Acoustic Energy Conservation Research Articles

Assessing Transfer Matrix Models and Measurements Using Acoustic Energy Conservation Principles

Abstract Acoustic transfer matrices are widely used in the analysis of combustion dynamics of gas turbines. The reliability of the analysis thus depends on the quality of the determination of the transfer matrices of the individual acoustic elements composing the system's acoustic network. These matrices are, in some simple cases, deduced analytically using one-dimensional acoustic modeling. For more complex elements, such as swirlers, perforated plates, or injection units, the transfer matrix has to be obtained experimentally using an impedance tube setup. There are, however, uncertainties in the experimental determination of the transfer matrix coefficients and in the modeling of key elements like injection units. It is thus worth examining experimental data and assessing models using acoustic energy conservation principles. The general idea is to consider the acoustic power flow in the element represented by the transfer matrix T and compare the power input to the power output. This is best accomplished by obtaining a representation in terms of a scattering matrix S, which may be deduced from the transfer matrix T. It is first shown that standard models like those corresponding to a constant area duct or an area change comply with acoustic energy conservation. This analysis is then employed to assess the L−ζ model, widely used to describe injection unit dynamics. Acoustic conservation principles are then used to assess transfer matrices of a family of injectors determined experimentally and check that the data complies with these principles.

Predicting the propagation of acoustic waves using deep convolutional neural networks

A novel approach for numerically propagating acoustic waves in two-dimensional quiescent media has been developed through a fully convolutional multi-scale neural network following a spatio-temporal auto-regressive strategy. This data-driven method managed to produce accurate results for long simulation times with a database of Lattice Boltzmann temporal simulations of propagating Gaussian Pulses, even in the case of initial conditions unseen during training time, such as the plane wave configuration or the two initial Gaussian pulses of opposed amplitudes. Two different choices of optimization objectives are compared, resulting in an improved prediction accuracy when adding the spatial gradient difference error to the traditional mean squared error loss function. Further accuracy gains are observed when performing an a posteriori correction on the neural network prediction based on the conservation of acoustic energy, indicating the benefit of including physical information in data-driven methods. Finally, the method is shown to be capable of relaxing the classical time-step constraint of LBM and to provide parallel properties suitable for accelerating significantly acoustic propagation simulations.

On a time domain boundary integral equation formulation for acoustic scattering by rigid bodies in uniform mean flow.

It has been well-known that under the assumption of a uniform mean flow, the acoustic wave propagation equation can be formulated as a boundary integral equation. However, the constant mean flow assumption, while convenient for formulating the integral equation, does not satisfy the solid wall boundary condition wherever the body surface is not aligned with the assumed uniform flow. A customary boundary condition for rigid surfaces is that the normal acoustic velocity be zero. In this paper, a careful study of the acoustic energy conservation equation is presented that shows such a boundary condition would in fact lead to source or sink points on solid surfaces. An alternative solid wall boundary condition, termed zero energy flux boundary condition, is proposed that conserves the acoustic energy and a time domain boundary integral equation is derived. Furthermore, stabilization of the integral equation by Burton-Miller type reformulation is presented. The stability is studied theoretically as well as numerically by an eigenvalue analysis. Numerical solutions are also presented that demonstrate the stability of the current formulation.

Simulating acoustic coherent backscattering enhancement from random aggregations of omnidirectional scatterers.

Coherent backscatter enhancement (CBE) is a multiple scattering phenomenon that can lead to a doubling of the backscattered field intensity from a random aggregation of scatterers. It may be useful for remote sensing of scatterer aggregations, such as fish schools. This paper presents simulations of acoustic CBE from randomly placed omnidirectional point scatterers based on Foldy's field equations. The simulations are verified and validated through comparisons with Bragg scattering and Foldy's effective-medium theory, assessments of acoustic energy conservation, and comparisons with prior optical and acoustical CBE results. To make CBE comparisons with prior optics results, a CBE coherence function was postulated to account for resolution differences between the optics and simulation studies. For the higher-resolution optics studies, the postulated coherence function yields a CBE of 1.68, which matches optical CBE measurements. For the lower-resolution simulations, the same coherence function yields a CBE of 1.034, which agrees with appropriately extrapolated CBE simulation results, 1.030 ± 0.005. Assuming comparable resolution, the acoustics experiment and simulations both produce a CBE of approximately 1.5. The CBE peak is found to increase approximately monotonically with (k(2)σs)(1/4)(ks)(-1), where k is the wave number, s is the average spacing between scatterers, and σs is a scatterer's cross section.

