Characteristics of penalty mode scattering by rigid splices in lined ducts

In lined ducts, incident modes are scattered by axially and circumferentially nonuniform impedance. Experiments and numerical calculations have proved that this mode scattering can reduce the liner performance in some cases. This paper is devoted to the characterization of the penalty mode scattering excited by hard-walled splices which often exist in lined ducts. It is shown that, in the range of small splice angles, the transmission loss may decrease sharply with increasing splice angle when one mode, which is near cut-off or has high azimuthal order, is incident. When the incident sound field is composed of several acoustical modes, the phase interferences of incident modes are important for the penalty mode scattering. The effects of other parameters, e.g., liner length, mode quasiresonance on the penalty mode scattering are also presented.     

Multi-mode sound transmission in ducts with flow

Exhaust mufflers, large exhaust stacks, and turbofan engines are common examples of ducted noise. The most useful measure of the sound produced by these noise sources is the sound power transmitted along the duct. When airflow is present, sound power flow can no longer be uniquely determined from the usual measurements of acoustic pressure and particle velocity.One approach to sound power determination from in-duct pressure measurement, and the one discussed in this paper, is to predict the relationship between the sound power and pressure based upon an assumed mode amplitude distribution. This paper investigates the relationship between acoustic pressure and power for a family of idealized source distributions of arbitrary temporal and spatial order. Incoherent monopole and dipole sources uniformly distributed over a duct cross-section can be obtained as special cases. This paper covers the sensitivity of the pressure-power relationship to source multipole order, frequency and, in particular, flow speed. It is shown that the introduction of flow in a hard-walled duct can have a substantial effect on the behavior of the pressure-power relationship for certain source distributions. Preliminary experimental results in a no-flow facility are presented in order to verify some of the main results.     

On the prediction of “buzz-saw” noise in acoustically lined aero-engine inlet ducts

Aero-engines operating with supersonic fan tip speeds generate an acoustic signature containing energy spread over a range of harmonics of the engine shaft rotation frequency. These harmonics are commonly known as the “buzz-saw” tones. The pressure signature attached to a supersonic ducted fan will be a sawtooth waveform. The non-linear propagation of a high-amplitude irregular sawtooth upstream inside the inlet duct redistributes the energy amongst the buzz-saw tones. In most modern aero-engines the inlet duct contains an acoustic lining, whose properties will be dependent on the mode number and frequency of the sound, and the speed of the oncoming flow. Such effects may not easily be incorporated into a time-domain approach; hence the non-linear propagation of an irregular sawtooth is calculated in the frequency domain, which enables liner damping to be included in the numerical model. Results are presented comparing noise predictions in hard-walled and acoustically lined inlet ducts. These show the effect of an acoustic liner on the buzz-saw tones. These predictions compare favourably with previous experimental measurements of liner insertion loss (at blade passing frequency), and provide a plausible explanation for the observed reduction in this insertion loss at high fan operating speeds.     

Bulk reaction modeling of ducts with and without mean flow

A general formulation for analysis of sound field in a uniform flow duct lined with bulk-reacting sound-absorbing material is presented here. Presented theoretical model predicts the rate of attenuation for symmetric as well as asymmetric modes in rectangular duct lined with loosely bound (bulk-reacting) sound-absorbing material, which allows acoustic propagation through the lining. The nature of attenuation in rectangular ducts lined on two and four sides with and without mean flow is discussed. Computed results are compared with published theoretical and experimental results. The presented model can be used as guidelines for the acoustic design of silencers, air-conditioning ducts, industrial fans, and other similar applications.     

Comparison of measured broadband noise attenuation spectra from circular flow ducts and from lined engine intakes with predictions

Sound propagation in lined circular ducts is investigated in the presence of uniform and sheared flow. The modal solutions are obtained by solving an eigenvalue equation which, in the case of sheared flow, is derived by using finite differences and by matching the pressure and the radial component of the particle velocity at the interface of the regions of uniform and sheared flow. For the uniform flow region, standard Bessel function solutions are used. The attenuation of acoustic energy at a given frequency and for a given liner length is computed on the assumption that at the inlet to the lined duct, the acoustic energy is equally distributed among the propagating modes. The total number of propagating modes is determined from the hard wall “cut off” condition. The failure to find some of the modal solutions on the attenuation computed in this way is discussed. It is shown that the reliability of this method of computing liner attenuation depends on the ability to successfully compute most of the modal solutions over a large range of frequencies, flow conditions and duct wall impedance values. A numerical technique is developed which uses a fraction of the total number of solutions to compute the total attenuations without appreciable loss of accuracy. Measured attenuation spectra from a flow duct facility and from lined intake ducts of the RB.211 engine are compared with predictions. In general very good agreement between predictions and measurements is obtained.     

