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Ueber den Einfluss der Wärmeleitung in einem Gase auf die Schallbewegung
... Furthermore, the BEM formulation implicitly satisfies the far-field radiation condition, making it a good fit for exterior acoustics. 23,24 Starting from the linearized Navier-Stokes, continuity, and Kirchhoff-Fourier equations describing the conservation of momentum, mass, and energy, 25,26 Cutanda Henríquez et al. 14 developed a first direct collocation BEM including the viscous and thermal effects. While this early version of the viscothermal BEM was tailored to axisymmetric geometries, later publications generalize the concept for arbitrary two-and three-dimensional problems. ...
... Unlike the finite element method, the BEM cannot directly evaluate the fundamental equations governing the acoustic wave propagation in the presence of viscosity and thermal conduction. Instead, Kirchhoff's dispersion relation 26 and its later extension 25 is adopted. Kirchhoff proposed to decompose the physical problem into three modal wave fields coupled solely on the domain boundary. ...
... While the local gradients in the coupling conditions are now accessible via Eq. (26), the mismatch between the locally defined velocities in the no-slip and null-divergence equation (Eqs. (16)- (19)) and the global definition in the discrete BEM equations (Eq. ...
The Helmholtz equation is a reliable model for acoustics in inviscid fluids. Real fluids, however, experience viscous and thermal dissipation that impact the sound propagation dynamics. The viscothermal losses primarily arise in the boundary region between the fluid and solid, the acoustic boundary layers. To preserve model accuracy for structures housing acoustic cavities of comparable size to the boundary layer thickness, meticulous consideration of these losses is essential. Recent research efforts aim to integrate viscothermal effects into acoustic boundary element methods (BEM). While the reduced discretization of BEM is advantageous over finite element methods, it results in fully populated system matrices whose conditioning deteriorates when extended with additional degrees of freedom to account for viscothermal dissipation. Solving such a linear system of equations becomes prohibitively expensive for large-scale applications, as only direct solvers can be used. This work proposes a revised formulation for the viscothermal BEM employing the Schur complement and a change of basis for the boundary coupling. We demonstrate that static condensation significantly improves the conditioning of the coupled problem. When paired with an iterative solution scheme, the approach lowers the algorithmic complexity and thus reduces the computational costs in terms of runtime and storage requirements. The results demonstrate the favorable performance of the new method, indicating its usability for applications of practical relevance in thermoviscous acoustics.
... Stokes [1] suggested that the main mechanism for the absorption of sound waves is internal friction (viscosity) that occurs during wave propagation. Later, Kirchhoff [2] showed that wave attenuation also occurs due to heat conduction. Based on these assumptions, the classical sound dispersion and attenuation theory was developed, expressing the attenuation coefficient α in terms of the viscosity and thermal conductivity coefficients ...
... In this study, we consider continuum relaxation models based on the generalized Chapman-Enskog method [24] and suitable for extended Navier-Stokes-Fourier equations. The paper is organized as follows: (1) we start with recalling the main peculiarities of the one-temperature (1T) description of nonequilibrium flows and compare bulk viscosity coefficients of diatomic and polyatomic species obtained in the frame of the 1T-model; in particular, we discuss the effect of splitting different energy modes on the bulk viscosity coefficient (Section 3); (2) in Section 4, we describe briefly a two-temperature (2T) model of a nonequilibrium flow with slow vibrational relaxation; (3) in the next sections, we derive the dispersion relation for a single-component gas in the frame of 2T and 1T models and analyze attenuation coefficients obtained using different approaches (Sections 5 and 6); (4) in Section 7, we derive the dispersion equation for a mixture with multiple vibrational temperatures and give preliminary estimates for the mixture attenuation coefficient; (5) the concluding remarks are given in Section 8. ...
... is the total constant-volume specific heat, c rot and c vibr are the rotational and vibrational specific heats, R = k B /m is the gas constant. The collision integrals Ω (1,1) , Ω (2,2) can be calculated using known interaction potentials [19,69]. The integral bracket β int is associated with the internal energy variation in inelastic collisions and has the form ...
Bulk viscosity and acoustic wave propagation in polyatomic gases and their mixtures are studied in the frame of one-temperature and multi-temperature continuum models developed using the generalized Chapman–Enskog method. Governing equations and constitutive relations for both models are written, and the dispersion equations are derived. In the vibrationally nonequilibrium multi-component gas mixture, wave attenuation mechanisms include viscosity, thermal conductivity, bulk viscosity, diffusion, thermal diffusion, and vibrational relaxation; in the proposed approach these mechanisms are fully coupled contrarily to commonly used models based on the separation of classical Stokes–Kirchhoff attenuation and relaxation. Contributions of rotational and vibrational modes to the bulk viscosity coefficient are evaluated. In the one-temperature approach, artificial separation of rotational and vibrational modes causes great overestimation of bulk viscosity whereas using the effective internal energy relaxation time yields good agreement with experimental data and molecular-dynamic simulations. In the multi-temperature approach, the bulk viscosity is specified only by rotational modes. The developed two-temperature model provides excellent agreement of theoretical and experimental attenuation coefficients in polyatomic gases; both the location and the value of its maximum are predicted correctly. One-temperature dispersion relations do not reproduce the non-monotonic behavior of the attenuation coefficient; large bulk viscosity improves its accuracy only in the very limited frequency range. It is emphasized that implementing large bulk viscosity in the one-temperature Navier–Stokes–Fourier equations may lead to unphysical results.
... 17 However, in this model, it is still very difficult to obtain equivalent parameters such as characteristic impedance and propagation constant of porous materials. Kirchhoff 18 and Rayleigh 19 regarded porous materials as smooth circular tubes arranged in parallel and uniformly and gave an accurate solution of sound propagation in these tubes considering the viscous effect and thermal effect. However, this theory is too complex and only considers the case of circular section, so it is rarely used to solve practical problems. ...
... (4) and (5). Then, the sound absorption coefficient can be derived from Eqs. (15)- (18). ...
... Using the five transport parameters in Table IV, the JCAL equivalent fluid model can be determined, and finally, the sound absorption coefficient of the porous material can be calculated by Eq. (18). As shown in Fig. 3, the black, green, blue, red lines, and symbols represent the theoretical predictions and numerical simulation results of porous materials containing parallel arranged smooth tubes, circumferential rough tubes, axial rough tubes, and two-dimensional rough tubes, respectively. ...
In this paper, a theoretical model for predicting the sound absorption performance of two-dimensional rough tube porous materials is established based on the Johnson-Champoux-Allard-Lafarge (JCAL) equivalent fluid model. The shape of the two-dimensional rough tube is approximated by trigonometric functions, and the theoretical expressions of its fluid transport parameters are given, including viscous permeability, thermal permeability, tortuosity, viscous characteristic length, and thermal characteristic length. In addition, the influence of shape factor is considered when calculating the thermal permeability and the viscous characteristic length, and its theoretical expression is given. The theoretical model is verified by a numerical simulation model based on the multi-scale asymptotic method, and good agreement is achieved. Compared with smooth tubes, circumferential rough tubes and axial rough tubes, the two-dimensional rough tubes not only enhances the viscous dissipation effect, but also enhances the thermal dissipation effect during the propagation of sound waves, thus realizing the high-efficiency sound absorption at lower frequencies. This work further develops the sound absorption theory of porous materials considering the roughness effect, and enriches the research and design ideas of porous materials.
... Stokes' law applies in an isotropic and homogeneous medium without taking into account heat conductivity P r = ∞. A generalization of this study taking into account the thermal conductivity was proposed by G.Kirchhoff in 1868 [1]. The Stokes-Kirchhoff relation expresses the sound attenuation in function of the characteristics of the fluid, namely the density ρ, the dynamic viscosity µ, the thermal conductivity ν, and the sound wave frequency ω. ...
... In the next section, we derive the Stokes-Kirchhoff attenuation rate from the linearized version of the Navier-Stokes system (1). The obtained formula coincides with the one of Stokes-Kirchhoff [1], but we also provide the sound dispersion i.e. the variation of the speed of propagation of the sound wave with the frequency. The original paper of Kirchhoff is only available in German, a modern and detailed presentation can be found in [8]. ...
... The above relation is exactly the Stokes-Kirchhoff attenuation derived in [1]. ...
