Statistical Inference Method for Liner Impedance Eduction with a Shear Grazing Flow

Abstract

This paper presents a statistical inference method for impedance eduction in a flow-duct facility. The acoustic impedance is recast into a random variable, and Bayes's theorem is used to obtain the posterior probability density function of both its real and imaginary parts, thus expressing the knowledge/uncertainty one has on the impedance value, given a certain experimental data uncertainty. An evolutionary Markov chain Monte Carlo technique is selected to explore the probability space, and a surrogate model based on the method of snapshots is employed to speed up the calculations. The linearized Euler equations are solved using a two-dimensional discontinuous Galerkin scheme, accounting for the presence of a grazing flow. The inference process is first validated on published NASA Grazing Incidence Tube results, in which acoustic-pressure measurements on the wall opposite the liner are used as inputs. Then, the same procedure is applied to educe the impedance of a conventional single degree-of-freedom liner in the ONERA-The French Aerospace Lab B2A acoustic bench, in which a laser Doppler velocimetry (LDV) technique is used to measure the two components of the acoustic-velocity fields above the liner. The primary conclusion of the study is that the Bayesian inference method allows for consistent impedance eductions, as compared to a classical deterministic eduction approach, for both microphone and LDV measurements. Furthermore, it yields the credibility intervals of the identified impedance, which represent the uncertainty on the identified impedance values, given an uncertain measurement. The identified parameters are less correlated using an LDV-based inference than a microphone-based inference, which might be due to the more limited number of data.

Liner Impedance Eduction Under Shear Grazing Flow at a High Sound Pressure Level

“…A statistical inference method developed by Roncen et al in 2018 can be also applied to compute uncertainties on the educed values. 7 This method is based on a Bayesian approach, and uses a MCMC (Markov Chain Monte Carlo) algorithm instead of the CMAES technique. Coupled with the bootstrap method for estimating the errors on LDA measurements, it gives a good appreciation of the confidence intervals on the educed impedance.…”

“…The constant educed resistance for case #8 is also plotted (dashed line) for comparison. The error margins computed using the statistical method developed by Roncen et al 7 are also added for the two constant cases (the inference method is not implemented with the affine boundary condition). One can notice that the educed values are close to each other, and that the variations all stay within the uncertainty margin.…”

“…The main challenge is to ensure the best reliability of the eduction process and high accuracy of the results. 6 Recently, Roncen et al 7 dealt with a new approach based on Bayesian inference in order to have an evaluation of the uncertainties on the impedance results. For a classical perforate-overhoneycomb liner, it is usually assumed that the liner impedance is constant along the streamwise direction.…”

Wavenumber-Based Impedance Eduction with a Shear Grazing Flow

“…£ Indirect eduction methods [19][20][21][22] consist of measuring an acoustic field (pressure or velocity) and minimizing the difference between experimental and numerical fields obtained by solving partial differential equations (Convected Helmholtz Equations CHE or Linearized Euler Equations LEE) coupled with an impedance boundary condition.…”

Liner Impedance Eduction under shear grazing flow for high sound pressure level

“…The uncertainty in the measurement of the velocity field is estimated with a Bootstrap approach which allows to evaluate the variance associated to the amplitude and phase of the measured field. 7…”