On the effect of viscosity and thermal conductivity on sound propagation in ducts: A re-visit to the classical theory with extensions for higher order modes and presence of mean flow

The dispersion equation for the axisymmetric modes of viscothermal acoustic wave propagation in uniform hard-walled circular ducts containing a quiescent perfect gas is classical. This has been extended to cover the non-axisymmetric modes and real fluids in contemporary studies. The fundamental axisymmetric mode has been the subject of a large number of studies proposing approximate solutions and the characteristics of the propagation constants for narrow and wide ducts with or without mean flow is well understood. In contrast, there are only few publications on the higher order modes and the current knowledge about their propagation characteristics is rather poor. On the other hand, there is a void of papers in the literature on the effect of the mean flow on the quiescent modes of propagation. The present paper aims to contribute to the filling of these gaps to some extent. The classical theory is re-considered with a view to cover all modes of acoustic propagation in circular ducts carrying a real fluid moving axially with a uniform subsonic velocity. The analysis reveals a new branch of propagation constants for the axisymmetric modes, which appears to have escaped attention hitherto. The solution of the governing wave equation is expressed in a modal transfer matrix form in frequency domain and numerical results are presented to show the effects over wide ranges of frequency, viscosity and mean flow parameters on the propagation constants. The theoretical formulation allows for the duct walls to have finite impedance, but no numerical results are presented for lined ducts or ducts carrying a sheared mean flow.     

Acoustic propagation in partially choked converging-diverging ducts

A computer model based on the wave-envelope technique is used to study acoustic propagation in converging-diverging hard walled and lined circular ducts carrying near sonic mean flows. The influences of the liner admittance, boundary layer thickness, spinning mode number, and mean Mach number are considered. The numerical results indicate that the diverging portion of the duct can have a strong reflective effect for partially choked flows.     

Sound attenuation in acoustically lined curved ducts in the absence of fluid flow

A study has been made of the sound attenuation in a lined curved duct with rectangular cross-section. In this study, the derivation of the eigenvalue equation was based on the continuity of the normal component of the particle displacement and the matching of the acoustic pressure on the acoustic lining surface. The sound attenuation was calculated by using the acoustic energy expression for the waves propagating in a curved duct. For a given duct geometry and known acoustic lining impedances, a computer program was developed to solve for the eigenvalues and to obtain the sound attenuation of the propagating waves in the lined curved duct. It was found that in the case studied here the fundamental mode was least attenuated. The total sound attenuation was calculated on the assumption that the amplitudes for all propagating waves were equal at a given frequency. Effects of aspect ratio, bend angle and the acoustic impedance on the sound attenuation were investigated in the present work.     

An improved multimodal method for sound propagation in nonuniform lined ducts

An efficient method is proposed for modeling time harmonic acoustic propagation in a nonuniform lined duct without flow. The lining impedance is axially segmented uniform, but varies circumferentially. The sound pressure is expanded in term of rigid duct modes and an additional function that carries the information about the impedance boundary. The rigid duct modes and the additional function are known a priori so that calculations of the true liner modes, which are difficult, are avoided. By matching the pressure and axial velocity at the interface between different uniform segments, scattering matrices are obtained for each individual segment; these are then combined to construct a global scattering matrix for multiple segments. The present method is an improvement of the multimodal propagation method, developed in a previous paper [Bi et al., J. Sound Vib. 289, 1091-1111 (2006)]. The radial rate of convergence is improved from O(n(-2)), where n is the radial mode indices, to O(n(-4)). It is numerically shown that using the present method, acoustic propagation in the nonuniform lined intake of an aeroengine can be calculated by a personal computer for dimensionless frequency K up to 80, approaching the third blade passing frequency of turbofan noise.     