In this paper, we first present an approach to derive the dispersion relation for sound waves in viscous and thermally conductive (mono) fluids. In particular, we derive a dispersion expression (i.e. the variation of the sound velocity as a function of frequency), which is of the second order of magnitude with respect to Knudsen numbers, as in the Stokes case corresponding to a non-conducting fluid (infinite Prandtl number).This formula completes the classical attenuation relation called Stokes-Kirchhoff. In the second part, we present some models for two fluids mixtures, and derive the effect of viscosity on the dispersion and attenuation of sound. In addition, we show how this approach can be used to discriminate multi-phase models that do not reproduce physically meaningful sound propagation velocities.
... Stokes' law applies in an isotropic and homogeneous medium without taking into account heat conductivity P r = ∞. A generalization of this study taking into account the thermal conductivity was proposed by G.Kirchhoff in 1868 [1]. The Stokes-Kirchhoff relation expresses the sound attenuation in function of the characteristics of the fluid, namely the density ρ, the dynamic viscosity µ, the thermal conductivity ν, and the sound wave frequency ω. ...
... The above relation is exactly the Stokes-Kirchhoff attenuation derived in [1]. We note that the attenuation in the medium of propagation varies with the frequency.Concerning sound wave dispersion, we calculate the phase velocity ...
... In this section we present the derivation of the Stokes-Kirchhoff attenuation formula according to the classical Kirchhoff paper [1]. This derivation addresses first order terms only and hence does not show the sound waves dispersion. ...
In this paper, we derive a dispersion relation for sound waves in viscous and heat conducting fluids. In particular this dispersion (i.e. variation of speed of sound with frequency) is shown to be of second order of magnitude, w.r.t. Knudsen numbers, as in the Stokes [2] case, corresponding to non-conductive fluid (Prandtl number P r = $\infty$). This formula completes the classical attenuation relation called Stokes-Kirchhoff. We represent in a simplified manner the Kirchhoff approach to derive this attenuation [1], starting from the 3D compressible Navier-stokes system. The classical Stokes-Kirchhoff formula has been questioned recently in [3] and a different (and incorrect) formula was proposed. We point out the non-trivial assumptions that are violated in the new derivation in [3] to reestablish the classical Stokes-Kirchhoff formula. Finally, we give an explanation to differences in dispersion and attenuation formulae that one may find in the literature through analysing the form of the considered attenuated solutions.
... In Schulz et al. [18] (Appendix B) it is shown that in most practical cases the contribution of / to the shear stress is small compared to the effect of the exchange of momentum. In addition, according to the common boundary layer approximation [6,19] we assume the pressure to be independent of the wall-normal coordinate in the shear stress layer. § A closer inspection of Eqs. ...
... There are several ways to connect the effective wall admittance to the momentum transfer impedance. We choose the way via the added displacement according to Eq. (19). By use of Eq. (12) excluding the pressure term, add is related to the wall shear stress by add w = 2 0 1 + ...
... which is in accordance with Starobinski [2] and Rebel and Ronneberger [13]. According to Eqs. (1), (19) ...
The acoustic shear stress effects at perforated liners with rigid but sound-permeable face
sheets and with superimposed grazing mean flow are investigated. The focus is on the soundinduced exchange of streamwise momentum between the mean flow and the wall. Two kinds of liner face sheets are to be distinguished, the homogeneously permeable wall and the wall with clearly separated and macroscopic openings, which is a more realistic model of technically feasible walls. To begin with, the focus is on the shear stress that drives the dynamics of the shearing mean flow over the homogeneously permeable wall. This is analyzed by means of a simple mathematical model of shear stress diffusion, which consists on a pair of differential equations for the acoustic shear stress and the wall-normal displacement. The physical analysis is concentrated on the relation between shear stress and the wall-normal displacement of the fluid elements, which determines the effective admittance of the wall. The shear stress is parametrized by the momentum transfer impedance which is defined as the ratio between the acoustic wall shear stress and the in-wall particle velocity evaluated at the wall. It turns out that the strong increase of the acoustic wall shear stress due to transfer of mean flow momentum to the wall is the dominating mechanism which affects the effective admittance of the wall. Nevertheless, the suitability of the momentum transfer impedance as part of a complete boundary condition of the wall is questioned. Furthermore, the disagreement between results predicted by the homogeneous wall model and some rare experimental data
obtained with real perforated liners indicates that further, still unknown mechanisms are
invoked by the inhomogeneity of real liners.
... The existence of a microscopic characteristic length l, determined by the size of local heterogeneities or the period size of the material, and a macroscopic characteristic length L = λ/2π, where λ = c/f is the sound wavelength with c the speed of sound, and their highly distinct values, i.e. l/L≪1, permit defining a Representative Elementary Volume, see Fig. 1 (Auriault et al., 2010). Therefore, under scale separation between the wavelength and the characteristic size of the pore (as highlighted by Kirchhoff (Kirchhoff, 1868) for r 3/2 W f < 10 − 6 cm⋅s − 3/2 , with r w the pore size being greater than 10 − 3 cm), momentum and energy equations allow to describe the visco-inertial and heat diffusion effect separately, together with the hypotheses of rigid and isothermal walls of the material solid skeleton, Ω fs : v 1 = 0, T 1 = 0onΩ fs , general solutions of Eqs. (1), (2) can be written in terms of complex density and bulk modulus, following (Allard and Atalla, 2009) ...
... Other than fibrous, granular and foam material, further typologies of micro-structures have been analysed in literature. Pin array lattice structure He et al., 2021;Horoshenkov et al., 2016;Horoshenkov et al., 2019;Auriault et al., 2010;Kirchhoff, 1868;Pompoli, 2023; arranged in tetragonal body cell have been studied and correlations for non-acoustical parameters have been presented in terms of dimensionless cell size. He at al. (He et al., 2021) pointed out the influence of roughness of the sound absorption (then on transport parameters) by investigating on a petal-shaped micro channel. ...
Transport parameters play a key role in characterizing the thermo-viscous behaviour of the microgeometry. Semi-phenomenological models provide valuable tools to establish a connection between the dynamic behaviour of porous materials and these transport parameters. However, each model has its limitations in terms of the frequency range and material types it can accurately represent. One of the most used semi-phenomenological acoustic models in the literature is the Johnson-Champoux-Allard-Lafarge (JCAL) model [J. This model requires the knowledge of six transport parameters, known as the porosity φ, airflow resistivity σ, thermal characteristic length Λ ʹ , viscous characteristic length Λ, high-frequency limit of tortuosity α ∞ , and static thermal permeability k ʹ 0 , which establish a connection between the micro-geometrical features of the porous material and its macroscopic behaviour when subjected to sound waves. The JCAL model is applicable to all types of porous materials, and the required transport parameters can be measured using suitable devices. With recent advancements in additive manufacturing, it is now possible to create porous materials with precise and controlled geometries. Therefore, understanding the relationships between microgeometry and transport parameters is crucial for designing porous materials with specific acoustic properties. This study provides a comprehensive overview of all the transport parameters involved in characterizing the JCAL model. It synthesizes various direct, indirect, and inverse measurement techniques used to assess these parameters. Additionally, computational approaches for evaluating the transport parameters from representative elementary volumes (REV) of materials are presented. Finally, the study compiles the existing correlations between transport parameters and the microgeometry of the unit cell from the available literature.
... The propagation of the sound waves in the tube illustrated in Fig. 1 is dissipative because of visco-thermal losses at the walls of the duct. In the present work, the damping of the acoustic wave without airflow is accounted for by modifying the wavenumber k o ¼ x=c o to k o C 0 where the propagation constant C 0 is expressed as [9,17] ...
... In Eq. (17), there are four unknown components of the transfer matrix and four equations. By solving this system, the components of the transfer matrix are obtained, they are given by [14,24,[34][35] T 11 ¼ p 0a u db À p 0b u da p da u db À p db u da ð18Þ T 12 ¼ p 0b p da À p 0a p db p da u db À p db u da ð19Þ ...
A transfer matrix method is proposed for experimental characterization of acoustic materials in the presence of airflow at high sound pressure levels. This method uses two experimental measurements with two different termination loads under flow to derive the transfer matrix coefficients of the tested material. The acoustic pressure and velocity fields upstream and downstream of the material are decomposed into forward and backward traveling waves. Complex models of the wavenumbers are used to account for the damping effect of acoustic waves in the presence of turbulent flow. The components of the transfer matrix of the material are given as function of the pressures and velocities at both faces of the material, which are obtained from the two measurements. The proposed method is validated by comparison with two-source method for different experimental measurements with airflow at high sound pressure levels up to 150 dB and good agreements are obtained. Thus, the proposed method can be used to estimate experimentally the acoustic properties of materials at higher sound pressure excitations in the presence of airflow.
