A comparison of impedance boundary conditions for flow acoustics

doi:10.1016/j.jsv.2012.10.014

Journal of Sound and Vibration

Volume 332, Issue 4, 18 February 2013, Pages 714-724

https://doi.org/10.1016/j.jsv.2012.10.014 Get rights and content

Abstract

Acoustic liners remain a key technology for reducing community noise from aircraft engines. The choice of optimal impedance relies heavily on the modeling of sound absorption by liners under grazing flows. The Myers condition assumes an infinitely thin boundary layer, but several impedance conditions have recently been proposed to include a small but finite boundary layer thickness. This paper presents a comparison of these impedance conditions against an exact solution for a simple benchmark problem and for parameters representative of inlet and bypass ducts on turbofan engines. The boundary layer thickness can have a significant impact on sound absorption, although its actual influence depends strongly on the details of the incident sound field. The impedance condition proposed by Brambley seems to provide some improvements in predicting sound absorption compared to the Myers condition. The boundary layer profile is found to have little influence on sound absorption.

Introduction

Acoustic liners are the most common technology used to reduce noise emissions from aircraft engines. Yet further improvements in their design will be required to support future evolutions of turbofans such as ultra-high bypass ratios and shorter nacelles. When predicting the efficiency of acoustic treatments for such applications, one has to model not only the interaction of the liner with the sound field but also the effects of the boundary layer of the grazing flow. This modifies the propagation of sound and can induce hydrodynamic oscillations and instabilities that interact with the liner and the sound field.

The impedance condition derived by Ingard [1] and generalized by Myers [2] has been the standard model to describe the effects of an infinitely thin boundary layer (BL) on sound absorption. But several limitations have become apparent. In parallel with experimental observations of instabilities developing over liners with grazing flows [3], [4], the properties and stability of surface waves described by the Myers condition were also studied [5], [6]. This led to the observation that the Myers condition is in fact ill-posed in the time domain due to the unbounded growth rate of the instability at high frequencies [7]. In addition, comparison with solutions with a finite boundary layer thickness has shown that this parameter can be significant [8], [9], [10]. Indeed, measurable discrepancies have been observed between experimental data and theoretical predictions (for instance in the context of impedance eduction methods [11]), suggesting that the accuracy of the Myers condition might not be sufficient for some practical applications.

In response to these findings, modified Myers conditions have recently been proposed to address the well-posedness issue. The model proposed by Rienstra and Darau [12], [13] includes a small but finite boundary layer thickness $δ$ and is derived by neglecting compressibility. Independently Brambley [14] derived a different impedance condition by using matched asymptotic expansions based on the small parameter $δ$ . These two models are well posed in the time domain and provide improved descriptions of the hydrodynamic stability of the boundary layer (see also the recent discussion by Marx [15]). The introduction of the boundary layer thickness $δ$ as an additional parameter offers the potential for more accurate predictions of the sound absorption. This is the topic of the present paper which aims to compare the Myers and modified Myers conditions against an exact solution to discuss the importance of the boundary layer thickness in practical applications and to assess how well the modified impedance conditions can capture these effects.

The next section describes the benchmark problem used for this comparison. Section 3 introduces some special cases to assess the consistency of the impedance conditions. Section 4 presents and discusses the results of the comparison.

Section snippets

Plane wave reflection by a lined surface

We consider a three-dimensional problem in the half-space $y > 0$ with a uniform, subsonic mean flow, with Mach number M in the x direction, as illustrated in Fig. 1a. The sound field has an $e^{+ i ω t}$ time dependence. A boundary with uniform impedance Z is located at y=0. All variables are non-dimensionalized using the sound speed $c_{\infty}$ , the mean flow density $ρ_{\infty}$ and a length scale L. The velocity potential $ϕ$ satisfies the convected Helmholtz equation $\frac{d_{0}^{2} ϕ}{d t^{2}} - \nabla^{2} ϕ = 0,$ where $d_{0} / d t = i ω + M \partial / \partial x$ is the material

Special cases

Several special cases are now considered to provide some insight into the impedance boundary conditions and to assess their consistency.