Redirection of sound in straight fluid channel with elastic boundaries

A fluid channel clad between two solid plates is an acoustic waveguide where excitation of elastic waves at the channel boundaries has been usually neglected. This work develops a rigorous theory of scattering of sound by a finite-length fluid channel which takes into account excitation of elastic eigenmodes of two plates acoustically coupled through a fluid channel. The theory predicts an evidently contradictory result that the transmission and reflection coefficients of a nondissipative channel do not sum up to one. Moreover, they both exhibit deep minima at the same series of frequencies. It is shown that conservation of acoustic energy occurs due to redirection of sound, since part of the acoustic flux escapes into the solid plates. This scattering becomes possible because the uniform flatness of the boundaries of a straight channel is broken by vibrations. Theoretical predictions are supported by the experiments with ultrasound transmission through a narrow slit obtained between two brass or aluminum plates submerged in water. Measured transmission spectra exhibit deep minima exactly at the frequencies where the theory predicts strong redirection of sound.

Simulating coherent backscatter enhancement from aggregations of underwater scatterers

Remote classification of scatterers is an enduring priority in active sonar applications. For aggregations of marine life, typically schools or shoals of fish, remote classification may be possible when there is coherent backscattering enhancement (CBE) from the aggregation. CBE is a multiple scattering phenomenon that occurs in optics and acoustics when wave propagation paths within the aggregation are likely to be traversed in both directions. For a plane wave illuminating a half-space of randomly-placed, omnidirectional scatterers, CBE may lead to a doubling of the scattered field intensity in the direction opposite that of the incident wave (Akkermans et al. 1986). This presentation describes acoustic CBE simulations for finite-size spherical aggregations of point scatterers. The results are developed from the classical multiple-scattering equations of Foldy (1945), and are checked to ensure appropriate rotational symmetries and acoustic energy conservation when scatterer locations are random and when they are structured. Variations in the magnitude of the CBE peak and its angular width are presented for different frequencies, aggregation sizes, scatterer densities, and scatterer properties. Extension of these results to sonar-pulse scattering from schools of fish will be discussed. [Supported by the Office of Naval Research.]

Disturbance energy norms: A critical analysis

The question of which norm should be used to characterize the fluctuating disturbance energy in studies on thermoacoustic instabilities remains a topic of continued debate. In this paper we formulate a strategy for developing mathematically consistent norms that do not support spurious growth of disturbance energy in the absence of ‘physical’ sources of energy. A critical analysis of various disturbance energy norms existing in literature is conducted and important conclusions regarding positive definitiveness and susceptibility to exhibiting unphysical growth are drawn. It is shown that the energy norm proposed by Cantrell and Hart [Interaction between sound and flow in acoustic cavities: mass, momentum and energy considerations, Journal of Acoustical Society of America 36 (1964) 697–706] and Myers' first order disturbance energy norm [Transport of energy by disturbances in arbitrary steady flow, Journal of Fluid Mechanics 226 (1991) 383–400] are positive definite if M0<1 and M0<1/γ, respectively, where M0 is the magnitude of the local mean flow Mach number, γ is the ratio of specific heats at constant pressure and volume and the stipulated conditions should be met at every point in the flow domain. Our analysis shows that the disturbance energy norm proposed by Chu [On the energy transfer to small disturbances in fluid flow (part I), Acta Mechanica 1 (1965) 215–234] does not exhibit unphysical growth. It is also shown that this property is not unique to Chu's disturbance energy norm and there exists a family of norms which satisfy this requirement. The analysis also leads to an interesting interpretation of the acoustic energy conservation principle of Cantrell and Hart. The potential held by various disturbance energy norms for exhibiting fictitious growth is quantified using tools from nonmodal stability theory. It is concluded that if the mean flow Mach number is small, Myers' norm is a suitable measure of the disturbance energy.

A review of time-domain impedance modelling and applications

Time-domain impedance boundary conditions are reviewed and similarities between the models are shown. The extended Helmholtz resonator model is used with a time-domain impedance boundary condition for impedance eduction and simulation of the noise radiation from a lined aeroengine nozzle. Model parameters obtained by impedance eduction from flow duct measurements and then used to predict the noise radiation for realistic aeroengine nozzle flow and geometry. The prediction of sound radiation from a coaxial jet nozzle at approach conditions with the educed model parameters is performed in 2D and 3D. Various features of the numerical method that enable it to be used for realistic applications are presented. This includes the suppression of flow instabilities in the linear inviscid solution and analysis of the solution to prove the global conservation of acoustic energy in each individual result. It is found that the current approach is too time consuming for optimisation, even for a 2D simulation using parallel processing. Thus, finally the potential of porting a CAA method to the graphics processing unit (GPU) is shown in one example. The GPU based computation is about 100 times faster. The results are encouraging, even though the GPU variant needs further validation.