... The exact solution in a rigid cylinder was derived by Kirchhoff [25]; however, its numerical computation requires an iterative solving of the implicit dispersion relation. The linearized Navier-Stokes equations may directly be solved in arbitrary geometries using 3D finite elements [26]. ...
... Various authors have focused on the special case of infinite right circular cylinders with rigid and isothermal walls, and have derived approximate models in various regimes of frequency and/or pipe size [25,30,31,32,33], often giving rise to fractional-order derivatives. A review of many such models was done by Tijdeman [1]. ...
Viscothermal acoustic propagation in gases contained in rigid straight or conical tubes is considered. Under the assumption that the wavelength is much larger than both the boundary layer thickness and the tube radius, pressure and flow are shown to be solutions of a pair of coupled 1D differential equations, formulated as transmission line equations involving complex loss coefficients. The derivation of these loss coefficients, which is usually accomplished in cylinders, is generalized here to conical geometries. In the well-known case of circular cylinders, the Zwikker–Kosten (ZK) theory is recovered. For circular cones, the expression of the loss coefficients is derived. It involves complex-order spherical harmonics, instead of Bessel functions for circular cylinders, and makes the hydraulic radius appear as a natural relevant geometrical parameter. We show that replacing the classical radius by the hydraulic radius in the ZK theory provides an affordable and accurate approximation of the analytic model derived for cones. The proposed formulas are used to compute the input impedance of a cone, and compared with a 3D reference. In an ideal setting, using the spherical harmonics or the hydraulic radius in the 1D method accurately approximates the full 3D method, and allows to increase accuracy by approximately two orders of magnitude compared to the ZK theory.
... While early attempts to model viscothermal acoustic fields had been mainly of analytical nature 183,184 , consecutive approaches were suited for coupling with numerical methods but still had substantial limitations due to the underlying assumptions. These concerned the geometry of the acoustic domain and the frequency range (cf. ...
... Kirchhoffs dispersion relation serves as starting point for boundary element based approaches. 183 In his pioneering work, Kirchhoff studied the attenuation of sound waves in cylindrical tubes due to viscous and thermal dissipation. He concluded that the viscothermal acoustic field can be split into three superimposed modal wave fields, which are solely coupled on the domain boundary: the acoustic mode, the entropy mode, and the vortex mode. ...
The modern scope of boundary element methods (BEM) for acoustics is reviewed in this paper. Over the last decades the BEM has gained popularity despite suffering from shortcomings, such as fictitious eigenfrequencies and poor scalability due to its dense and frequency-dependent coefficient matrices. Recent research activities have been focused on alleviating these drawbacks to enhance BEM usability across industry and academia. This paper reviews what is commonly known as direct BEM for linear time-harmonic acoustics. After introducing the boundary integral formulation of the Helmholtz equation for interior and exterior acoustic problems, recommendations are given regarding the boundary meshing and treatment of the non-uniqueness problem. It is shown how frequency sweeps and modal analyses can be carried out with BEM. Further extensions for efficient modeling of large-scale problems, including fast BEM and solutions methods, are surveyed. Additionally, this review paper discusses new application areas for modern BEM, such as viscothermal wave propagation, surface contribution analyses, and simulation of periodically arranged structures as found in acoustic metamaterials.
... Plusieurs des modèles acoustiques semi-phénoménologiques et empiriques les plus significatifs seront expliqués ci-après. Kirchhoff, 1868 Le premier modèle acoustique développé fut (Kirchhoff, 1868). Il fut le premier à adresser la propagation et l'absorption des ondes acoustiques dans les milieux poreux. ...
... Plusieurs des modèles acoustiques semi-phénoménologiques et empiriques les plus significatifs seront expliqués ci-après. Kirchhoff, 1868 Le premier modèle acoustique développé fut (Kirchhoff, 1868). Il fut le premier à adresser la propagation et l'absorption des ondes acoustiques dans les milieux poreux. ...
En France, la réglementation thermique pour les bâtiments évolue pour faire face aux enjeux climatiques. La loi Grenelle 2 et le Plan de Rénovation Energétique de l’Habitat établissent des exigences qui motivent la recherche de solutions novatrices pour l’isolation de bâtiments à fortes déperditions thermiques. C’est le cas du patrimoine vernaculaire, dont la bio-rénovation énergétique est au coeur de ce projet de thèse. Dans ce contexte, la filière des agro-bétons connaît actuellement un essor poussé par les avantages économiques et environnementaux de l’exploitation de déchets agricoles et de la production locale de ressources. Ce travail cherche à caractériser des bétons à base de chaux et de moelle de tournesol et de maïs, deux sous-produits agricoles disponibles en grande quantité et dont les propriétés ont été peu étudiées. A cette fin, une étude des caractéristiques mécaniques, hygrothermiques et acoustiques, comparées aux propriétés du béton de chanvre, est menée, en mettant l’accent sur l’impact des couples liant-granulat. Cette campagne expérimentale a le double objectif d’explorer de nouvelles méthodes de caractérisation des propriétés macroscopiques. En outre, un modèle mathématique, qui prend en considération le couplage des effets thermiques et hygroscopiques, est proposé afin de décrire la réponse hygrothermique des bétons étudiés à l’échelle paroi. L’étude expérimentale a permis de constater que les bétons de moelle à faible densité présentent des caractéristiques mécaniques relativement faibles, les classant à la limite du seuil pour les applications de type « mur » des Règles Professionnelles Construire en Chanvre. Toutefois, ses propriétés hygrothermiques intéressantes, dont la variation avec l’humidité a été déterminée, le rendent apte à l’utilisation en tant qu’isolant intérieur, qui est l’application principale envisagée par le projet. La campagne a également mis en évidence l’ampleur de l’impact des interactions entre la moelle et le liant sur les propriétés et l’importance d’étudier la compatibilité entre agrégats et liants lors du développement de nouveaux bétons. Lors de cette campagne, un nouveau dispositif de mesure de la conductivité thermique des parois a été mis en place. L’étude croisée des propriétés a débouché en une contribution à la détermination de la conductivité thermique et de la perméabilité à la vapeur à partir de mesures acoustiques. D’autre part, les résultats de l’étude numérique soulignent l’influence du climat sur la réponse de la paroi, qui détermine le choix du matériau isolant, et ont révélé que la présence de moelle ne garantit pas un degré d’hygroscopicité du béton plus important que la présence de chènevotte. Cette hygroscopicité a été prouvée avoir un impact non négligeable sur les flux thermiques en surface. Enfin, le modèle numérique proposé est utilisé pour quantifier l’impact de la présence de différents types de fluxmètres sur le flux thermique traversant une paroi lors d’un essai au laboratoire sous des sollicitations hygrothermiques maîtrisées.
... Stokes' law applies in an isotropic and homogeneous medium without taking into account heat conductivity Pr = ∞. A generalization of this study taking into account the thermal conductivity was proposed by G. Kirchhoff 1 in 1868. The Stokes-Kirchhoff relation expresses the sound attenuation in function of the characteristics of the fluid, namely the density ρ, the dynamic viscosity µ, the thermal conductivity ν, and the sound wave frequency ω. ...
... As perspective to this work, we may take into account the conductivity for the bi-fluid case, and also address the case of cylindrical and spherical waves. 1 ...
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ABSTRACT
We present a new formula for dispersion relation for sound waves in viscous and heat conducting (mono)fluids. In particular, this dispersion (i.e. variation of speed of sound with frequency) is shown to be of second order of magnitude, with respect to Knudsen numbers, as in the Stokes case, corresponding to non-conductive fluid (Infinite Prandtl number). This formula completes the classical attenuation relation called Stokes–Kirchhoff. In the meanwhile, we represent in a simplified manner the Kirchhoff approach to derive this attenuation, starting from the 3D compressible Navier-stokes system and emphasis the hypothesis that are essential to its validity. In a second part of the presentation we deal with the case of mixture of fluids (multi-fluids) and present how viscosity and thermal conductivity have effects on the dispersion and attenuation of sound. In addition, we show how the approach may be used to discriminate models in multi-phase modeling that do not reproduce physically meaningful sound propagation speeds.
... The results are summarized in Table 5 (Section 4.4), where the expressions for these quantities are the same provided that relevant values are taken for the speed of sound at infinite length scale (k = 0) 3 and for the equivalent dynamic viscosity. Concerning the single-fluid case, reference is made to the classical works of Stokes [47], Kirchhoff [28] and Fletcher [14] and asymptotic developments are presented with the specific physical conditions under which the different dispersion and attenuation formulas apply. ...
