessitates progress in impedance measurement methods by including the effect of the grazing flow. For this objective, impedance eduction techniques have been developed during the past three decades, e.g., [1][2][3][4][5][6][7]. The acoustical waves in the experimental duct are normally computed with potential flow, while both the effect of the boundary layer and the effect of the lined wall are taken into account as a boundary condition for the computation.…”

On the Use of Experimental Methods to Improve Confidence in Educed Impedance

“…One such method is the mode-matching method, [11][12][13] in which complex acoustic pressures are measured in the hardwall sections upstream and downstream of the liner, and are then used to determine the modal content in these hardwall sections. This modal information is combined with a multi-modal duct propagation code to determine the liner impedance.…”

Bayesian identification of acoustic impedance in treated ducts

“…1 The optimization problem was also solved with the adjoint-based method 3 without the usual cumbersome calculations necessary to find the optimum. Other inverse methods were proposed [4][5][6] where the cost functions were based either on the sound pressures at the walls, 5 the scattering matrix coefficients, 4 or the power dissipated by the liner, 6 while the direct models rely either on a mode matching method, 4,5,7 or an axisymmetric finite element method. 6 In all these studies the impedance is always adjusted step by step at each frequency.…”