Fluid dynamics of a flow excited resonance,Part II: Flow acoustic interaction

This is the second of two companion papers in which the physics and detailed fluid dynamics of a flow excited resonance are examined. The approach is rather different from those previously used, in which stability theory has been applied to small wavelike disturbances in a linearly unstable shear layer, with an equivalent source driving the sound field which provides the feedback. In the approach used here, the physics of the flow acoustic interaction is explained in terms of the detailed momentum and energy exchanges occurring inside the fluid. Gross properties of the flow and resonance are described in terms of the parameters necessary to determine the behaviour of the feedback system. In this second paper it is shown that two relatively distinct momentum balances can be considered in the resonator neck region. One can be identified with the vortically induced pressure and velocity fluctuations and the other with the reciprocating potential flow. The fluctuating Coriolis force caused by the interaction of the potential and vortical flows is shown to be the only term in the linearized momentum equation which is not directly balanced by a fluctuating pressure gradient. This force provides the mechanism for the exchange of the mean energies associated with the mean and fluctuating momenta, respectively. A source and sink of energy are identified in which mean energy associated with fluctuating momentum is extracted from and returned to the mean flow, respectively. The imbalance between the source and sink is responsible for both the radiated acoustic power and the power carried away by the vortices as they convect downstream. This radiated acoustic power and vortically convected power, and the source and sink powers, are all of the same order of magnitude. With the vortex shedding and reciprocating potential flow “phase locked” the amplitude of the steady state oscillations is determined by the condition that the net power produced in the resonator neck (the source power less the sink power) is equal to the sum of the radiated acoustic power and that carried by the vortices.     

Nature of inlet turbulence and strut flow disturbances and their effect on turbomachinery rotor noise

Results of an investigation in which turbomachinery rotor sound spectra were correlated with aerodynamic measurements of the inlet turbulence, strut wake, and vortex flow strengths are reported. Aerodynamic measurements included mean velocity profiles, turbulence intensity, and axial length scales. Inlet turbulence data indicate that the major effect of flow contraction appears to be the elongation of turbulent eddies. Eddies of this size dominate the blade passing frequency (BPF) tones. Decreasing eddy size by use of a grid revealed vortex flow strength to be the second major sound source. A doubling of vortex flow strength produced a 6 dB increase in the SPL of the first BPF. The sound pressure level showed less than a 2 dB change with doubling of strut wake turbulence intensity or velocity defect. A discussion of the relative importance of various sources of noise due to flow non-uniformities at the inlet is given.     

Einfluß der Umklapp-Prozesse auf die hydrodynamischen Bewegungsformen der Phononen in Kristallen

The influence of U-processes on second sound and Poiseuille flow of the phonons in dielectric solids is investigated. Longitudinal and transverse phonons are allowed for throughout. In the first two sections the mean free time approximation is used. Various ratios of the mean free times for U- and N-processes are considered. In the third section the linearized Boltzmann-Peierls equation is solved by means of a double perturbation calculation, whereU-frequencies as well as hydrodynamic frequencies are considered as small, and the damping of second sound is calculated.     

Improved Statistically Optimal Nearfield Acoustical Holography in subsonically moving fluid medium

Statistically Optimal Nearfield Acoustical Holography (SONAH) can be used to reconstruct three-dimensional sound fields by projecting two-dimensional data measured on a “small” aperture that partially covers a composite sound source in a “static” fluid medium. Here, an improved SONAH procedure is proposed that includes the mean flow effects of a moving fluid medium while the sound source and receivers are stationary. The backward projection performance of the proposed procedure is further improved by using a wavenumber filter to suppress subsonic noise components. Through numerical simulations at Mach 0.6, it is shown that the improved procedure can accurately reconstruct sound source locations and radiation patterns: e.g., the spatially averaged reconstruction errors of the conventional and improved SONAH procedures are 15.40 dB and 0.19 dB, respectively, for a monopole simulation and 21.60 dB and 0.19 dB for an infinite-size panel. The wavenumber filter further reduces spatial noise, e.g., decreasing the reconstruction error from 1.73 dB to 0.19 dB for the panel simulation. An existing data measured in a wind tunnel operating at Mach 0.12 is reused for the validation. The locations and radiation patterns of the two loudspeakers are successfully identified from the sound fields reconstructed by using the proposed SONAH procedure.     