Two-dimensional analysis

The impedance boundary conditions are now compared using a series of two-dimensional test cases ( $φ = 0 °$ or 90°) with parameters representative of turbofan engines. These parameters are listed in Table 1. Case A corresponds to the inlet of a typical turbofan engine at the blade passing frequency (BPF) and with Mach number M=0.55 and impedance $Z = 5 - i$ (the curvature of the duct is neglected). The boundary layer thickness $δ$ is 1.4 percent of the fan radius which is similar to what would be observed

Conclusions

Two modified impedance conditions have been compared to the standard Myers condition and to an exact solution for the test case of a plane wave reflected off a flat lined surface, in two and three dimensions. The main observations are as follows:

•
The effect of a finite thickness boundary layer can be significant, and the standard Myers condition can lead to significant errors when predicting sound absorption.
•
The impedance condition proposed by Rienstra–Darau yields results that can be quite

References (17)

M. Myers
On the acoustic boundary condition in the presence of flow
Journal of Sound and Vibration
(1980)
Y. Aurégan et al.
Experimental evidence of an instability over an impedance wall in a duct with flow
Journal of Sound and Vibration
(2008)
D. Marx et al.
PIV and LDV evidence of hydrodynamic instability over a liner in a duct with flow
Journal of Sound and Vibration
(2010)
S. Rienstra
A classification of duct modes based on surface waves
Wave Motion
(2003)
E. Brambley et al.
Classification of aeroacoustically relevant surface modes in cylindrical lined ducts
Wave Motion
(2006)
E. Brambley
Fundamental problems with the model of uniform flow over acoustic linings
Journal of Sound and Vibration
(2009)
A. Nayfeh et al.
Effect of mean-velocity profile shapes on sound transmission through two-dimensional ducts
Journal of Sound and Vibration
(1974)
A. McAlpine et al.
“Buzz-saw” noisea comparison of modal measurements with an improved prediction method
Journal of Sound and Vibration
(2007)

There are more references available in the full text version of this article.