Acoustic intensity based data analysis and assessment in computational aeroacoustics

A methodology for assessing the quality and analysing the sound field of a numerical solution for an aeroacoustic problem is presented using an in-duct propagation problem as example. Namely an analysis of the acoustic energy conservation provides insight about the solution quality; sources of numerical error as well as physical source of sound are located by an analysis of the acoustic intensity; a mode analysis provides information about the sound field in the duct and a visualisation of the average pathway of sound by the acoustic intensity helps the interpretation of the modal amplitude data. The problem chosen for the presentation deals with the propagation of a single acoustic mode upstream through the tapering inlet duct of a centrifugal compressor. The propagation of tone noise is a typical problem for computational aeroacoustics. The numerical approach employs high-order finite-difference discretisation schemes with structured body-fitted meshes. An overset grid approach allows to overcome the geometrical complexity of the problem. The conservation of the acoustic energy is found to be violated by less than 3% in case of a potential base flow. In this case major sources of error are found in the interpolation between overset and background mesh. In case of a non-potential flow with boundary layers at the wall, the conservation of acoustic energy cannot be claimed. Consequently, it shows an increase of the acoustic energy, for which the accelerated flow in the nozzle is identified as source. It is shown that the filtering scheme affect the solution more than the spatial discretisation, if the grid resolution is relatively high. A strong scattering of energy into higher radial modes is detected. The fifth and sixth radial mode are found to be preferred in the current example. They are excited in the nozzle with a similar level. For these high radial modes, the transport of acoustic energy takes place further away from the duct wall. This explains the relatively large drop of about 20 dB for the average sound pressure level at the wall over the inlet nozzle.

On the Technology of Acoustic Utility. Acoustic Information and Energy. Study on Conservation Law of Acoustic Energy.

To explain the mechanism of sound generated by flow field, this paper proposes an equation of acoustic energy conservation which consists in general flow field. The equation of conservation expresses law of acoustic energy conversions. From this equation, it bacame clear that following things generate or dissipate sound. (1) propagation (2) work by viscosity (3) interaction between pressure and heat variation (4) work generated by heterogeneous of mean flow velocity (5) high degree action generated by turbulent perturbation (6) dissipation accompanied by thermal expansion (7) dissipation accompanied by pressure loss (8) interaction between pressure loss and propagation of sound (9) action by non linerity of low degree sound. This conversion law and incompressible numerical simulation will make clear mechanism of sound generation.

ON CALCULATION OF ACOUSTIC POWER

The previous work on sound power calculation in the presence of a mean flow has been focused on the casting of the basic acoustic equations in the form of an acoustic energy conservation law from which a definition of the acoustic intensity is extracted. The present paper shows that such definitions of sound intensity are deducible from the physical definition of sound power. Also presented is an expression or the sound power in a narrow pipe with a uniform mean flow.

Acoustic energy and momentum in a moving medium

By exploiting the mathematical analogy between the propagation of sound in a nonhomogeneous potential flow and the propagation of a scalar field in curved space-time, various wave "energy" and wave "momentum" conservation laws are established in a systematic manner. In particular, the acoustic energy conservation law due to Blokhintsev appears as the result of the conservation of a mixed covariant and contravariant energy-momentum tensor, while the exchange of relative energy between the wave and mean flow, first noted by Longuet-Higgins and Stewart in the context of ocean waves, appears as the covariant conservation of the doubly contravariant form of the same energy-momentum tensor.

Study on Sound Excited by Field. 2nd Report. Analysis on Rijke Tube.

For a thermoacoustic refrigerator or combustion oscillation, the secondary effect of oscillation is a ruling factor. A Rijke tube is made by the same effect and its structure is very simple. It has these characteristics : heat is released by heat conduction from a solid heat source, there is a stable flow, and it is an open system. The effects of excitation and dissipation for a Rijke tube are evaluated quantitatively. An equation of acoustic energy conservation which is a ruling equation for local area is integrated, then each term is solved. Analytic solution, transfer matrix method for vibration mode, and numerical calculation for the area around the heat source are used. There is a big density change in the area around the heat source, so the method used is adapted accordingly. A prediction of oscillation growth rate calculated by numerical simulation is compared with the experimental result. They are the same order.

