Viscothermal Losses Research Articles

Numerical modeling of acoustic visco-vhermal losses: simplified implementations and their limitations

Several numerical adaptations have been proposed to account for viscous and thermal losses of acoustic waves. This is a physical effect that is mostly relevant in a region very close to boundaries, the so-called boundary layers, in the micrometer range. It is therefore relevant when modeling small devices such as acoustic transducers and acoustic metamaterials. An often used modeling technique collapses the viscous and thermal losses into a boundary layer impedance (BLI), which is used in a calculation where the regular wave equation with no losses is discretized. However, this method has two shortcomings: i) the boundary layers may not overlap, and ii) it assumes flat surfaces. The BLI can be used either with the Finite Element Method (FEM) or the Boundary Element Method (BEM). BEM and FEM implementations with visco-thermal losses having no such restrictions also exist, at the cost of a heavier computational burden. This contribution discusses the shortcomings and advantages of the simulations using BLI as compared with full implementations and points to possible ways to overcome them. The BEM is employed to illustrate the analysis.

A comparison of modeled and measured impedance of brass instruments and mouthpieces

The impedance of a brass instrument has an important influence on its playability and sound timbre. The geometry of the mouthpiece has various features, such as the cup volume and shape, opening diameter, and length, that determine the characteristics of the overall impedance of the instrument-mouthpiece combination. Brass instruments, and especially mouthpieces, are designed for specific purposes, and horns or mouthpieces are chosen depending on the musical requirements. In order to investigate the relationship between the physical parameters and the impedances of instruments and mouthpieces, they have been modeled with transfer matrix and finite element model techniques, and the results are compared with impedance measurements of instruments, mouthpieces, and combinations of instruments and mouthpieces. Trumpets, flugelhorns, (French) horns, trombones, and the corresponding mouthpieces have been used. A detailed analysis of the estimation of the viscothermal losses has been performed, as the loss estimation in the narrow throat of the mouthpiece and in the flaring part of the brass instrument bell departs from the ordinary transfer matrix calculations. The effect of varying the physical parameters of mouthpieces and instruments is investigated by means of impedance considerations and sound synthesis, and the resulting influences on intonation, playability, and timbre are presented.

Numerical modeling of a saxophone and comparison with experimental measurements

Saxophones making has long relied on craftsmanship, associated with an empirical knowledge of their acoustic functioning. The design process is now mainly based on the study of the input impedance, accessible experimentally or computationally. The numerical approaches are currently limited either by the accuracy (analytical models) or by the computation time (numerical resolution of PDEs). The challenge is to propose a numerical method combining both, to access the acoustic pressure and velocity fields in the instrument. Our objective was to develop a high-performance parallel numerical tool based on FEM to model accurately (a few cents on the resonances) the acoustic behavior of the resonator under playing conditions. The computation time was reduced by an order of magnitude compared to the brute force approach, using different modeling strategies specific to wind instruments: First, the modeling of the visco-thermal losses at the walls; then, the reduction of the size of the linear systems and finally, the reuse and model reduction to limit the effect of the multiple resolutions imposed by the wide frequency range of interest. The model is validated by comparison with experimental measurements. An impedance measurement system allowing acquisitions under controlled flow has been developed to reproduce realistic playing conditions.

Near-perfect sound absorption using hybrid resonance between subwavelength Helmholtz resonators with non-uniformly partitioned cavities

We present near-perfect sound absorption using a metasurface composed of meta-atoms (MAs) which are subwavelength Helmholtz resonators (HRs) with cavities non-uniformly partitioned by membranes. By embedding the membranes at different horizontal locations in the cavities, we break geometrical symmetry between the MAs so as to derive hybrid resonance between the MAs at our target frequency. The resonance frequency of each MA is determined by delicately adjusting the locations of the membranes, resulting in perfect absorption at the target frequency which is different from the resonance frequencies of MAs. The metasurface is designed to satisfy impedance matching conditions with air at one or more target frequencies with the aid of a theoretical model for frequency-dependent effective acoustic impedance. The theoretical model is established with physical reality by considering the higher-order eigenmodes of the membrane, the visco-thermal losses in narrow orifices, and the end corrections of the subwavelength HR. The designed metasurface is fabricated and its absorption performance is verified experimentally in an impedance tube. Near-perfect absorption of sound is achieved at the target frequency of 500 Hz, which is 12.3% lower than that of near-perfect absorption by previous metasurfaces inducing hybrid resonance between HRs without membranes.

