Elsevier

Journal of Sound and Vibration

Volume 483, 29 September 2020, 115430
Journal of Sound and Vibration

On spatially varying acoustic impedance due to high sound intensity decay in a lined duct

https://doi.org/10.1016/j.jsv.2020.115430Get rights and content

Abstract

Direct numerical simulations (DNS) based on advanced computational aeroacoustic (CAA) methods are conducted to study the acoustic impedance of a uniformly distributed multi-slit liner under grazing incident sound waves at different frequencies and sound pressure levels (SPL), in the absence of mean-flow. The liner consists of a series of fifteen identical slit Helmholtz resonators. Numerical results show that a portion of the slit resonators in the upstream works in a non-linear regime and the others in the downstream work in a linear regime when high intense waves near the resonance are excited. It is a direct evidence that the acoustic impedance is non-uniform and spatially varying over the liner length due to the different local incident sound pressure levels. With the numerical data, two methodologies are adopted to determine the acoustic impedance of the liner, one is referred to as the Prony's method and the other one is based on an inverse method. The impedance value is validated by a time-domain linearized Euler equation (LEE) solver. Numerical results indicate that representing the liner with a homogenized impedance leads to considerable errors in the reconstructed sound fields when the liner experiences a transition from non-linear regime to linear regime. On the other hand, in cases where such a transition is absent, the homogenized constant impedance is adequate to estimate the acoustic response of the liner. Therefore, a spatially piece-wise function is derived and calibrated by the DNS results to take into consideration both the resistance and reactance variation in stream-wise direction. Comparison shows that the prediction of the sound field is improved by the space-dependent impedance boundary.

Introduction

Acoustic liner is recognized as one of the most effective passive control strategy in commercial transportation industry, such as nacelle liners for a jet-engine or silencers for a car. One kind of commonly used liners are locally reacting which consists of honeycomb cells covered with a perforated facing sheet and a rigid backing sheet. Since the orifices on the liner surface are very tiny in size and great in number, it is impossible to determine the acoustic performance of a liner by evaluating the sound energy absorption and reflection of each small resonators one by one. For industrial applications, it is usually assumed that all the Helmholtz resonators exhibit the same acoustic behaviour, which indicates the liner in a duct or a nacelle could be treated macroscopically as a homogenized impedance boundary at each specific frequency.

The impedance is an intrinsic parameter of acoustic liners which characterizes the sound energy reflection and absorption. It is defined as the ratio of the complex sound pressure and the particle velocity:Z(ω)=R(ω)+iX(ω)=p˜(ω)u˜(ω)nwhere ω is the angular frequency, R is referred as resistance and X is reactance, u˜(ω)n is the complex amplitude of the normal acoustic velocity pointing into the lined wall. Commonly, the acoustic impedance is normalized by ρc.ζ(ω)=Z(ω)ρc=r(ω)+iχ(ω)

The frequency-dependent acoustic impedance is normally either deduced from an experiment (a normal impedance tube or a grazing flow impedance tube) or predicted by a semi-empirical model. Using the concept of acoustic impedance, the evaluation of the acoustic performance of a liner seems disarmingly simple. However, it is worth noting that using a homogenized impedance is not always accurate enough to represent the acoustic behaviour of a liner.

Acoustic impedance is strongly dependent on liner geometric parameters. Besides the liner structure, the flow condition and the incident sound intensity can also affect the impedance significantly. The non-linear acoustic response of an orifice was first found by Sivian [1] and Bolt et al. [2]. Ingård & Ising [3] investigated the acoustic non-linearity of an orifice in a plate with a series of experiments. Since 2000, the direct numerical simulation (DNS) method has been successfully adopted to investigate the physical procedure of resonant liners exciting by sound waves [[4], [5], [6], [7], [8]]. With numerical data the flow and sound fields could be reproduced in details, which is critical for understanding the absorption mechanism of liners. The most important finding from the simulation is that high intense incoming waves trigger a vortex shedding phenomenon resulting in an energy transformation from acoustic energy to fluid kinetic energy. Nowadays, it is widely accepted that the Helmholtz resonator could work in a linear regime or a non-linear regime depending on the excitation amplitude of the sound wave. Many semi-empirical models [[9], [10], [11]] include the sound pressure level as a factor for predicting the acoustic impedance.

In practical applications, a liner section is usually inserted in a duct to suppress the noise propagating thought the duct. Apparently, the sound pressure level gradually decays along the liner surface due to the sound energy absorption and reflection by the liner. Consequently, the resistance and reactance at different parts of the liner could differ because of the different local incident amplitudes. And the acoustic impedance of the liner could gradually change and the variation of the impedance in space may affect the acoustic performance eventually. However, the whole locally reacting liner panel in a duct or a nacelle is usually treated as a homogenized impedance boundary with a single impedance value at each specific frequency [[12], [13], [14]].