Study on Sound Excited by Field. 1st Report. Proposal of Equation of Acoustic Energy Conservation.

Study on Sound Excited by Field. 1st Report. Proposal of Equation of Acoustic Energy Conservation.

Multisensor integration for underwater scene classification

We describe a new approach for the classification of a seafloor that is imaged with high frequency sonar and optical sensors. Information from these sensors is combined to evaluate the material properties of the seafloor. Estimation of material properties is based on the phenomenological relationship between the acoustical image intensity, surface roughness, and intrinsic object properties in the underwater scene. The sonar image yields backscatter estimates, while the optical stereo imagery yields surface roughness parameters. These two pieces of information are combined by a composite roughness model of high-frequency bottom backscattering phenomenon. The model is based on the conservation of acoustic energy travelling across a fluid-fluid interface. The model provides estimates of material density ratio and sound velocity ratio for the seafloor. These parameters serve as physically meaningful features for classification of the seafloor. Experimental results using real data illustrate the usefulness of this approach for autonomous and/or remotely operated undersea activity.

Estimation of the inverse acoustic transmission operator of a heterogeneous region directly from its reflection operator

A method is derived for estimating the inverse acoustic transmission operator of a heterogeneous subsurface region directly from its reflection operator (without determining the subsurface model itself). As such, it is an alternative way of formulating the seismic near-surface inverse problem and is an extension of widely used processing techniques for removing near-surface effects from seismic data (like gapped-deconvolution methods or methods for static corrections). The method is deterministic and based on two general concepts: the conservation of acoustic energy and an auxiliary relation derived from causality arguments. The different steps are illustrated by considering the case of a depth-dependent near-subsurface model for which the reflection response due to an impulsive line source is known. It is shown that a meaningful estimate is obtained for the inverse transmission operator, the most important step in the estimation procedure turns out to be the solution of a linear set of equations with a Levinson recursion scheme. In this way, near-surface effects can be removed without first having to estimate the near-surface model itself (this estimation would be a nonlinear, and in many cases, ill-posed problem). The example presented serves as an illustration and a 'proof of principle'. In order to fully assess the method, its sensitivity to imperfect knowledge of the source wavelet, to noise and to imperfect knowledge of the reflection operator has to be investigated.

Aeroacoustic fluid dynamic equations and their acoustic energy conservation corollary with O 2 and N 2 vibrational relaxation effects included

A formulation of fluid dynamic equations for air is given based on concepts of irreversible thermodynamics. An explicit expression for the instantaneous entropy function is derived from statistical physical principles and is approximated for the case when only O2 and N2 molecular vibrations are not in internal equilibrium. The fluid model includes an entropy balance equation based on this instantaneous entropy function. The model is complete; numerical values and formulas are included for all coefficients. In the latter part of the paper, an acoustic energy conservation theorem is derived which includes a non-negative expression for acoustic energy dissipation per unit time and volume. The latter is shown to be a lowest order approximation of temperature times the irreversible entropy production term in the entropy balance equation, even though this term does not appear in the linear acoustic equations. An independent derivation of the absorption coefficient of plane waves is given in which the energy theorem is utilized.

Acoustic conservation principles and an application to plane and modal propagation in nozzles and diffusers

Acoustic conservation principles are reviewed in some detail. The energy flow in the case of plane wave transmission across a shear discontinuity is analyzed and this problem serves as a convenient test for conservation. In this situation the energy principle developed by Blokhintsev is shown to be applicable. This principle is then used to study the propagation of plane and modal acoustic waves in nozzles and diffusers. The results obtained in an earlier work from a detailed analysis of the unsteady behavior of a supercritical nozzle are here completely retrieved. For modal propagation the duct is assumed to have a slowly varying section; conservation of acoustic energy then provides the axial distribution of pressure amplitude. An interesting cut-off property of nozzles and diffusers then is presented. It is shown that these elements have a last turning section at Mach number 0·51. A mode which passes this section cannot be turned or cut-off and will keep propagating.

Acoustic energy in non-uniform flows

The concept of acoustic energy is extended to non-uniform fluid flows. In general, the resulting energy balance equation exhibits acoustic energy production or dissipation within the flow, but in an important special case the production term is zero, implying conservation of acoustic energy. The acoustic energy equations used in classical and geometric acoustics are recovered from the general formulation by making the appropriate assumptions.