Complex-valued impedance tiles to reduce noise emanating through openings in mechanical systems

We present acoustic metasurfaces to achieve high noise reduction for an open structure, which refers to a structure with openings, by manipulating complex impedances along unit cells called impedance tiles. These acoustic metasurfaces are applied to inner walls of the open structure to suppress noise emanating through openings. Each impedance tile is made up of subwavelength Helmholtz resonators (HRs), and its effective impedance is tuned to minimize the noise level outside the open structure at an arbitrary target frequency by adjusting the geometrical parameters of those HRs. Rather than absorbing incident waves, the proposed metasurfaces significantly affect the magnitude and phase of scattered waves, causing the sound energy to be trapped near their surfaces and dissipated by visco-thermal losses in narrow orifices of the impedance tiles. As a result, the metasurfaces enable more effective noise reduction for the open structure compared to impedance-matching conditions, which absorb sound waves impinging on the surface. Experimental results show that the sound pressure level reduction exceeding 12.6 dB is verified for a single source at a target frequency of 360 Hz with λ/31-thick metasurfaces fabricated via three-dimensional printing. Furthermore, we demonstrate that the proposed metasurfaces are also useful in reducing noise when the number of target frequencies is more than one or when multiple sources are distributed in different locations, thereby offering a potential solution for noise reduction in a wide range of mechanical systems with open structures.

Impact of viscothermal loss on modulation instability and rogue waves in left-handed nonlinear diffractive acoustic transmission line metamaterials

In this study, the transmission line approach is used to describe the studied acoustic metamaterial model. Through Kirchoff’s pressure and volume-velocity laws and using multiple scales method, nonlinear coupled Schrödinger equations are obtained. Then, the amplitude disturbance method is applied to these equations to obtain and plot the modulational instability gain curves. Analytically, the impact of viscothermal loss on the modulational instability gain is studied. The similarity technique is used to derive integrable Manakov’s equations. First and second-order rational rogue wavelike solutions of coupled nonlinear Schrödinger are deduced. The results indicate that the modulational instability gain and Rogue wave intensities depend on the viscothermal parameter. This parameter can be considered in the design of nonlinear acoustic metamaterials to minimize the damage caused by the dynamics of freak waves.

Topology optimization of a waveguide acoustic black hole for enhanced wave focusing.

The waveguide acoustic black hole (WAB) effect is a promising approach for controlling wave propagation in various applications, especially for attenuating sound waves. While the wave-focusing effect of structural acoustic black holes has found widespread applications, the classical ribbed design of waveguide acoustic black holes (WABs) acts more as a resonance absorber than a true wave-focusing device. In this study, we employ a computational design optimization approach to achieve a conceptual design of a WAB with enhanced wave-focusing properties. We investigate the influence of viscothermal boundary losses on the optimization process by formulating two distinct cases: one neglecting viscothermal losses and the other incorporating these losses using a recently developed material distribution topology optimization technique. We compare the performance of optimized designs in these two cases with that of the classical ribbed design. Simulations using linearized compressible Navier-Stokes equations are conducted to evaluate the wave-focusing performance of these different designs. The results reveal that considering viscothermal losses in the design optimization process leads to superior wave-focusing capabilities, highlighting the significance of incorporating these losses in the design approach. This study contributes to the advancement of WAB design and opens up new possibilities for its applications in various fields.

Revising the Boundary Element Method for Thermoviscous Acoustics: An Iterative Approach via Schur Complement

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.

Complex frequency analysis and source of losses in rectangular sonic black holes

The amount and practical source of losses required by rectangular sonic black holes in air to effectively absorb low-frequency sound are analyzed numerically and experimentally. In the sole presence of viscothermal losses, only high-order (high frequency) Fabry–Perrot resonances are likely to be critically coupled in realistic rectangular sonic black holes. This results in sharp absorption peaks that do not reach unity at low frequencies, because the quality factors of the associated resonances are high. To avoid these drawbacks, slits of rectangular sonic black holes are partially filled with porous materials so that a profile of porous filled slits is superimposed on the sonic black hole profile itself. The improvement in the absorption coefficient is significant, particularly at frequencies below the viscous/inertial transition frequency of the porous material. This transition frequency is assumed to be the limit of possible perfect absorption of the bulk porous material layer. The numerical results are supported by experimental results, which also show that the vibrations of the plates forming the acoustic black hole must be considered with caution.

Noise reduction in an open structure by tailoring complex impedances on acoustic metasurfaces

In this study, we design acoustic metasurfaces to the inner surfaces of the open structure to mitigate noise emanating through the opening by manipulating complex impedances along unit cells called impedance tiles. The impedance tile is comprised of subwavelength Helmholtz resonators, which have a theoretical model that accurately predicts the effective impedance, considering visco-thermal losses in narrow orifices. By adjusting thegeometrical parameters of these resonators, the surface impedance of the impedance tiles can be finely tuned to minimize noise escaping from the open structure at any desired target frequency. To evaluate the noise reduction performance, the designed metasurfaces were fabricated via 3D printing, and the sound pressure levels (SPLs) were measured before and after applying the metasurfaces inside the open structure. The experimental results show that the λ/31-thick metasurfaces achieve SPL reduction exceeding 12.6 dB at the target frequency of 360 Hz.