管道后传声数值模拟的不稳定波抑制

在管道后传声的数值模拟中,必须考虑平均流剪切层的散射效应,然而在非均匀剪切流动下时域求解线化欧拉方程会面临Kelvin-Helmholtz不稳定波产生和放大的难题。已有的不稳定波抑制技术通常很难获得令人满意的结果。本文采用一种混合方法,首先引入有限时段的宽频声源波包将声波和不稳定波分离,进而采用声源滤波器技术对不稳定波进行抑制。数值验证算例选择半无限长轴对称环形硬壁直管道,采用计算气动声学方法时域求解2.5维线化欧拉方程,无背景流动的数值解与解析解符合很好,验证了程序的精度与可靠性,非均匀流动算例则表明所采用波包加声源滤波器混合方法对不稳定波抑制效果明显,对声场影响很小,充分显示了该方法的精度与可行性。     

Simple sound power measurements

A simple method of measuring sound power is described. Noise is added from a second calibrated source until the room sound pressure level is increased by 3 dB. The noise output of the device is then equal to the added power. A simple loudspeaker, modified to reduce the effect of the environment on the sound power output, is used as the calibrated source of sound power. Checks show that the sound power readings are substantially independent of the room in which they are made.     

Wavelet packet based analysis of sound fields in rooms using coincident microphone arrays

This paper presents a passive analysis method for determining the spatio-temporal characteristics of sound fields in small rooms. The analysis finds an approximate directional reflectogram (ADR) which reveals the approximate arrival directions, time delays and amplitudes of the direct sound and early reflections without using a special or known sound source. A coincident microphone array is used to obtain directional recordings. The recordings are analysed by wavelet packet decomposition to determine the direction of the sound source and select wavelet packet coefficients to reconstruct the estimate of the direct sound. ADR is then computed via deconvolution using this estimate. Experiments have been carried out using synthesized recordings that were obtained from actual room impulse responses measured in two rooms for various source locations. The method estimates the source direction with a mean absolute error of about 7°. Calculated ADRs provide a good estimate of the time delays and arrival directions of acoustical reflections, whereas the amplitudes differ slightly.     

The active minimization of harmonic enclosed sound fields,part II: A computer simulation

This paper is Part II in a series of three papers on the active minimization of harmonic enclosed sound fields. In Part I it was shown that in order to achieve appreciable reductions in the total time averaged acoustic potential energy, E_p, in an enclosed sound field of high modal density then the primary and secondary sources must be separated by less than one half wavelength, even when a relatively large number of secondary sources are used. In this report the same theoretical basis is used to investigate the application of active control to sound fields of low modal density. By the use of a computer model of a shallow rectangular enclosure it is demonstrated that whilst the reductions in E_p which can be achieved are still critically dependent on the source locations, the criteria governing the levels of reduction are somewhat different. In particular it is shown that for a lightly damped sound field of low modal density substantial reductions in E_p can be achieved by using a single secondary source placed greater than half a wavelength from the primary source, provided that the source is placed at a maximum of the primary sound field. The problems of applying this idealized form of active noise control are then discussed, and a more practical method is presented. This involves the sampling of the sound field at a number of discrete sensor locations, and then minimizing the sum of the squared pressures at these locations. Again by use of the computer model of a shallow rectangular enclosure, the effects of the number of sensors and of the locations of these sensors are investigated. It is demonstrated that when a single mode dominates the response near optimal reductions in E_p can be achieved by minimizing the pressure at a single sensor, provided the sensor is at a maximum of the primary sound field. When two or three modes dominate the response it is found that if only a limited number of sensors are available then minimizing the sum of the squared pressures in the corners of the enclosure gives the best reductions in E_p. The reasons for this behaviour are discussed.     

