Time-domain impedance boundary conditions (TDIBCs) can be enforced using the impedance, the admittance, or the scattering operator. This article demonstrates the computational advantage of the last, even for nonlinear TDIBCs, with the linearized Euler equations. This is achieved by a systematic semi-discrete energy analysis of the weak enforcement of a generic nonlinear TDIBC in a discontinuous Galerkin finite element method. In particular, the analysis highlights that the sole definition of a discrete model is not enough to fully define a TDIBC. To support the analysis, an elementary physical nonlinear scattering operator is derived and its computational properties are investigated in an impedance tube. Then, the derivation of time-delayed broadband TDIBCs from physical reflection coefficient models is carried out for single degree of freedom acoustical liners. A high-order discretization of the derived time-local formulation, which consists in composing a set of ordinary differential equations with a transport equation, is applied to two flow ducts.

A methodology to design broadband time-domain impedance boundary conditions (TDIBCs) from the analysis of acoustical models is presented. The derived TDIBCs are recast exclusively as first-order differential equations, well-suited for high-order numerical simulations. Broadband approximations are yielded from an elementary linear least squares optimization that is, for most models, independent of the absorbing material geometry. This methodology relies on a mathematical technique referred to as the oscillatory-diffusive (or poles and cuts) representation, and is applied to a wide range of acoustical models, drawn from duct acoustics and outdoor sound propagation, which covers perforates, semi-infinite ground layers, as well as cavities filled with a porous medium. It is shown that each of these impedance models leads to a different TDIBC. Comparison with existing numerical models, such as multi-pole or extended Helmholtz resonator, provides insights into their suitability. Additionally, the broadly-applicable fractional polynomial impedance models are analyzed using fractional calculus.

Characteristic boundary conditions for the Navier-Stokes equations (NSCBC) are implemented for the first time with discontinuous spectral methods, namely the spectral difference and flux reconstruction. The implementation makes use of the resolution by these methods of the strong form of the Navier-Stokes equations by applying these conditions through a flux balance regularization which takes the form of a generalized element-compact correction polynomial. It is shown to be at least as effective as similar implementations in finite volume solvers, and sustains arbitrarily-high orders of accuracy on hexahedral-based unstructured meshes. Further, Navier-Stokes time-domain impedance boundary conditions are derived and implemented as a NSCBC sub-class. They account for the diffusive process at the wall and are shown to properly resolve broadband impedance models under normal and grazing flow conditions. The ability of these NSCBC in preventing the appearance of spurious reflections at the boundaries is demonstrated through a varied series of bench-marking simulations. They effectively shield the inner computational domain from any far-field unphysical contamination. Overall, this work enables the use of strong discontinuous spectral methods to study unsteady problems on complex geometries.

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A Parabolized Stability Equation (PSE) method is applied to hot inviscid Mach 0.7 and Mach 2 axisymmetric jets. The Parabolized Stability Equations are derived from the linearized Euler equations. Spatial development of pressure perturbations is computed in the vicinity of the jet, and the associated radiated noise is obtained by solving the wave equation. A Large Eddy Simulation is performed on the subsonic jet and compared with the results obtained by the PSE analysis of the LES-computed mean flow. Good agreement is found for the spatial growth of pressure instability waves, the spatial damping being slightly underestimated in the PSE analysis. Only LES predicts acoustic radiation, which may thus be created by the turbulence cascade rather than by the Kelvin-Helmholtz instability waves. PSE method is then applied to the supersonic jet and compared to solutions of Linearized Euler Equations. The common mean flow is analytical. A very good agreement is found for pressure perturbation evolution and for directivity and levels of acoustic radiation.

The purpose of this paper is to present a method for the ultrasonic characterization of air-saturated porous media, by solving the inverse problem using only the reflected waves from the first interface to infer the porosity, the tortuosity, and the viscous and thermal characteristic lengths. The solution of the inverse problem relies on the use of different reflected pressure signals obtained under multiple obliquely incident waves, in the time domain. In this paper, the authors propose to solve the inverse problem numerically with a first level Bayesian inference method, summarizing the authors' knowledge on the inferred parameters in the form of posterior probability densities, exploring these densities using a Markov-Chain Monte-Carlo approach. Despite their low sensitivity to the reflection coefficient, it is still possible to extract the knowledge of the viscous and thermal characteristic lengths, allowing the simultaneous determination of all the physical parameters involved in the expression of the reflection operator. To further constrain the problem and guide the inference, the knowledge of a particular incident angle is used at one's advantage in order to more precisely define the thermal length, by effectively yielding a statistical relationship between tortuosity and characteristic length ratio.

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.