... and (111) generalizes Vekstein [53] to the case of arbitrary divariant fluids. Note also that the zero order term in ω ± I k is the same as the thermal contribution in the Stokes-Kirchhoff [28] attenuation: ...
Study of the propagation of sound in a single non-ideal fluid originates with Stokes in 1845 and Kirchhoff in 1868. The situation is much more complex in the case of two-fluid flow, both from the physical point of view, as the configuration of the flow matters greatly, and from the analytical point of view. The principle two-fluid models currently in use for CFD are the focus of this article. It is shown that analytical expressions for the speed of sound depend heavily on the chosen model. These sound speed expressions are compared with experimental values. The consequences for CFD models are discussed in the final section of this paper. It is found that numerical models with inaccurate wave speeds lead to incorrect numerical solutions, despite the accuracy of the numerical scheme.
... The results are summarized in Table 5 (Section 4.4), where the expressions for these quantities are the same provided that relevant values are taken for the speed of sound at infinite length scale (k = 0) 3 and for the equivalent dynamic viscosity. Concerning the single-fluid case, reference is made to the classical works of Stokes [47], Kirchhoff [28] and Fletcher [14] and asymptotic developments are presented with the specific physical conditions under which the different dispersion and attenuation formulas apply. ...
... and (111) generalizes Vekstein [53] to the case of arbitrary divariant fluids. Note also that the zero order term in ω ± I k is the same as the thermal contribution in the Stokes-Kirchhoff [28] attenuation: ...
Study of the propagation of sound in a single non-ideal fluid originates with Stokes in 1845 and Kirchhoff in 1868. The situation is much more complex in the case of two-fluid flow, both from the physical point of view, as the configuration of the flow matters greatly, and from the analytical point of view. The principle two-fluid models currently in use for CFD are the focus of this article. It is shown that analytical expressions for the speed of sound depend heavily on the chosen model. These sound speed expressions are compared with experimental values. The consequences for CFD models are discussed in the final section of this paper. We outline the fact that any physical model, (on which numerical models could be based through numerical methods) should propagate the pressure-waves in a physically meaningful manner coherent with experimentation. Hence we question the relevance of physical models that imply two sound speeds (e.g.one per fluid). Whatever is the numerical scheme, it is not reasonable to assume that it could correct the physical model.
... The propagation of sound in a uniform, circular narrow, and wide tube can be solved by Kirchhoff's theory of sound propagation [19], which accounts for the effect of air viscosity and thermal conductivity. More recently, Zwikker and Kosten [20] introduced a simpler method by considering the effect of viscosity and thermal conductivity separately in terms of complex density and compressibility. ...
This study employs a series of multi-coiled quarter wave resonators (QWRs) to diminish low-frequency noise in exhaust systems. To optimize space utilization, these QWRs are coiled into a helical structure. The absorption mechanism of the absorber is comprehensively explained through analytical derivation and numerical simulation. Six QWRs are organized in parallel to absorb a broad spectrum of ultra-low frequency sound (75 to 100 Hz). This study also investigates the impact of background flow within the system, which alters the acoustic properties of the resonators. The thickness of the resonators is kept within the subwavelength limit.
... For the reflections we assume a value of R = 1 upstream and R = 0.9 downstream. The visco-thermal damping is based on the Kirchoff [11] damping, but corrected for the total wall length. Here a model similar to porous media is used [12]. ...
Internally corrugated flexible pipes and hoses are prone to generate high amplitude tonal pressure pulsations (singing, also called FLIP). This is a common experience in flexible risers and flowline used in the offshore oil & gas industry. Models to predict the onset conditions when this singing occurs have been developed for this market. Similar flexible lines are and will be used for CO2 transport, for instance in CCS projects. The transport conditions will be mainly in the supercritical region. To verify if singing occurs in the supercritical region and if the prediction models are still valid, small scale singing experiments have been done with a 1.37m long 6mm ID hose with pure CO2 at conditions ranging from 2 to 180 bara.
Clear singing was shown to occur at all pressures with similar characteristics (Strouhal number and amplitudes) between CO2 operations as low pressure air conditions. The onset velocity could be matched based on an energy balance method and on a feedback criterion. The current tools employed for natural gas transport do need adaptions to be valid for CO2 transport calculations.
The main conclusion is that flow induced pulsations in corrugated flexibles are generated in CO2 at supercritical conditions and that when these type of flexibles are used for CO2 transport, the same screening is advised as is common for the use of flexibles for natural gas transport.
... 348) derived a fourth-order differential equation describing single-frequency temperature disturbances of small amplitude in an otherwise uniform thermoviscous fluid, whose equilibrium pressure and specific internal energy are functions pðq; TÞ; eðq; TÞ of density and temperature. Rayleigh's derivation extended an earlier study by Kirchhoff (1868) in which the fluid was assumed to be an ideal gas. Viscous stresses and heat conduction were assumed to obey the Navier-Stokes-Fourier equations in both studies, with bulk viscosity included by Rayleigh (1894). ...
The Kirchhoff-Helmholtz representation of linear acoustics is generalized to thermoviscous fluids, by deriving separate bounded-region equations for the acoustic, entropy, and vorticity modes in a uniform fluid at rest. For the acoustic and entropy modes we introduce modal variables in terms of pressure and entropy perturbations, and develop asymptotic approximations to the mode equations that are valid to specified orders in two thermoviscous parameters. The introduction of spatial windowing for the mode variables leads to surface source and dipole distributions as a way of representing boundary conditions for each mode. For the acoustic mode the boundary source distribution is expressible in terms of the fluid normal velocity, the normal heat flux, and the vector ω×n̂, where ω is the vorticity on the boundary and n̂ is the unit normal; only the first of these is present in the usual lossless-fluid version of the Kirchhoff-Helmholtz representation. Use of the generalized thermoviscous representation to project exterior sound fields from surface data, where the data may contain contributions from all three linear modes, is shown to be robust to cross-modal contamination. The asymptotic limitations of the thermoviscous modal equations are discussed in an appendix.
... Considering the attenuation of ultrasonic waves in air, the relation between the ultrasound pressure at the sensor (P sensor ) and that at the ultrasonic transducer (P PZT ) is P sensor (ω) = e −α(ω)d P PZT (ω), where d is the distance between the ultrasonic transducer and the sensor, which is kept to be approximately 1 cm in our experiment. α(ω) is the frequency-dependent acoustic attenuation coefficient, which is obtained from the Stokes-Kirchhoff formula [54,55]: ...
Owing to their dual-resonance enhanced sensitivity, cavity optomechanical systems provide an ideal platform for ultrasound sensing. In this work, we realize high-sensitivity air-coupled ultrasound sensing from kilohertz to megahertz frequency range based on whispering gallery mode microcavities. Using a 57-μm-diameter microtoroid with high optical Q factor (approximately 107) and mechanical Q factor (approximately 700), we achieve sensitivities of 46μPa Hz−1/2–10 mPa Hz−1/2 in a frequency range of 0.25–3.2 MHz. Thermal-noise-limited sensitivity is realized around a mechanical resonance at 2.56 MHz, in a frequency range of 0.6 MHz. We also observe the second- and third-order mechanical sidebands, and quantitatively study the intensities of each mechanical sideband as a function of the mechanical displacement. Measuring the combination of signal-to-noise ratios at all sidebands has the potential to extend the dynamic range of ultrasound sensing.
... As a result, heat is collected by the air during the period of maximum condensation, and heat is given off by it during the phase of highest rarefaction, and therefore there is a propensity to sustain the vibrations. Kirchhoff et al. [11] developed equations that provided an account for both heat attenuation of sound, viscous effects and effect of a large-radius tube. Rott's et al. [12] established an effective general linear theory of thermoacoustics to construct and examine thermoacoustic devices such as refrigerators and engines (see Table 1). ...
Thermo-Acoustics is a phenomenon that involves the interplay of sound and thermal engines. Its most recent application is in the development of heat engines and pumps/refrigerators. One such occurrence is thermo-acoustic refrigeration, which employs high-intensity sound waves in a pressured gas tube to transport heat from one location to another and generate a cooling effect. In this paper, the design covers stack dimension, acoustic driver selection, and acoustic resonator selection. Acoustic resonator selection, stack dimension and acoustic driver selection are discussed in this paper. The investigation revealed that, the temperature control by thermal acoustics was found to be feasible, but inefficiency was not achieved because of material constraints. For this basic device, we acquired a temperature differential of 3 degrees Celsius. However, based on these constraints, various recommendations for improving the performance of thermoacoustic refrigerators were developed. In experiments, superior materials, such as materials with high heat capacity, and working fluids, such as inert gases, have been shown to improve efficiency.