Acoustic metasurface for perfect absorption using helmholtz resonators with non-uniformly partitioned cavities by membranes

We propose an acoustic metasurface consisting of sub-wavelength Helmholtz resonators whose cavities are non-uniformly partitioned by membranes, aiming at perfect absorption of low-frequency sound with the aid of hybrid resonance. Compared to the existing unit cells based on Helmholtz resonators, the proposed ones with embedded membranes have lower resonance frequencies, which implies that it is possible to achieve perfect absorption of sound by using a thinner metasurface. This study provides a theoretical model for fast and accurate design considering the visco-thermal losses in narrow orifices and the higher-order modes of the membrane. By optimizing geometrical parameters and locations of the embedded membranes, we design the metasurface for perfect absorption based on the theoretical model. Experimental validation is performed in impedance tube via fabrication.

Broadband sound attenuation and absorption by duct silencers based on the acoustic black hole effect: Simulations and experiments

Wideband reduction of both sound reflection and transmission using compact non-intrusive liners remains a challenging issue for duct noise control applications. This work focuses on the acoustical performance limitations of cylindrical silencers made up of annular ring resonators with axial gradient of their cavity depths. It is shown from theoretical, numerical and experimental studies that a sub-wavelength silencer with optimized acoustic black hole (ABH) properties can be designed through which incident sound waves are retarded and fully dissipated within the activated resonant cavities. Such ABH-type silencer requires a suitable interplay between the wall impedance axial variations and the visco-thermal losses within the cavities. Key parameters that influence this balance are the axial growth of cavity depths and the wall porosity. A causal-based criterion has been proposed that maximizes the total integrated dissipated power, thereby leading to the optimal value of the wall porosity and to the ultimate bandwidth-to-length ratio that can be achieved by an ABH-type silencer given a target dissipation value. A space-frequency region has been numerically and experimentally identified over which the ABH effect is prominent. The causal-based criterion has been extended to account for the low-frequency ultimate performance of ABH silencers with coiled cavities and for the effects of a low-speed grazing flow.

Topology optimization of wave-focusing waveguide acoustic black holes

The acoustic black hole, as first proposed by Mironov for beams and plates, provides broadband wave focusing, exploitable for energy harvesting, for weak signal sensing, and as a strategy to minimize the amount of damping material needed for efficient, broadband damping. There have been a few suggestions aimed at providing an analogous device for acoustics in air. However, although these waveguide devices may work fine as broadband absorbers, their operational mechanism is decidedly different from their structural counterparts. Instead of providing wave focusing, all published devices seem to rely on the creation of resonances occurring increasingly closer to the waveguide entrance for increasing frequency. To explore a much larger class of possible designs than previously examined, we here apply a gradient-based material distribution topology optimization method. The optimization objective is a broadband maximization of power in a small region towards the end of the waveguide. We demonstrate that with this tool, it is indeed possible to design broadband true wave-focusing waveguides. These are geometrically much more complex than the previously proposed ones, a complexity that nevertheless efficiently can be handled by a recently developed gradient-based optimization method, able to accommodate also visco-thermal boundary-layer losses.

Description of sound absorption by a flat resonator stacking metamaterial with double porosity model

Acoustic metamaterials can be designed by inserting along the path of a sound wave periodically spaced side resonators. An example of efficient design was recently proposed consisting of a perforated stacking of flat annular cavities (the pancake resonator), the perforation allowing the propagation of sound waves. The pancake resonator is used in absorber mode and the theoretical description of sound absorption can be achieved with the help of the theory of sound propagation in fluid saturated porous media in which two porosities are considered: the main porosity associated with the perforation and a porosity associated with the flat cavity volumes. Considering a perforation diameter and flat cavity thickness ranging from submillimetric values to a few millimeters allows a wide range of material permeabilities and permeability contrasts between main pore and stacking of cavities. The relatively small values of diameter and cavity thickness also results in the existence of viscous and thermal boundary layers in the main pore (the perforation) and in the flat cavities. This metamaterial makes simultaneous use of viscothermal losses and periodicity in order to achieve low frequency sound absorption for an overall small absorber thickness. Experimental results are also presented for the validation of the model.