On long-wave sound scattering by a Rankine vortex: Non-resonant and resonant cases

The well-known two-dimensional problem of sound scattering by a Rankine vortex at small Mach number M is considered. Despite its long history, the solutions obtained by many authors still are not free from serious objections. The common approach to the problem consists in the transformation of governing equations to the d’Alembert equation with right-hand part. It was recently shown [I.V. Belyaev, V.F. Kopiev, On the problem formulation of sound scattering by cylindrical vortex, Acoustical Physics 54(5) (2008) 603-614] that due to the slow decay of the mean velocity field at infinity the convective equation with nonuniform coefficients instead of the d’Alembert equation should be considered, and the incident wave should be excited by a point source placed at a large but finite distance from the vortex instead of specifying an incident plane wave (which is not a solution of the governing equations).Here we use the new formulation of Belyaev and Kopiev to obtain the correct solution for the problem of non-resonant sound scattering, to second order in Mach number M. The partial harmonic expansion approach and the method of matched asymptotic expansions are employed. The scattered field in the region far outside the vortex is determined as the solution of the convective wave equation, and van Dyke's matching principle is used to match the fields inside and outside the vortical region. Finally, resonant scattering is also considered; an O(M²) result is found that unifies earlier solutions in the literature. These problems are considered for the first time.     

Sound propagation above an inhomogeneous plane: Boundary integral equation methods

A boundary integral equation method is used to compute the sound pressure emitted by a harmonic source above an inhomogeneous plane. First, the theoretical aspects of the problem (behaviour of the pressure around the discontinuities,…) are studied. Then, a comparison between theoretical levels and experimental levels obtained in an anechoic room is presented. It shows that the boundary integral equation (BIE) method is quite convenient for solving this kind of problem. Two interesting results are pointed out: (i) if only a prediction of maximum sound levels is needed, the attenuation is the same for a cylindrical source, a spherical source and N spherical sources, and so it is possible to transform some three-dimensional problems into two-dimensional ones; (ii) a numerical method of computation of the sound field above an inhomogeneous plane does not provide a correct prediction if each part of the plane is not accurately described by the boundary condition chosen.     

Accurate experimental two-port analysis of flow generated sound

In the study of in-duct aero-acoustic phenomena two-port analysis based upon measurements has become an important method for the plane wave region. However even at moderate Mach numbers (0.2-0.3) the errors in the results can be hard to suppress. Ways of dealing with this include the use of over-determination methods and methods of obtaining more accurate wavenumbers. Most of the previous published work on this subject deals with the passive (scattering) part of the two-port. In this paper, both the passive and active (sound generating) parts of the two-port are addressed, and different methods for the determination of the source data and the scattering matrix are evaluated. For the source data in the form of a cross spectrum matrix an over-determination method is introduced. Additionally, a method of obtaining the mean Mach number from experimentally determined wavenumbers is described.In order to evaluate the methods measurements are conducted at a mean Mach number of 0.2 for two test cases—an empty duct and a mixer plate inside the duct. The main improvements in the scattering matrix results are achieved by discarding measurements from an abundant set, based upon the measured coherence between excitation and fluctuating pressures. For the source part, it is shown that the error in the magnitude of the source cross spectrum matrix can be significantly suppressed by having additional (>2) reference microphones on each side of the two-port.The mean Mach number obtained from an experimentally determined wavenumber yields more accurate scattering matrix results in both phase and magnitude, than those based upon flow velocity measurements at one point and an assumed flow profile.     

