Low Helmholtz Research Articles

Radiation efficiency varying equivalent radiated power.

In this paper, an improved version of the classical equivalent radiated power (ERP) approximation is proposed based on principled physical arguments. A geometry-, frequency-, and vibration pattern-dependent approximation of radiation efficiency is developed and used as a corrective factor for the classical ERP approximation. The proposed method called "radiation efficiency varying equivalent radiated power" (revERP), is shown to greatly improve the accuracy of classical ERP at low Helmholtz numbers, while attaining the accuracy of classical ERP at high Helmholtz numbers. The revERP method is exemplified on several components of industrial complexity, and shows a valuable alternative to full vibro-acoustic finite element analysis for early-stage assessment of acoustic performance.

Resolvent model for aeroacoustics of trailing edge noise

This study presents a physics-based, low-order model for the trailing edge (TE) noise generated by an airfoil at low angle of attack. The approach employs incompressible resolvent analysis of the mean flow to extract relevant spanwise-coherent structures in the transitional boundary layer and near wake. These structures are integrated into Curle’s solution to Lighthill’s acoustic analogy to obtain the scattered acoustic field. The model has the advantage of predicting surface pressure fluctuations from first principles, avoiding reliance on empirical models, but with a free amplitude set by simulation data. The model is evaluated for the transitional flow (Re=5e4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Re} = 5e4$$\\end{document}) around a NACA0012 airfoil at 3 deg angle of attack, which features TE noise with multiple tones. The mean flow is obtained from a compressible large eddy simulation, and spectral proper orthogonal decomposition (SPOD) is employed to extract the main hydrodynamic and acoustic features of the flow. Comparisons between resolvent and SPOD demonstrate that the physics-based model accurately captures the leading coherent structures at the main tones’ frequencies, resulting in a good agreement of the reconstructed acoustic power with that of the SPOD (within 4 dB). Discrepancies are observed at high frequencies, likely linked to nonlinearities that are not considered in the resolvent analysis. The model’s directivity aligns well with the data at low Helmholtz numbers, but it fails at high frequencies where the back-scattered pressure plays a significant role in directivity. This modeling approach opens the way for efficient optimization of airfoil shapes in combination with low-fidelity mean flow solvers to reduce TE noise.Graphical abstract

Near-field modeling of the head-related transfer function by using the dummy head sound source

Various approaches based on direct measurement have been used to obtain the precise headrelated transfer function (HRTF). However, several problems persist, such as scattering of sound source and distance effect, which are related to the measurement configuration. In this work, the modified dummy head having two sound sources located at the ear positions is applied as the sound source reciprocally to overcome the trouble. By using the near-field acoustic holography, the sound sources located in the ear holes are reconstructed from the measured data in the near-field hologram. One can predict the acoustic transfer function between the ear holes and the field positions at any point using the recovered source information described in spherical harmonics. The obtained transfer function is valid for predicting the binaural response from any sound source position. This configuration is possible to measure the near-field response with the relatively small scattering effect because of the low Helmholtz number of microphones. The far-field reconstruction of HRTF is also possible with the near-field measurement data using acoustical holography. The proposed method is demonstrated with the modified B&K HATS system, in which the microphones in the ear holes are replaced by pipe sources connected to the loudspeaker.

Tailored acoustic metamaterials. Part II. Extremely thick-walled Helmholtz resonator arrays.

We present a solution method which combines the technique of matched asymptotic expansions with the method of multipole expansions to determine the band structure of cylindrical Helmholtz resonator arrays in two dimensions. The resonator geometry is considered in the limit as the wall thickness becomes very large compared with the aperture width (the extremely thick-walled limit). In this regime, the existing treatment in Part I (Smith & Abrahams, 2022 Tailored acoustic metamaterials. Part I. Thin- and thick-walled Helmholtz resonator arrays), with updated parameters, is found to return spurious spectral behaviour. We derive a regularized system which overcomes this issue and also derive compact asymptotic descriptions for the low-frequency dispersion equation in this setting. We find that the matched-asymptotic system is able to recover the first few bands over the entire Brillouin zone with ease, when suitably truncated. A homogenization treatment is outlined for describing the effective bulk modulus and effective density tensor of the resonator array for all wall thicknesses. We demonstrate that extremely thick-walled resonators are able to achieve exceptionally low Helmholtz resonant frequencies, and present closed-form expressions for determining these explicitly. We anticipate that the analytical expressions and the formulation outlined here may prove useful in designing metamaterials for industrial and other applications.