Low frequency attenuation of acoustic waves using sound-soft scatterers

The attenuation of acoustic waves by silencers is typically achieved through the employment of rigidly backed cavities, connected to a main waveguide by a perforated panel. This invokes a resonant response at frequencies determined by the dimensions of the perforation and the rigidly backed cavity. A limiting factor in this approach is that to achieve low frequency attenuation, either large cavity depths are required, which is often impractical, or narrow neck regions need to be used, where performance is limited due to the viscothermal losses. Within this paper, a non-rigidly backed perforated pipe is presented where the pressure release condition at each perforation creates an acoustic sound-soft boundary condition. By the introduction of sound-soft boundary conditions, a low frequency band gap is created as the first order mode is shifted to a non-zero value, the frequency of which is determined by the dimensions and separation of the perforations. Experimental results displaying this band gap are presented along with numerical models verifying the phenomenon and an ideal analytical model based upon the transfer matrix method.

Amplitude-dependent modal coefficients accounting for localized nonlinear losses in a time-domain integration of woodwind model

This article develops the design of a sound synthesis model of a woodwind instrument by modal decomposition of the input impedance, taking into account viscothermal losses as well as localized nonlinear losses at the end of the resonator. This formalism has already been applied by Diab et al. [Journal of Sound and Vibration 528 (2022) 116892] to the study of forced systems. It is now implemented for self-oscillating systems. The employed method extends the definition of the input impedance to the nonlinear domain by adding a dependance on the RMS acoustic velocity at a geometric discontinuity. The poles and residues resulting from the modal decomposition are fitted as a function of this velocity. Thus, the pressure-flow relation defined by the resonator is completed by new equations which account for the dependence with the velocity at the end of the tube. To assess the ability of the model to reproduce a real phenomenon, comparisons with the experimental results of Atig et al. [PhD thesis, Université du Maine (2004)] and Dalmont and Frappé [Journal of the Acoustical Society of America 122(2) (2007) 1173–1179] were carried out. Simulations show that the model reproduces these experimental results qualitatively and quantitatively.

On the validity of numerical models for viscothermal losses in structural optimization for micro-acoustics

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.

How the waveguide acoustic black hole works: A study of possible damping mechanisms.

The acoustic black hole (ABH) effect in waveguides is studied using frequency-domain finite element simulations of a cylindrical waveguide with an embedded ABH termination composed of retarding rings. This design is adopted from an experimental study in the literature, which surprisingly showed, contrary to the structural counterpart, that the addition of damping material to the end of the waveguide does not significantly reduce the reflection coefficient any further. To investigate this unexpected behavior, we model different damping mechanisms involved in the attenuation of sound waves in this setup. A sequence of computed pressure distributions indicates occurrences of frequency-dependent resonances in the device. The axial position of the cavity where the resonance occurs can be predicted by a more elaborate wall admittance model than the one that was initially used to study and design ABHs. The results of our simulations show that at higher frequencies, the visco-thermal losses and the damping material added to the end of the setup do not contribute significantly to the performance of the device. Our results suggest that the primary source of damping, responsible for the low reflection coefficients at higher frequencies, is local absorption effects at the outer surface of the cylinder.

Realization of an acoustic metalens exhibiting broadband high transmission

Acoustic metamaterials and acoustic metasurfaces have recently been exploited to design lenses that have extraordinary sound focusing capabilities. Among these so-called metalenses, many thin and planar ones are composed of space-coiled structures, which are widely employed structures as unit cells for the lenses. Although it is known that visco-thermal losses in narrow channels and elastic deformation of thin structures severely affect the power transmission coefficients and refractive indices of the structures, the losses are often neglected and the structures are assumed to be rigid. In this study, we realize a planar acoustic lens with broadband high transmission by taking into account both the visco-thermal losses and elastic deformation in the design procedure. Experiments verify that the lens has a broad bandwidth of 2.5–5.5 kHz and focuses sound waves into a narrow region with a full width at half maximum of 0.56λ within the bandwidth, amplifying the intensity by a factor of almost six at the center frequency of 4 kHz.

Influence of Higher Order Viscous and Thermal Effects on an Ultrasonic Wave Reflected from the First Interface of a Porous Material.

Ultrasound propagation in porous materials involves some higher order physical parameters whose importance depends on the acoustic characteristics of the materials. This article concerns the study of the influence of two parameters recently introduced, namely, the viscous and thermal surfaces, on the acoustic wave reflected by the first interface of a porous material with a rigid structure. These two parameters describe the fluid/structure interactions in a porous medium during the propagation of the acoustic wave in the high-frequency regime. Both viscous and thermal surfaces are involved in Laurent expansion, which is limited to the dynamic tortuosity and compressibility to a higher order and corrects the visco-thermal losses. A sensitivity study is performed on the modulus of the reflection coefficient at the first interface as a function of frequency and on the waveforms reflected by the porous material in the time domain. The results of this study show that highly absorbent porous materials are the most sensitive to viscous and thermal surfaces, which makes the consideration of these two parameters paramount for the characterization of highly absorbent porous materials using the waves reflected from the first interface.