Prediction model of flow duct constriction noise

The scaling law for aerodynamic dipole type of sound from constrictions in low speed flow ducts by Nelson and Morfey is revisited. A summary of earlier published results using this scaling law is presented together with some new data. Based on this, an effort to find a general scaling law for the sound power for components with both distinct and non-distinct flow separation points are made. Special care is taken to apply the same scaling to all data based on the pressure drop. Results from both rectangular and circular ducts, duct flow velocities from 2 to 120 m/s and sound power measurements made both in ducts and in reverberation chambers are presented. The computed sound power represents the downstream source output in a reflection free duct. In particular for the low frequency plane wave range strong reflections from e.g. openings can affect the sound power output. This is handled by reformulating the Nelson and Morfey model in the form of an active acoustic 2-port. The pressure loss information needed for the semi-empirical scaling law can be gained from CFD simulations. A method using Reynold Average Navier Stokes (RANS) simulations is presented, where the required mesh quality is evaluated and estimation of the dipole source strength via the use of the pressure drop is compared to using the turbulent kinetic energy.     

On longitudinal acoustic propagation in convergent and divergent nozzle flows

The longitudinal propagation of sound in quasi-one-dimensional low Mach number nozzle flow is considered in section 1. The solution in the ray approximation (section 2.1) is used to transform the wave equation into a Schrödinger form, which is studied for the family of power-law ducts, including, as a particular case, the conical nozzle. It is shown that the coincidence of flow sources/sinks with sound sources can lead to appearance of essential singularities (section 2.2), which can be removed by using a Riccati transformation (section 2.4). The exact solutions of the acoustic equations for the parabolic (Figure 1) and hyperbolic (Figure 2) nozzles are obtained in terms of Bessel functions (section 2.3), respectively of complex order and argument. The general formulas, together with limiting forms in the compactness, ray and asymptotic approximations (section 3.1), are used to establish properties of the acoustic velocity and pressure (section 3.2), kinetic and compression energies, and energy flux and wave action (section 3.3); for example, it is shown that the equipartition of energies for moderate variations in cross-section, gives way (Table 2) to a predominance of kinetic/compression energies respectively near blockages/openings. The effects of non-uniform mean flow (Table 1) are discussed by comparing horns with nozzles (section 3.4): e.g., it is shown that the duality principle, in three alternative forms, does not extend from horn to nozzles, and the acoustic equations have no elementary solutions for the latter, in contrast with the former.     

Taking account of loudness constancy for the loudness criterion for concert halls

One of the surprises from analysis of results of an objective and subjective study of British concert halls (1988 Acustica 66, 1–14) was that the subjective judgement of loudness in concert halls is influenced not only by sound level but also by the source–receiver distance. This response implies that the same sound level is judged louder at positions further from the orchestra platform. Whereas level decreases with distance in actual halls, loudness is judged more-or-less independent of position in average halls (except at positions close to the platform and seats overhung by balconies). As an observation it ties in with evidence from experimental psychologists for loudness constancy throughout a space. The sound strength G is the sound level in an auditorium normalised to the sound power level of the source; the traditional criterion of acceptability for level is that G ? 0 dB. The paper proposes that, on the basis of subjective evidence and objective behaviour in auditoria, the criterion for G should not be a unique value of G but rather a function of source–receiver distance.     

Patch nearfield acoustic holography in a moving medium

To realize the accurate reconstruction of sound field in a moving medium under the condition of limited holographic aperture, a patch nearfield acoustic holography (NAH) in a moving medium is proposed. The proposed method not only reduces the influence caused by the limited aperture effects through sound field extrapolation, but also perfectly suits for sound field reconstruction in a moving medium by improving the shape of the modified Tikhonov regularization filter and the noise estimation method in accordance with flow effects. In the method, two cases that the flow direction is parallel to and perpendicular to the hologram surface are considered. Especially in the perpendicular case, the expression of the wavenumber component in the z direction is improved to make the proposed method suitable for the moving medium at a high Mach number. Simulations are investigated to examine the performance of the proposed method and show its advantages by comparing with NAH in a moving medium and the conventional patch NAH. It is found that, the proposed method is effective and robust at different flow velocities of the medium and different frequencies of the sound source.     