Green's function of the rigid half cylinder with soft free surface for application to ship noise

The present study addresses an experimental and analytical investigation of the scattering of the tonal noise of low-speed propeller by a half-cylinder and a soft free surface. The interest is marine propeller noise of surface ships at the first harmonics of the blade passing frequency. Indeed, installed marine propellers correspond to very compact configurations. Both sizes blade tip radius and the distance to the ship hull are small compared to the wavelength. Thus, in view of the extremely low Helmholtz numbers, some amplification due to an asymptotic scattering regime is expected. Measurements made in air in similar conditions, in anechoic chamber confirmed that noise from small-scale isolated propeller is dramatically intensified near a half cylinder with reflecting plane. The problem is formulated analytically by using the formalism of source-modes near a scattering half cylinder combined with the effect of the free surface in two dimensions. In this configuration, the diffraction is found to induce an amplification but with a different wave pattern to what the diffraction is with a cylinder only. The results show that the installation effect is crucial for the analysis of marine propeller noise at the very low frequencies.

Experimental evaluation on acoustic impedance and sound absorption performances of porous foams with additives with Helmholtz number

Porous foams may be classified into various sub-categories basing on their different microstructures nature. Each type of foams is involved with different acoustic damping capabilities and physical properties. In this work, we experimentally study and compare the acoustic damping performances of different types of porous foams with different densities and constitutions. A sample from each type of porous foams is tested in an acoustic impedance tube. Here the sound absorption coefficient, the reflection coefficient and the acoustic impedance are defined to quantify the noise damping performances of these samples over a broad Helmholtz number He range. The effects of 1) the porosity, 2) the materials density and 3) the thickness of the materials are examined one at a time. For this, a total of 13 samples are experimentally tested and comparison is then made. It is found that increasing the material density does not lead to an improvement on noise damping at Helmholtz number He below 0.17. In addition, increasing the thickness of the same samples gives rise to dramatic different noise damping behaviours at low Helmholtz number range (<0.35). However, the sound absorption coefficients of different thickness samples at higher He follow the similar trends. The thickness is shown to be affect a little on the noise damping performances. Ceramic fibre sample is shown to involve with a more broad effective He range than that of the samples of the normal concrete and polyurethane foam. To further enhance the noise damping performances, triphenylene is introduced an additive and combined into the pure silicone material. The additives effect is then evaluated. It is found that the sound absorption coefficient is increased dramatically, as the triphenylene is applied. The maximum absorption coefficient is increased from approximately 0.2 to 0.87. In general, the present work sheds light on how to optimize the acoustic damping performances of porous foams.

Combined microphone array for precise localization of sound source using the acoustic intensimetry

Various applications require a compact system for acoustic source localization. Intensimetry methods based on three-dimensional (3D) acoustic intensity are favorable for miniaturizing localization array systems because of the small errors at low Helmholtz numbers. However, in practice, conventional 3D intensimetry has not performed well at high Helmholtz numbers because of large spatial and spectral bias errors. In this work, the spatial bias error of 3D acoustic intensimetry is analyzed with respect to the irregularities in directivity. The magnitude of spatial bias error is found to be proportional to the gradient of the array directional response. Therefore, the maximum spatial bias error occurs in the vicinity of the direction corresponding to the maximum or minimum value of the array directional response. For smoothing the spatial array directional response, a combination of array modules with mutual symmetry in directivity is proposed as a compensation method. A stellated octahedral shape, and its truncated form, can regularize the array directional response while maintaining a small number of microphones. Source localization is tested using single and combined modules. The root-mean-square error of the bearing angle when using the combined modules is 1.0° in the Helmholtz number range of 2.0–3.0, which is far smaller than the error of 10.1° when using the single modules. Moreover, the truncated stellated octahedral probe, which is composed of a combination of tetrahedral and hexahedral arrays, is the most efficient array configuration for source localization and achieves high precision at high Helmholtz numbers with only five microphones.

Characterization and suppression of the hydrodynamic instability in the time domain for acoustic propagation in a lined flow duct

The gradient term suppression (GTS) method for removing the hydrodynamic instability appearing in the time-domain solutions of the linearized Euler equations (LEE) along a lined flow duct is assessed. For this, the characterization of a convective instability in the time domain, with the aid of a complementary modal analysis, is first presented. The effect of the mesh size and spatial filtering on the instability is investigated. In particular, a convergence of the instability in the time domain is achieved for a small enough grid size. The consequence of suppressing the mean flow gradient term on the modes is then investigated. It is shown that the unstable modes are indeed removed, but also that acoustic modes are significantly modified, especially for low Helmholtz numbers. The GTS method is finally applied to the NASA grazing impedance tube benchmark. It is found that tuning the weight of the mean flow gradient term within the LEE can be effective for suppressing the instability while conserving a reasonable accuracy of the acoustic component.