The attenuation of lined plenum chambers in ducts: I. Theoretical models

Two types of theory are described, with the purpose of predicting the acoustic transmission loss of lined plenum chambers (which are sometimes used as attenuators in air conditioning duct systems). The first kind of theory embodies a low frequency wave acoustic approach, and two separate models are evolved: one is for a single plenum chamber, and the second is for a plenum chamber incorporating one or more acoustically lined baffles. The other type of theory is valid at high frequencies, and is based upon geometrical (or “ray”) acoustics. This is applied to a single chamber and to chambers containing either one or two lined baffles. Both the high frequency and low frequency results are reasonably simple. A limited amount of experimental data is also presented, as justification for the validity of part of the theory.     

Sound wave propagation in ducts whose walls are lined with a porous layer backed by cellular cavities

This paper concerns propagation and attenuation of sound waves through acoustically lined ducts. For a cylindrical duct whose liner consists of a point-reacting porous material layer backed by cellular cavities, the admittance formula derived by taking into account a wave motion within the liner is applied to an analysis of waves propagating downstream. For the point-reacting liner of fixed porous material properties, influences of the porous layer thickness, cellular cavity depth, mean flow profile, and three dimensionality of the duct (i.e., cylindrical or plane) on the attenuation are examined. The results show a significant role of the porous layer thickness. For the cylindrical duct, attenuation spectra evaluated from this analysis are compared with those given by the widely used semi-empirical formula.     

Non-linear attenuation of sound in circular lined ducts

The attenuation of high intensity sound in circular ducts lined with fibrous material has been investigated. With no mean flow, the sound pressure levels are varied to illustrate the linear and non-linear absorption characteristics of the liner. Effects of liner thickness, perforation ratio of the duct wall and the $\text{d}\text{t}$ ratio are analysed.Optimum combinations of the perforation geometry and liner thickness are found to be of stable attenuation characteristics over a wide frequency range and at high sound levels.     

Effect of wall shear layers on the sound attenuation in acoustically lined rectangular ducts

This paper deals with the manner in which a shear layer proximate to the wall of an acoustically treated rectangular duct modifies the attenuation spectra. The restriction of this shear layer to the region near the lined duct walls is aimed at simulating boundary layer effects on the attenuation. Theoretical results show that shear significantly changes the peak attenuation, causing a frequency shift of this peak. For the inlet mode, i.e. flow against the direction of sound propagation, both results are a strong function of Mach number and layer thickness. For the exhaust mode, i.e. flow in the direction of propagation, these effects are relatively weak.     

Sound generation and transmission in flow ducts with axial temperature gradients

This article describes a one-dimensional, linearized, analysis of fundamental mode sound generation and propagation in rigid-walled flow ducts with axial temperature variation. An acoustic wave equation, including damping effects and volume sources, is derived and its solution (in the absence of sources) by a numerical technique and an approximate analytical method is discussed. The “forced” wave equation is then solved (the existence of an oscillating solution to the “unforced” equation being assumed) for sound generation by a side-branch volume source in an infinite duct, and the results are applied to a duct of finite length. Reasonably good agreement is obtained between measurements and predictions of the sound pressure field in a flow duct, away from the source region.     

Flow-resonant sound interaction in a duct containing a plate,part I: Semi-circular leading edge

The interaction between flow and flow-induced acoustic resonances near rigid plates with semi-circular leading edges located in a hard-walled duct is described. These plates generate acoustic resonances over flow velocity ranges depending on thickness, chord and trailing edge geometry, together with rigidity, internal dimensions, length of the working section and shape of the terminations of the working section. A potential flow model for the plate with a smooth leading edge is developed, and the acoustic power generated by vortices growing and shedding from the trailing edge is calculated. The rate of growth of the vortices is determined by an instantaneous Kutta condition applied over part of the cycle. This technique simulates the influence of the sound field on vortex growth.     