ON THE GENERALISED FANT EQUATION

An analysis is made of the fluid-structure interactions involved in the production of voiced speech. It is usual to avoid time consuming numerical simulations of the aeroacoustics of the vocal tract and glottis by the introduction of Fant's ‘reduced complexity’ equation for the glottis volume velocity Q [G. Fant, Acoustic Theory of Speech Production, Mouton, The Hague 1960]. A systematic derivation is given of Fant's equation based on the nominally exact equations of aerodynamic sound. This can be done with a degree of approximation that depends only on the accuracy with which the time-varying flow geometry and surface-acoustic boundary conditions can be specified, and replaces Fant's original ‘lumped element’ heuristic approach. The method determines all of the effective ‘source terms’ governing Q. It is illustrated by consideration of a simplified model of the vocal system involving a self-sustaining single-mass model of the vocal folds, that uses free streamline theory to account for surface friction and flow separation within the glottis. Identification is made of a new source term associated with the unsteady vocal fold drag produced by their oscillatory motion transverse to the mean flow.     

Vertical distribution of traffic noise—A preliminary study

This paper examines the effect of flight on the sound radiated by a high frequency source embedded in a constant area jet pipe in the presence of flow. Ray acoustics theory and classical results for sound transmission at an interface of relative motion are used. The diffraction of sound at the nozzle lips, the inhomogeneity and irregularity of the interface and the possibility of instability waves being triggered by the incident sound are neglected. Some of the waves characterized by wave-fronts pointing upstream are shown to be convected downstream by the flow and to illuminate the forward arc after refraction at the jet interface. The amount of energy emitted by the source, which is trapped inside the flow, depends only on internal jet pipe conditions. However, the portion of the forward arc which is illuminated by this energy, is a function of flight speed. The radiation into the ambient atmosphere at rest of a basically omnidirectional source peaks at the edge of the downstream zone of silence and falls off rapidly when the observation angle is increased. The flight to static comparison reveals an interesting forward arc amplification due to flight but this occurs in a range of angles where the radiation is basically rather feeble.     

Spherical harmonic analysis of the sound radiation from omnidirectional loudspeaker arrays

Omnidirectional sound sources are widely used in room acoustics. These devices are made up of loudspeakers mounted on a spherical or polyhedral cabinet, where the dodecahedral shape prevails. Although such electroacoustic sources have been made readily available to acousticians by many manufacturers, an in-depth investigation of their vibroacoustic behavior has not been provided yet. In order to fulfill this lack, this paper presents a theoretical study of the sound radiation from omnidirectional loudspeaker arrays, which is carried out by using a mathematical model based on the spherical harmonic analysis. Eight different loudspeaker arrangements on the sphere are considered: the well-known five Platonic solid layouts and three extremal system layouts. The latter possess useful properties for spherical loudspeaker arrays used as directivity controlled sound sources, so that these layouts are included here in order to investigate whether or not they could be of interest as omnidirectional sources as well. It is shown through a comparative analysis that the dodecahedral array leads to the lowest error in producing an omnidirectional sound field and to the highest acoustic power, which corroborates the prevalence of such a layout. In addition, if a source with less than 12 loudspeakers is required, it is shown that tetrahedra or hexahedra can be used alternatively, whereas the extremal system layouts are not interesting choices for omnidirectional loudspeaker arrays.     

Effect of randomly varying impedance on the interference of the direct and ground-reflected waves

A randomly varying ground impedance is introduced into the solution for the sound field produced by a point source in a homogeneous atmosphere above a flat ground. The results show that in general the ground with a random impedance cannot be represented by an effective, non-random impedance. The behavior of the solution is studied with a relaxation model for the impedance in which porosity and the static flow resistivity are random variables. Mean values and standard deviations are adopted from measurements of two types of ground surfaces. For both surfaces, the mean intensity of the sound field above a random-impedance ground deviates only slightly from the intensity above a non-random impedance. The normalized standard deviation of intensity fluctuations can, however, be greater than one, thus indicating that for a particular realization of the random impedance, the sound intensity might significantly deviate from the intensity for a non-random impedance.