Heat is turned into sound energy in a thermoacoustic engine, and its energy is useful for practical work. By using this device, heat is transferred from a higher-to a lower-temperature sink. The preceding process is reversed by a thermo-acoustic refrigerator, which absorbs heat from a low-temperature medium and rejects it to a high-temperature medium by using acoustic power.
... As is often true in applications, we assume that the mean flow is slow compared to the speed of sound so that it is justified to linearize the compressible Navier-Stokes equations around a fluid state at rest, which leads to a considerable simplification of the resulting equations. Since Kirchhoff [14] devised a semi-analytical solution for lossy sound propagation in a cylindrical pipe, much effort has been devoted to reformulations, approximate models, or numerical schemes that may reduce the computational cost of viscothermal acoustic simulations. Chabassier and Thibault [6] provide a comprehensive review of existing approaches with emphasis on simulations of wind instruments in frequency domain. ...
Accurate simulations of sound propagation in narrow geometries need to account for viscous and thermal losses. In this respect, effective boundary conditions that model viscothermal losses in frequency-domain acoustics have recently gained in popularity. Here, we investigate the time-domain analogue of one such boundary condition. We demonstrate that the thermal part of the boundary condition is dissipative in time domain as expected, while the viscous part, unexpectedly, may lead to an infinite instability. A finite-difference-time-domain scheme is developed for simulations of sound propagation in a duct with only thermal losses, and the obtained transmission characteristics are found to be in excellent agreement with frequency-domain simulations.
... Such acoustic systems are predominantly dominated by two different forms of pressure distortion, resonance and attenuation, depending on the pressure frequency f . [107] and Rayleigh [108]) and approaches introducing one or more simplifying assumptions. The following brief theoretical discussion deals with the latter. ...
In the aerodynamic examination of objects in wind tunnels, engineers can choose from
a wide range of different measurement tools for quantifying the flow. The choice of
the appropriate measurement methodology is made based on the characteristics of the
possible tools. Hence, subject of the present work is the development and testing of
a multi-hole probe for the measurement of unsteady aerodynamic phenomena, which
tries to combine properties of the different measurement systems and is also called fast-response aerodynamic probe (FRAP). For this purpose, the characteristics of the probe
and the incorporated pressure sensors are determined and unresolved concerns with
respect to the associated metrological possibilities and limitations are addressed. Using
miniaturized sensors in the pressure probe body, a competitive measurement solution is
to be found.
Therefore, the working principles of multi-hole probes and pressure sensors are briefly
discussed. Special focus lies on newly developed fiber-optic pressure sensors. Various
methods for the spatial calibration of the probe are presented. Furthermore, the machine
learning approach, Gaussian process regression, is applied to reduce the calibration time.
Additionally, a temporal calibration characterizes the acoustic system inside the probe.
Various tests that determine the sensitivity of the pressure sensor with respect to external
effects are explained and the sensor performance is evaluated. The multi-hole probe
behavior is further analyzed in numerical investigations of the flow around the probe
and of the line-cavity system inside the probe.
In addition, multiple measurement campaigns are carried out in the low-speed wind
tunnel facilities at the Chair of Aerodynamics and Fluid Mechanics of the Technical University of Munich (TUM-AER). The determination of the spatial and temporal resolution
of the probe in grid-generated turbulence is of special interest and aspects concerning
uncertainty quantification of the probe measurements are given. To demonstrate the capabilities of the fully characterized probe, the quantification of unsteady flow phenomena in different measurement scenarios are discussed, a) in the wake of a circular cylinder and b) in the wake of a dynamically actuated wind turbine. By additionally acquiring a synchronization signal, independent FRAP measurements in the wind turbine wake are synchronized in the phase-locking postprocessing procedure. The FRAP investigations unveil that the actuated cases show an earlier mixing and entrainment of high energetic fluid already near the rotor plane. Consequently, the actuated control strategy is a promising approach that can contribute to a synergistic interaction of multiple wind turbines, as found in wind farms.
Measurements with the FRAP show a very robust and easy to use handling with a
fast and cost-effcient setup. The incorporation of advanced (post-) processing routines,
allows time-averaged, phase-locked and transient analyses of the flow field patterns.
... The study of the viscous and boundary layers was first explored by Kirchhoff [88] and expanded by Rayleigh [89]. Also, there are many studies on the effects of the thermal and viscous boundary layers in acoustics [90][91][92][93]. ...
The advent and development of acoustic metamaterials and metasurfaces in recent years has overturned conventional means in all aspects of acoustic waves propagation and manipulation. In the context of the sound absorption and noise conrtol, it has offered an unprecedented expansion of our ability to attenuate the low-frequency sound beyond the classical physical limits. The main aim of this PhD dissertation is to conceive and design acoustic metasurfaces for extreme low-frequency absorption (<100Hz). First, the concept of multicoiled metasurface absorber (MCM) is proposed and discussed. The effectiveness of its physical mechanism is theoretically, numerically, and experientially demonstrated. The presented multicoiled metasurface is capable of fully absorbing acoustic energy at extreme low-frequency of 50Hz with a deep subwavelength thickness (λ/527). To circumvent the conventional physics- and rule-based approaches and accelerate the design process, novel deep learning-based framework is introduced in this dissertation. Specifically, the convolutional neural network (CNN) and conditional generative adversarial networks (CGAN) are implemented to simulate and optimize complex metasurface absorber structures. The developed deep learning-based framework for the acoustic metasurface absorbers can be potentially extended to the design and optimization of other acoustic devices and structures. This dissertation provides a new way for deep learning-enabled acoustic metasurface designs that will allow acoustics community to focus more on truly creative and pragmatic ideas. This will be led to solving complex design problems that have yet to be explored by the machine, rather than on tedious trial and error processes.
... The Stokes-Kirchoff classic attenuation equation 170,304 predicts the acoustic attenuation of a plane wave due to viscous dissipation and heat conduction as a function of frequency: ...
Combustion noise is relevant to current aviation, rocket, and ground-based gas turbine engines, as it contributes to environmental noise pollution and can trigger thermoacoustic instabilities. These consequences are particularly prevalent in lean, premixed, prevaporised combustors, which are designed to reduce nitrous oxide (NOx) emissions. As a result, there is a need to better understand the mechanisms that drive sound generation in such systems. There are two components to combustion noise: direct noise – generated by the unsteady heat release of a flame – and indirect noise – produced by the acceleration of entropic, vortical, or compositional inhomogeneities. Separation of the respective contributions has proven to be complex to achieve in real engines – for this purpose, model experiments have been developed. These are non-reacting experiments that use unsteady, synthetic perturbations to emulate the fundamental physics of combustion acoustics processes and provide clear data for comparison with theory. Indirect noise models have been theorised for compositional perturbations and experimental validation has been provided via the measurement of acoustic waves (i.e. the output), while assuming a constant compositional perturbation (i.e. the input). This thesis follows on from such experiments by simultaneously measuring both acoustic and compositional waves in a model setup, making use of numerical, analytical, and experimental studies. It first builds upon a previous model experiment through a numerical investigation on the generation, mixing, and convection of entropic and compositional waves generated by heat addition and gas injection. The computed temperature and mass fraction fields are compared with experimental results and inform the design of a new model setup – the Canonical Wave Rig (CWR). The CWR is then used to study direct and indirect noise under simplified, well-controlled conditions. Subsonic and sonic (choked) conditions are investigated for a convergent-divergent nozzle. Acoustic, entropic, and compositional perturbations are generated via the co-flow injection of air or methane into a low Mach number mean flow of air. Spontaneous Raman spectroscopy (1.5 kHz) is employed for the time-resolved measurement of the local concentration upstream of the nozzle. Single pulse experiments in the infra-sound range are used to validate the derived analytical model for direct noise due to co-flow injection. The measurement of non-reverberated indirect noise is made for the first time and is contrasted with results obtained via dereverberation (i.e. removing the effect of pressure build up due to acoustic reflections). Indirect noise transfer functions are calculated using the acoustic and compositional measurements, and issues pertaining to the methods applied are highlighted. Lastly, the pulse burst injection of methane at frequencies up to 250 Hz is presented. The goal of these experiments is to provide data at more realistic frequencies and amplitudes.
... 4 However, the two approaches are equivalent as shown by Dragonetti et al. 5 Numerous theoretical models exist to predictK andq for acoustical porous materials with a motionless skeleton. For elementary uniform cross-section geometries, such as slits, circular pores, and pin arrays, analytical descriptions of these quantities are available in the literature, 6,7 whereas for materials such as fibres and foams, semiempirical models have been developed. The simplest mono-parameter empirical model was proposed by Delany and Bazley 8 and further revisited by Miki,9 where the only airflow resistivity r is required as the input parameter. ...