This simplification generally holds true with the identical geometry feature of all the resonators and the locally reacting assumption. However, a homogenized impedance may fail to represent the acoustic behaviour of a liner under specific conditions. Eversman [15] studied the effect of local impedance variation associated with local lining non-linearity and introduced an additional step in the liner design process. A previous numerical investigation [16] showed that the impedance variation occurred due to the rapid SPL decrease over the liner length at the resonant frequency which indicated the treatment of the liner as a uniform impedance boundary is inaccurate. And the acoustic performance near the resonant frequency is no doubt a key feature for the liner design. A piecewise function was used to fit the resistance spatial variation [17] by which the sound field can be predicted more precisely. Most recently, Méry [18] et al. and Lafont et al. [19] developed a new strategy for impedance eduction to take into account the effect of incident sound pressure level by including a space-dependent variable term. As suggested in the aforementioned investigations, the local impedance variation should be considered for either impedance eduction or impedance optimization under high intensity incident sound wave which is a typical condition for a nacelle liner.

This paper aims to explore the acoustic impedance variation along a liner panel in a grazing incident tube by numerical simulations, in the absence of mean-flow. A two-dimensional multi-slit-resonator liner mounted on the bottom wall of a grazing incident rectangular tube is investigated. Firstly, direct numerical simulations are conducted to resolve the physical procedures of tonal sound waves propagating through the liner section. The acoustic responses of the liner to different sound pressure levels and frequencies of the incident sound waves are analysed subsequently. Two distinct impedance eduction methods are adopted to extract the acoustic impedance of the liner panel from the DNS results. The accuracy of the impedance is estimated through comparing the sound fields solved by a linearized Euler equation (LEE) solver with the results from the direct numerical simulations.

The rest of this paper is as follows. In Section 2, the computational models used in the numerical simulations are provided, followed by a description of the governing equations and corresponding numerical algorithms in Section 3. Then, the numerical results are analysed and discussed in the fourth section. Primary conclusions resulting from this investigation are given in the last section.

Section snippets

Computational model for the direct numerical simulation

Overview of the computational domain and boundary conditions for the simulations are depicted in Fig. 1. The duct is L = 900 mm long and the height of the duct is H = 52 mm. On the bottom wall of the rectangular duct an acoustic liner is installed. The liner consists of a series of 15 identical slit Helmholtz resonators arranged evenly side by side. The geometric parameters of the resonator are given in Fig. 2. The thickness of the partition wall between two cavities is tw = 2 mm. Therefore,

Governing equations

The two-dimensional compressible Navier-Stokes (N–S) equations are employed as the governing equations, which can be represented in a conservation form as,Ut+Ex+Fy=Evx+Fvywhere,U=ρ,ρu,ρv,ρeT,E=ρu,ρu2+p,ρuv,(ρe+p)uT,F=ρv,ρuv,ρv2+p,(ρe+p)vT,Ev=0,τxx,τxy,qx+uτxx+vτxyT,Fv=0,τyx,τyy,qy+uτyx+vτyyT,qx=μ(γ1)PrReTx,qy=μ(γ1)PrReTy,τxx=2μReux,τyy=2μRevy,τxy=τyx=μReuy+vxp=γ1ρeu2+v22,T=pγρρ denotes the density, u and v are the velocity components, p is the pressure, T is the

DNS results

From the perspective of the flow around the aperture introduced by the sound wave, the difference between linear regime and non-linear regime is that vortex shedding occurs when the sound pressure level is high enough. Fundamentally, the high flow velocity and the corner-shape at the resonator opening cause the shedding of vortices. As a consequence, the linear theory is violated. For instance, Fig. 5 illustrates the instantaneous vorticity field. The sound pressure level of the incoming wave

Conclusions

Direct numerical simulation of the sound propagation through a lined duct without grazing flow is conducted. With the numerical data, the acoustic performance of each resonator is determined. It is found that different parts of the liner may work in non-linear or linear regime due to the variation of the local incident sound intensity. Under a specific situation in which the SPL decays significantly from a high value along the duct, a homogenized impedance boundary could no longer represent the

CRediT authorship contribution statement

Chao Chen: Methodology, Software, Formal analysis, Writing - original draft. Xiaodong Li: Conceptualization, Writing - review & editing, Funding acquisition. Fangqiang Hu: Methodology, Writing - review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work is supported by grants from the National Key Research and Development Project (No. 2018YFA0703300), the National Natural Science Foundation of China (No.91752204, 91952302) and the National Science and Technology Major Project (No. 2017-II-0008-0022). The authors would also like to thank the anonymous reviewers for the helpful comments and suggestions.

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