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Acoustic lining treatments are widely used in modern aircraft engines for noise reduction. The key to the successful application of this technique is to establish a link between the geometry parameters (thickness, hole diameters, etc.) and the acoustic impedance. To achieve this goal, the impedance eduction technique is widely used by the aircraft engine noise research community. The technique iteratively finds an impedance value that leads to the best fit between the predicted and experimental results. More recently, comparative studies have shown that, in the presence of flow, the acoustic impedance educed from a liner test-rig with an upstream acoustic excitation is different from that of the same liner subject to a downstream acoustic excitation. This discrepancy is surmised to be attributed to the Ingard-Myers boundary condition. The current work attempts to provide more information about the influence of boundary layer effect on the educed impedance when the acoustic source is exerted either upstream or downstream. A duct acoustic model for a non-uniform flow with a thin but finite-thickness sheared boundary layer over the liner is developed for impedance eduction. An alternative boundary condition is derived from the Pridmore-Brown equation using the perturbation method that gives rise to an effective impedance to be enforced at the interface between the sheared boundary layer and core flow region carrying a uniform mean flow. Then, a modified impedance eduction technique procedure is developed and used to obtain the liner impedance using a flow duct facility and the result is compared with that based on the Ingard-Myers condition. Results show that considering the sheared flow effect in the impedance eduction makes no significant improvement to the educed impedance for liners used in the experiment.
Experimentally testing impedance boundary conditions for acoustic liners with flow: Beyond upstream and downstream
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Impedance eduction experiments on acoustic liners with flow have systematically shown the educed impedance depending on the direction of the incident wave. Recent attempts to model this dependence include impedance boundary conditions with an additional degree of freedom. In this case, both upstream and downstream acoustic sources must be used to educe both the liner impedance and the extra degree of freedom of the model, which implies two different axial wavenumber conditions are used, and always result in a perfect collapse of the educed impedances; this would be true whether or not the model itself is correct. In this work, we describe a novel experimental setup that allows for four different axial wavenumbers per frequency to be measured: two upstream and two downstream. We use this experiment to investigate three different impedance boundary conditions, namely inviscid sheared, viscous, and momentum transfer conditions, in addition to the classical Ingard–Myers boundary condition, using an inverse eduction technique based on the mode matching method. The additional degree of freedom in each of the first three models is best fitted to experimental data, and then compared to the theoretically predicted values. Unphysical or unrealistic values are found at certain frequencies for each model, and therefore the validity of such models is questionable. The predictive capabilities of the models are tested and compared by means of the plane-wave scattering matrix, and no model is found to be truly predictive. Under certain conditions, educed impedances using the Ingard–Myers boundary condition show similar accuracy in terms of transmission coefficients when compared to the other models.
Optimum acoustic impedance in circular ducts with inviscid sheared flow: Application to turbofan engine intake
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Acoustic liners are widely employed in turbofan aero-engines as acoustic treatment. Their performance is determined by its acoustic impedance in the presence of flow. In recent years, much effort has been devoted to find the “optimum impedance” i.e. the impedance that results in the maximum attenuation. Although such analysis can be carried out by means of numerical simulations, analytical expressions can also be derived in order to predict the optimum impedance based on exceptional points i.e. when two modes coalesce, resulting in a maximum modal decay rate. Previous works focus on the optimum impedance of higher order modes in rectangular ducts with uniform flow. In this work, the analysis is expanded to circular ducts for both uniform and sheared inviscid flows. Typical turbofan engine intake geometry and flight conditions are considered, corresponding to certification points: sideline, cut-back and approach. It is shown that, for sideline and cut-back conditions, the optimum impedance and the maximum attenuation are affected even by the presence of a small boundary layer. For approach condition, uniform and shear flows lead to essentially the same results. Limitations of the analytical expression in the presence of multi-modal propagation is also discussed.
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Meanwhile, Rienstra and Darau [14] derived a modified boundary condition from a two-dimensional incompressible model to investigate the ill-posed problem in the time domain. The two boundary conditions were examined by Gabard [15] in the scenario of a lined surface subject to a plane wave incidence. The absorption and reflection coefficients so obtained together with those obtained from the Ingard-Myers boundary condition were compared with a benchmark solution computed from the Pridmore-Brown equation.
A duct acoustics model is an essential component of an impedance eduction technique and its computation cost determines the impedance measurement efficiency. In this paper, a model is developed for the sound propagation through a lined duct carrying a uniform mean flow. In contrast to many existing models, the interface between the liner and the duct field is defined with a modified Ingard-Myers boundary condition that takes account of the effect of the boundary layer above the liner. A mode-matching method is used to couple the unlined and lined duct segments for the model development. For the lined duct segment, the eigenvalue problem resulted from the modified boundary condition is solved by an integration scheme which, on the one hand, allows the lined duct modes to be computed in an efficient manner, and on the other hand, orders the modes automatically. The duct acoustics model developed from the solved lined duct modes is shown to converge more rapidly than the one developed from the rigid-walled duct modes. Validation against the experiment data in the literature shows that the proposed model is able to predict more accurately the liner performance measured by the two-source method. This, however, cannot be made by a duct acoustics model associated with the conventional Ingard-Myers boundary condition. The proposed model has the potential to be integrated into an impedance eduction technique for more reliable liner measurement.
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Eigenmodes of the linearised Euler equations are computed in order to study lined flow duct global stability. A simplified configuration is considered and the governing equations are discretised by means of the discontinuous Galerkin method. A biorthogonal technique is used to decompose the global eigenfunctions into local eigenmodes. The system dynamics switches from noise amplifier to resonator as the length of the liner is increased. The global mode in the liner region is shown to be mainly composed of a hydrodynamic unstable wave and a left-running acoustic mode.
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A comparison of impedance boundary conditions for flow acoustics

Abstract

Introduction

Section snippets

Plane wave reflection by a lined surface

Special cases

Two-dimensional analysis

Conclusions

Journal of Sound and Vibration

Journal of Sound and Vibration

Journal of Sound and Vibration

Wave Motion

Wave Motion

Journal of Sound and Vibration

Journal of Sound and Vibration

Journal of Sound and Vibration