In this work, an acoustic lumped element technique has been developed to measure the dynamic bulk modulus of porous materials in the low frequency range (f < 500 Hz). Based on the electroacoustic analogy of wave propagation inside a porous medium, an analytical derivation of the measurement method is given. Unlike other techniques, it requires the use of only two microphones placed in the cavity containing the sample being tested and in the loudspeaker box. The proposed method provides reliable results when the longitudinal viscous impedance within the medium is negligible with respect to the transversal thermal impedance. The upper limit of the frequency validity range can be determined from the relationship abs(kd)<=0:5, where k is the acoustic wavenumber of the porous material and d is the sample thickness. Furthermore, some practical aspects related to the measure are also reported. To validate the methodology, experimental campaigns have been performed on different typologies of materials (fibrous, uniform cross-section geometries, additive manufactured sample, and foam) in two laboratories.
The experimental results show good agreement with the theoretical results within the frequency validity range.
... ,, mk . The exact solution for 0 was derived by Kirchhoff [38]. The exact solution of this problem for arbitrary radial, azimuthal and axial modes is implemented in the open-source code ColESo [39], the implementation details can be found in [40]. ...
In this paper, the vertex-centered EBR schemes, originally developed for solving the Euler and Navier–Stokes equations on unstructured tetrahedral meshes, are generalized to unstructured mixed-element meshes. In these schemes, the convective flux is approximated using quasi-1D edge-oriented reconstructions. On hybrid mixed-element meshes with layers of highly anisotropic prismatic cells that are commonly used in simulations of external turbulent flows, the quasi-1D reconstructions may lead to significantly irregular stencils, which cause both larger approximation error and computational instability. To improve accuracy and robustness of the EBR schemes on these meshes, we propose to use curvilinear reconstructions of variables. We present algorithms of constructing curvilinear stencils and validate the resulting schemes on a series of 2D and 3D problems such as an acoustic wave within an infinite cylinder in the presence of viscosity and thermal conduction, turbulent flows around the NACA0012 airfoil and the Caradonna-Tung helicopter rotor. In all these cases, the curvilinear reconstructions improve accuracy of the numerical solutions with no extra computational cost.
... Considering the attenuation of ultrasonic waves in air, the relation between the ultrasound pressure at the sensor (P sensor ) and that at the ultrasonic transducer (P PZT ) is P sensor (ω) = e −α(ω)d P PZT (ω), where d is the distance between the ul-trasonic transducer and the sensor, which is kept to be ∼1 cm in our experiment. α(ω) is the frequency dependent acoustic attenuation coefficient, which is obtained from the Stokes-Kirchhoff formula [54,55] ...
Cavity optomechanical systems provide an ideal platform for ultrasound sensing, due to its dual-resonance enhanced sensitivity. Here we realize high sensitivity air-coupled ultrasound sensing in the megahertz (MHz) frequency range, using a microtoroid cavity. Benefitting from both the high optical Q factor (~10^7) and mechanical Q factor (~700), we achieve sensitivity of 46 {\mu}Pa/Hz^1/2 -10 mPa/Hz^1/2 in a frequency range of 0.25-3.2 MHz. Thermal-noise-limited sensitivity is realized around the mechanical resonance at 2.56 MHz, in a frequency range of 0.6 MHz. We also observe the second- and third-order mechanical sidebands, when driving the microcavity with an ultrasonic wave at the mechanical resonance and quantitatively study the intensities of each mechanical sideband as a function of the mechanical displacement. Measuring the combination of signal to noise ratios at all the sidebands has the potential to extend the dynamic range of displacement sensing.
... The finite element method (FEM) can be used to generate numerical solutions; however, the computational cost is high because the equations are strongly coupled with state fields. Kirchhoff [18] studied analytical solutions to the linearized Navier-Stokes equations defined in an infinite cylinder with a rigid isothermal wall as its surface. Tijdeman [19] validates Zwikker and Kosten's [20] simplified Kirchhoff solution, and the low reduced frequency (LRF) model was proposed. ...
This paper proposes a level set-based topology optimization method for designing acoustic structures with viscous and thermal boundary layers in perspective. It is known that acoustic waves propagating in a narrow channel are damped by viscous and thermal boundary layers. To estimate these viscothermal effects, we first introduce a sequential linearized Navier-Stokes model based on three weakly coupled Helmholtz equations for viscous, thermal, and acoustic pressure fields. Then, the optimization problem is formulated, where a sound-absorbing structure comprising air and an isothermal rigid medium is targeted, and its sound absorption coefficient is set as an objective function. The adjoint variable method and the concept of the topological derivative are used to obtain design sensitivity. A level set-based topology optimization method is used to solve the optimization problem. Two-dimensional numerical examples are provided to support the validity of the proposed method. In addition, the mechanisms that lead to the high absorption coefficient of the optimized design are discussed.
... Rigid Frame Models such as Kirchhoff [196] , Zwikker and Kosten [192] , Stinson [197] : For cylindrical pores: ...
Noise pollution impacts the well-being of millions of people on a daily basis and can lead to serious health issues such as hearing loss and stress. Developing efficient yet cost-effective sound absorbing materials for noise reduction in vehicles, buildings and large spaces has become an important research area. The present review focuses on the latest developments in sound absorbing products based on engineering materials solutions as well as tailored micro and nanostructures. In addition, modelling techniques for simulating sound wave propagation through porous media are briefly introduced. Various materials such as polyurethane foam, thermoplastic foams, textile fabrics and composites are reviewed with different design strategies and structures ranging from foam structures to micro-perforated panels summarized and compared. The effect of different types of micro- and nanofillers, hierarchical and sandwich structures and synergistic effects of combining multiple constituents with structural designs at different length scales to achieve the desired acoustic properties are discussed. Mechanisms of each are analysed with the aim of exploring new strategies based on existing knowledge. Opportunities and obstacles are identified, while engineering applications ranging from automotive to built environment are reviewed, together with their desired properties and functions to shed light on future research directions towards advanced acoustic materials.
... In the case of a cylindrical pipe, (Kirchhoff, 1868) derives an implicit analytical dispersion relation to the LNS equations that must be solved iteratively, which does not offer a practical modeling framework. The reduced 1D model derived from an approximation of LNS by (Zwikker and Kosten, 1949) has been shown to be valid for audible frequencies and assuming the reduced frequency k = ωR/c is small, where ω is the angular frequency, R the radius of the pipe and c the celerity of the wave (Tijdeman, 1975). ...
Accurate modeling of the acoustic propagation in tubes of varying cross section in musical acoustics must include the effects of the viscous and thermal boundary layers. Models of viscothermal losses are classically written in the frequency domain. An approximate time-domain model is proposed in which all of the physical parameters of the instrument, the bore shape or the wave celerity, are explicit coefficients. The model depends on absolute tabulated constants, which only reflect that the pipe is axisymmetric. It can be understood as a telegrapher's equations augmented by an adjustable number of auxiliary unknowns. A global energy is dissipated. A time discretization based on variational approximation is proposed along with numerical experiments and comparisons with other models.
Low-frequency noise attenuation is a challenge in diferent engineering applications. This study investigates the efectiveness of a labyrinth-type Helmholtz resonator to control noise in pipelines, whose confguration aims to efciently reduce low-frequency noise, occupying a small volume compared to the working wavelength. An analytical model was developed using the transfer matrix method, implementing simplifed assumptions of the resonator geometry. The results of this model are compared to results of numerical simulations obtained by the fnite element method, which indicated errors in the simplifed hypotheses. The Schwarz–Christofel mapping allowed a better representation of the duct, so that the curves in the analytical model were considered and subsequently the dissipative efects of air by the Stinson equivalent fuid model were also included in its formulation. Three resonator prototypes were built by 3D printing. Experimental tests were carried out with an impedance tube and the results compared to theoretical predictions showed an excellent agreement in the resonator
tuning frequency with a maximum error of 1%, and adequate agreement in the magnitude with a maximum error of 16%. The modifed labyrinthic resonator proved to be a viable alternative for controlling low-frequency noise, reaching attenuations
of around 28 dB at 182 Hz, with a total thickness 20 times smaller than the working wavelength.