Non-linear effect of wall linings on sound attenuation in ducts

The equivalent surface source method is extended to the analysis of high intensity sound propagation in a duct whose wall is partially treated with a sound absorbing material. The propagation of sound in the gas is assumed to be linear, but the acoustic resistance of the sound absorbing material is assumed to be a function of the normal acoustic velocity. The problem is reduced to a non-linear integro-differential equation for the fluid particle displacement at the lined wall surface, which can be solved by a successive approximation method. Numerical examples show that the non-linear effect decreases or increases the peak sound attenuation rate of the lowest mode depending upon the linear component of the resistance. The dependence of the attenuation spectrum on modal phase difference of multi-mode incident waves is heavily affected by the non-linear effect. In the case of incident waves of multi-circumferential modes, different circumferential modes are generated by the non-linear effect.     

Numerical solutions of the acoustic eigenvalue equation in the rectangular room with arbitrary (uniform) wall impedances

Two numerical procedures for finding the acoustic eigenvalues in the rectangular room with arbitrary (uniform) wall impedances are developed. One numerical procedure applies Newton's method. Here, starting with soft walls, the eigenvalues are found by increasing the impedances of each wall pair in small increments up to the terminal impedances. Another procedure poses the eigenvalue problem as one of homotopic continuation from a non-physical reference configuration in which all eigenvalues are known and obvious. The continuation is performed by the numerical integration of two differential equations. The latter procedure was found to be faster and finds all possible solutions. The set of eigenvalues allowed the room modal natural frequencies and damping constants to be obtained. From sound decays measured in a hard-walled rectangular room, and from the collective-modal-decay curve, the impedances of the hard walls are estimated. These are then used to find the reverberation times of the modes in the room with the floor lined with sound absorbing material of known acoustic impedance. It was found that a single reverberation time, for all modes, is only supported in the rectangular room with hard walls and at the higher frequency bands, consistent with Sabine's theory, which assumes a diffuse sound field. In the rectangular room with hard walls and at the lower frequency bands, and in the rectangular room with the floor lined with sound absorbing material and for all frequency bands, modes with rather distinctive reverberation times may produce sound decays not always consistent with Sabine's prediction.     

Application of the equivalent surface source method to the acoustics of duct systems with non-uniform wall impedance

The paper outlines the application of the equivalent surface source method to the analysis of the acoustic field in a partially lined duct with arbitrarily non-uniform wall impedance. Lined sections of the duct wall are represented by unsteady mass source singularities, the strengths of which are determined by solving integral equations. The method is applicable to lined walls of impedance which is non-uniform in the streamwise and/or circumferential direction. Numerical examples are given to show the effects of various design parameters on sound attenuation. Some advantageous features of circumferentially non-uniform wall impedance are demonstrated.     

“Buzz-saw” noise: Prediction of the rotor-alone pressure field

Public expectations of lower environmental noise levels, and increasingly stringent legislative limits on aircraft noise, result in noise being a critical technical issue in the development of jet engines. Noise at take-off, when the engines are at high-power operating conditions, is a key reference level for engine noise certification. “Buzz-saw” noise is the dominant fan tone noise from modern high-bypass-ratio turbofan aircraft engines during take-off. Rotor-alone tones are the key component of buzz-saw noise. The rotor-alone pressure field is cut-off at subsonic fan tip speeds; buzz-saw noise is associated with supersonic fan tip speeds, or equivalently, high power engine operating conditions. A recent series of papers has described new work concerning the prediction of buzz-saw noise. The prediction method is based on modelling the nonlinear propagation of one-dimensional sawtooth waveforms. A sawtooth waveform is a simplified representation of the rotor-alone pressure field. Previous validation of the prediction method focussed entirely on reproducing the spectral characteristics of buzz-saw noise; this was dictated at that time by the availability of spectral data only for comparison between measurement and prediction. In this paper, full validation of the method by comparing measurement and prediction of the rotor-alone pressure field is published for the first time. It is shown that results from the modelling based on a one-dimensional sawtooth waveform capture the essential features of the rotor-alone pressure field as it propagates upstream inside a hard-walled inlet duct. This verifies that predictions of the buzz-saw noise spectrum, which are in good agreement with the measured data, are based on a model which reproduces the key physics of the noise generation process. Validation results for the rotor-alone pressure field in an acoustically lined inlet duct are also shown. Comparisons of the measured and predicted rotor-alone pressure field are more difficult to interpret because the acoustic lining significantly modifies the sawtooth waveform, but there remains good agreement with the measured spectral data. The buzz-saw noise prediction code used to generate the simulations in this paper has been used by the Rolls–Royce Noise Department since 2004.