Моделируется рассеяние плоской звуковой волны на сферической границе раздела двух жидких или газообразных сред. Принимается во внимание влияние теплопроводности и вязкости; при этом используются результаты классической работы Г. Кирхгофа о распространении звука в вязкой и теплопроводящей среде. Сферическая поверхность может иметь любой волновой размер. Полученные результаты сравниваются с полем, рассеянным на твердой сфере, являющейся идеальным проводником тепла.
In this work, we demonstrate in a proof of concept experiment the efficient noise absorption of a 3-D printed panel designed with appropriately arranged space-coiling labyrinthine acoustic elementary cells of various sizes. The labyrinthine unit cells are analytically and numerically analysed to determine their absorption characteristics and then fabricated and experimentally tested in an impedance tube to verify the dependence of absorption characteristics on cell thickness and lateral size. The resonance frequency of the unit cell is seen to scale approximately linearly with respect to both thickness and lateral size in the considered range, enabling easy tunability of the working frequency. Using these data, a flat panel is designed and fabricated by arranging cells of different dimensions in a quasi-periodic lattice, exploiting the acoustic “rainbow” effect, i.e. superimposing the frequency response of the different cells to generate a wider absorption spectrum, covering the target frequency range, chosen between 800 and 1400 Hz. The panel is thinner and more lightweight compared to traditional sound absorbing solutions and designed in modular form, so as to be applicable to different geometries. The performance of the panel is experimentally validated in a small-scale reverberation room, and an absorption close to ideal values is demonstrated at the desired frequencies of operation. Thus, this work suggests a design procedure for noise-mitigation panel solutions and provides experimental proof of the versatility and effectiveness of labyrinthine metamaterials for tunable mid- to low-frequency sound attenuation.
We present a new stochastic simulation method for determining the long-wavelength effective dynamic bulk modulus of gases, such as ambient air, saturating porous media with relatively arbitrary microgeometries, i.e., simple enough to warrant Biot’s simplification that the fluid and solid motions are quasi-incompressible motions at the pore scale. The simulation method is based on the mathematical isomorphism between two different physical problems. One of them is the actual Fourier heat exchange problem between gas and solid in the context of Biot theory. The other is a diffusion-disintegration-controlled problem that considers Brownian motion of diffusing particles undergoing radioactive-type decay in the pore volume and instant decay at the pore walls. By appropriately choosing the decay time and the diffusion coefficient, the stochastic algorithm we develop to determine the average lifetime of the diffusing particles, directly gives the effective apparent modulus of the saturating fluid. We show how it leads to purely geometric stochastic constructions to determine a number of geometrical parameters. After validating the algorithm for cylindrical circular pores, its power is illustrated for the case of fibrous materials of the type used in noise control. The results agree well with a model of the effective modulus with three purely geometric parameters of the pore space: static thermal permeability divided by porosity, static thermal tortuosity, and thermal characteristic length.
Early detection of gas influx in boreholes while drilling is of significant interest to drilling operators. Several studies suggest a good correlation between ultrasound backscatter/attenuation and gas volume fraction (GVF) in drilling muds, and thereby propose methods for quantification of GVF in boreholes. However, the aforementioned studies neglect the influence of bubble size, which can vary significantly over time. This paper proposes a model to combine existing theories for ultrasound backscatter from bubbles depending on their size, viz. Rayleigh scattering for smaller bubbles, and specular reflection for larger bubbles. The proposed model is demonstrated using simulations and experiments, where the ultrasound backscatter is evaluated from bubble clouds of varying bubbles sizes. It is shown that the size and number of bubbles strongly influence ultrasound backscatter intensity, and it is correlated to GVF only when the bubble size distribution is known. The information on bubble size is difficult to obtain in field conditions causing this correlation to break down. Consequently, it is difficult to reliably apply methods based on ultrasound backscatter, and by extension its attenuation, for the quantification of GVF during influx events in a borehole. These methods can however be applied as highly sensitive detectors of gas bubbles for GVF ≥\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ge$$\end{document}1 vol%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\%$$\end{document}.
Sonochemistry in a thin fluid layer has advantages of no visible cavitation, no turbulence, negligible temperature changes (≲1 °C), low power transducers, and transmissibility (sound pressure amplification) of ≳106. Unlike sonochemistry in semi-infinite fluids, resonance and so constructive interference of sound pressure can be established in thin layers. Constructive interference enables substantial amplification of sound pressure at solid fluid interfaces. Fluid properties of sound velocity and attenuation, oscillator input frequency, and thin fluid layer thickness couple to established resonance in underdamped conditions. In thin layer sonochemistry (TLS), thin layers are established where ultrasonic wavelength and oscillator-interface separation are comparable, about a centimeter in water. Solution of a one dimensional wave equation identifies explicit relationships between the system parameters required to establish resonance and constructive interference in a thin layer.
Acoustic liners are an effective way to dampen aircraft noise. Conventional single-degree-of-freedom liners consist of a perforated facesheet backed with a honeycomb structure and a rigid end plate. Their damping excels near their resonance frequency, which is antiproportional to the cavity depth ([Formula: see text]-resonator) or the cavity volume (Helmholtz resonator). However, this is a challenge for low-frequency noise with long wavelengths due to the limited installation space. We therefore propose a resonator in which the back cavity is divided into two cavities by a flexible plate. The aim is to combine the damping mechanisms of the Helmholtz resonator with the material damping of the flexible plate. With carefully chosen parameters, this flexible plate resonates well below the Helmholtz frequency. We derived an analytic model based on waveguide theory to predict the impedance of the resonator concept. The Helmholtz equation was solved to (numerically) determine the scattering coefficients of a channel section in which one wall is lined with the predicted resonator impedance. The predicted dissipation agreed well with experimental data from measurements at the aeroacoustic wind tunnel DUCT-R.
There is much current interest in improving the efficiency gas turbines used in aircraft engines and thereby reducing their carbon emissions. The perforated combustor liners in gas turbine silencers absorb the sound associated with thermo-acoustic instabilities and thereby reduce them. A semi-empirical model has been developed to predict the absorption of perforated liners in the absence of bias flow as a function of frequency in terms of liner characteristic such as orifice diameter, thickness, spacing, orientation and perforation ratio and liner configuration which requires an additional model for the impedance of the cavity behind the liner. The expression used for impedance includes a cavity factor which corrects for the change in combustor liner diameter. Predictions of energy absorption coefficient spectrum are compared with data from measurements on several configurations of full-scale liners. It is found that predictions using the standard tangent term in the cavity impedance do not agree with data at frequencies below 500 Hz as well as predictions that use a cosine term instead. The resulting model, which is validated by the data comparisons, should be useful in optimising liner characteristics for manufacture.
The increasing interest in miniaturizing acoustic devices has made accurate and efficient models of acoustic viscous and thermal losses progressively more important. This is especially the case in micro-acoustic devices such as hearing aids, condenser microphones and MEMS devices. Using the full linearized Navier Stokes equations to numerically model losses comes at a high computational cost. An approximate boundary layer impedance boundary condition representing acoustic losses has therefore become popular due to its high computational efficiency. This is especially true in the context of optimization where an efficient numerical method is required due to the many repeated analyses needed. However, the boundary layer impedance is only valid in the computational region where boundary layers are non-overlapping. Applying the boundary layer impedance can therefore lead to poor optimization results or limit the possible design space if the optimization violates this limitation. Therefore, the benefit of losses in narrow regions cannot be exploited if the boundary layer impedance is used. This work investigates two shape optimization test cases for maximizing the absorption properties of Helmholtz-like geometries based on the Boundary Element Method. The test cases are used to compare and validate the boundary layer impedance against a full viscothermal implementation revealing the benefits of the boundary layer impedance but also its limitations in a structural optimization setting. Based on the numerical experiments it is recommend to avoid the use of the boundary layer impedance in cases where any theoretical boundary layer overlap exists or at least verify simulation and optimization results with a full-losses implementation.
Bei einer Vielzahl technischer Anwendungen von durchströmten Kanälen, wie z. B. Flugzeugtriebwerken, Lüftungsanlagen oder Abgasleitungen, werden akustische Wandauskleidungen, sogenannte Liner, als wirkungsvolles Mittel zur Lärmbekämpfung eingesetzt. Bei der Entwicklung solcher Liner ist es üblich, die Wand durch eine akustische Randbedingung, üblicherweise die Wandimpedanz, zu beschreiben. Die Schallabsorption eines beliebigen Liners kann dann leicht durch die reibungsfreien Schallfeldgleichungen samt Randbedingung vorhergesagt werden. Die- ser Zusammenhang verkompliziert sich allerdings erheblich, wenn die Wand wie in den o. g. technischen Anwendungen überströmt wird. Die im ruhenden Medium sehr kleinen Reibungseffekte, die sich in einer dünnen Wandgrenzschicht abspielen und von L. Cremer erstmals als Zusatz zur akustischen Randbedingung beschrieben worden sind, können drastisch anwachsen, da nun Zähigkeitskräfte entstehen, die das schallbedingt in die Wandöffnungen eindringende Medium abbremsen. Dieser Effekt wurde in den bisherigen Arbeiten zum großen Teil völlig vernachlässigt. Hier soll ein Einblick in die Etappen, Fortschritte und Schwierigkeiten bei der Modellierung dieses Problems gegeben werden.
ColESo is a set of subroutines that evaluate particular solutions of the Euler equations, the Navier – Stokes equations, and their linearized versions. These solutions were used by the author for the verification of high-order methods for the compressible flow simulation. It is convenient to use an exact solution when it is available for each point of the computational domain, accurate enough, and quick to evaluate. A solution given by a composition of elementary functions is usually a good choice, but the set of such solutions is too limited. ColESo adds to this set several solutions in integral form and provides their quick calculation with a good accuracy.
Program summary
Program Title: ColESo
CPC Library link to program files: https://doi.org/10.17632/z7mypchgfz.1
Developer's repository link: https://github.com/bahvalo/ColESo
Code Ocean capsule: https://codeocean.com/capsule/0365245
Licensing provisions: GPLv2.1
Programming language: C++ (C++03 standard)
Nature of problem: Verification of high-order methods for the simulation of compressible gas dynamics on curvilinear or unstructured grids.
Solution method: Change of variables, Fourier transformation, Gauss–Legendre and Gauss–Jacobi quadrature rules.
Additional comments including restrictions and unusual features: ColESo uses several special functions provided by Cephes Math Library and GNU LibQuadMath (for the quadruple precision). To define the Gauss – Jacobi quadrature rules, ColESo uses a code written by S. Elhay, J. Kautsky, J. Burkadrt. These libraries are open-access and distributable with compatible licenses.
Acoustic liners are an effective way to dampen aircraft noise. Conventional single-degree-of-freedom liners consist of a perforated facesheet backed with a honeycomb structure and a rigid end plate. Their damping excels near their resonance frequency which is anti-proportional to the cavity depth (λ/4-resonator) or the cavity volume (Helmholtz resonator). However, this is a challenge for low-frequency noise with long wavelengths due to the limited installation space. We therefore propose a resonator in which the back cavity is divided into two cavities by a flexible plate. The aim is to combine the damping mechanisms of the Helmholtz resonator with the material damping of the flexible plate. With carefully chosen parameters, this flexible plate resonates well below the Helmholtz frequency. We derived an analytic model based on waveguide theory to predict the impedance of the resonator concept. The Helmholtz equation was solved to (numerically) determine the scattering coefficients of a channel section in which one wall is lined with the predicted resonator impedance. The predicted dissipation agreed well with experimental data from measurements at the aero-acoustic wind tunnel DUCT-R.
This study proposes a level set-based topology optimization method for designing acoustic structures with viscous and thermal boundary layers in perspective. Acoustic waves propagating in a narrow channel are damped by viscous and thermal boundary layers. To estimate these viscothermal effects, we first introduce a sequential linearized Navier–Stokes model based on three weakly coupled Helmholtz equations for viscous, thermal, and acoustic pressure fields. Then, the optimization problem is formulated, where a sound-absorbing structure comprising air and an isothermal rigid medium is targeted, and its sound absorption coefficient is set as an objective function. The adjoint variable method and the concept of the topological derivative are used to approximately obtain design sensitivity. A level set-based topology optimization method is used to solve the optimization problem. Two-dimensional numerical examples are provided to support the validity of the proposed method. Moreover, the mechanisms that lead to the high absorption coefficient of the optimized design are discussed.
Aiming at the problem that the peak frequency of Fabry–Pérot (F–P) resonance metamaterials is shifted to high frequency due to the right-angle bend, we start from the mechanism of the frequency shift caused by the bend and propose a length correction theoretical model. Using the idea of equivalence, the effect of a corner is the first equivalent to the effective density. Then, the change in effective density is equivalent to the effective length, and the theoretical derivation is completed. This model can guide the length design of the F–P tube. Moreover, it can be used to predict the peak frequency of the F–P tube with a right-angle bend if its geometric dimensions are known. Through the analysis from theory, simulation, and experiment of two samples, the accuracy of the length correction theoretical model is verified. Additionally, by the power dissipation density and the dissipated energies, it is determined that the fundamental reason for the frequency shift is that the right-angle bend changes the distribution of power dissipation density in the tube. The work in this paper is of guiding significance for the frequency prediction and length design of F–P tubes.
This thesis addresses the mathematical aspects of thermoacoustics, a subfield within physical acoustics that comprises all effects in which heat conduction and entropy variations of the gaseous medium play a role. We focus specifically on the theoretical basis of two kinds of devices: the thermoacoustic prime mover, that uses heat to produce sound, and the thermoacoustic heat pump or refrigerator, that use sound to produce heating or cooling.
Two kinds of geometry are considered. The first one is the so-called standing-wave geometry that consists of a closed straight tube (the resonator) with a porous medium (the stack) placed inside. The second one is the so-called traveling-wave geometry that consists of a resonator attached to a looped tube with a porous medium (regenerator) placed inside. The stack and the regenerator differ in the sense that the regenerator uses thinner pores to ensure perfect thermal contact. The stack or regenerator can in principle have any arbitrary shape, but are modeled as a collecting of long narrow arbitrarily shaped pores. If the purpose of the device is to generate cooling or heating, then usually a speaker is attached to the regenerator to generate the necessary sound.
By means of a systematic approach based on small-parameter asymptotics and dimensional analysis, we have derived a general theory for the thermal and acoustic behavior in a pore. First a linear theory is derived, predicting the thermoacoustic behavior between two closely placed parallel plates. Then the theory is extended by considering arbitrarily shaped pores with the only restriction that the pore cross-sections vary slowly in longitudinal direction. Finally, the theory is completed by the inclusion of nonlinear second-order effects such as streaming, higher harmonics, and shock-waves. It is shown how the presence of any of these nonlinear phenomena (negatively) affects the performance of the device.
The final step in the analysis is the linking of the sound field in the stack or regenerator to that of the main tube. For the standing-wave device this is rather straightforward, but for the traveling-wave device all sorts of complications arise due to the complicated geometry. A numerical optimization routine has been developed that chooses the right geometry to ensure that all variables match continuously across every interface and the right flow behavior is attained at the position of the regenerator. Doing so, we can predict the flow behavior throughout the device and validate it against experimental data.
The numerical routine can be a valuable aid in the design of traveling-wave devices; by variation of the relevant problem parameters one can look for the optimal traveling-wave geometry in terms of power output or efficiency.
Millions of people are subjected to stress, particularly hearing losses due to the adverse impact of noise pollution. Noise mitigation demands inexpensive, efficient and feasible solutions to be developed in habitable spaces including long duration transport systems. The comprehensive review presented here focuses on different noise reducing materials being utilized presently, including recent developmental efforts towards noise mitigation. Sound absorption characterization and associated material parameters are presented initially. The material parameters affecting sound absorption are listed and defined subsequently. A summary about the foaming agents being widely utilized is presented next. The different materials like foams (open and closed cell), metamaterials, sandwiches, and microperforated panels are reviewed in detail before introducing the simulation studies of acoustic wave propagation in cellular structures. The applications are summarized before possible future trends and challenges in developing advanced smart, sustainable noise mitigating material.
The aim of this study is to investigate the role of base zone on rug sound absorption and to predict the acoustic behavior of typical Iranian rugs using a mathematical model. For this purpose, four rug samples (1.5 cm thickness) were produced with different base structures, so that the base warp yarns are divided into two equal groups and arranged from each other at a different adjustable distance (0.0, 0.1, 0.2 and 0.3 cm). Sound absorption measurements were carried out using an impedance tube. Then, in order to focus on the base zone and to separate the piles contribution from the global rug sound absorption, the pile yarns were shaved from all samples and the non-piled samples were measured again. The five-parameter model of Johnson-Champoux-Allard (JCA model) was fitted to experimental data, assuming the rug is a two-layer (pile + base zone) porous absorber. The agreement between the model and the experiments is satisfactory. This optimized model can be used to predict the sound absorption coefficient of Iranian rugs. The results indicate that the rug base plays an important role in the overall rug sound absorption and that the air gap between the two groups of warp yarn improves the rug sound absorption.