Analysis on surface mobility of an infinite beam-stiffened structure

Vibration transmission analysis is significant for the functionality, safety and amenity of many engineering structures. The present research aims to reveal the importance of applying the concept of surface mobility in predicting the vibration transmission of a beam-stiffened plate, which is one of the simplest but most important structural elements in engineering practice. First, a theoretical framework is established to analyze the surface mobility of an infinite beam-stiffened plate excited by a uniform contact force and then validated. With this scheme, one can easily reveal that the stiffened beam can reduce the vibration transmission of the plate with low Helmholtz number (<5) but may amplify it when that number increases. The present investigation also reveals that crossed and infinite equidistant uniformly-spaced parallel stiffened beams can reduce more vibration of the plate than parallel stiffened beams. Finally, analyses are conducted to determine when engineers should employ surface mobility, instead of point mobility, to describe the vibration transmission ability of beam-stiffened plates. Comparing with parallel layout of beams, the crossed layout of beams can decrease the difference between the point and surface mobility of the plate. Differences between the surface and point mobility of the plate become less obvious with the increase of the stiffness and the mass of the beam but it does not always decline with the increase of the number of beams.

Geometric shapes effect of in-duct perforated orifices on aeroacoustics damping performances at low Helmholtz and Strouhal number.

In this work, experimental studies are conducted to measure the aeroacoustics damping performances of 11 in-duct perforated plates in a cold-flow pipe with a variable Mach number. These in-duct plates have the same porosities but different number N and geometric shaped orifices. Here six shapes are considered, i.e., (1) triangle, (2) square, (3) pentagon, (4) hexagon, (5) star, and (6) circle. It is shown that the orifice shape has little influence on power absorption Δ and reflection coefficient R at a lower Helmholtz number He ≤ 0.0903. However, as He is increased, the in-duct plate with a star-shaped orifice is shown to be with much lower Δ in comparison with that of other plates with different shape orifices. In addition, the perforated orifice with the same shape and porosity but a larger N is shown to be associated with 20% more power absorption at approximately He = 0.1244. Δmax is observed to be approximately 85% at about He = 0.0244, as Ma≈0.029. To gain more insights, the quasi-steady model is applied, depending on the Strouhal number Sr. The transition from quasi-steady flow behaviors to unsteady behaviors occurs at approximately Sr = 0.45. The measured minimum reflection coefficient Rmin occurs at Ma ≈ 0.024. This experimental finding is consistent with the quasi-steady prediction.

Optimum Impedance in the Presence of an Inviscid Sheared Flow

In recent years, much effort has been devoted to find the “optimum impedance” (i.e., the impedance that results in the maximum modal decay rate in flow duct acoustics for a given frequency, Mach number, and azimuthal mode order). Although such analysis can be carried out by means of numerical simulations, analytical expressions can also be derived to predict the optimum impedance. Previous works have been concerned with the optimum impedance of higher-order modes in rectangular ducts with uniform flow. In this work, the analysis is expanded to circular ducts for both uniform and sheared inviscid flows. Focus is given to typical operating conditions found in turbofan engine intakes and vehicle exhaust systems. It is shown that, in certain conditions, the optimum impedance is affected even by the presence of a small boundary-layer thickness. It is also noted that, for low Helmholtz numbers, the optimum impedance may have a negative resistance.

Estimation of the source distance using the multiple three-dimensional acoustic intensimetry

A 3D acoustic intensimetry (AI) in tetrahedral configuration can be used for the source localization. Such a 3D AI module is compact, but it can also yield a precise bearing angle for the low Helmholtz numbers if some bias errors are compensated. In this study, besides the bearing angle, the source distance is estimated by using the multiple 3D AI probes with different acoustic centers. Because each AI module indicates a vector from the acoustic center to the source direction, the nearest point between any pair of line vectors can approximate the source position. The source position can be estimated by minimizing the least square error of the positions determined by multiple intensity vectors. Such idea is preliminarily tested with four 3D AI modules, of which each module is composed of 4 MEMS microphones separated by 30 mm. Four 3D AI modules are also arranged in a tetrahedron layout, and the source frequency range is 0.5–2 kHz. The distance between modules is varied for 150–250 mm range, and the source distance for 1–4 m range. Current test results reveal that the estimation error is less than about 2, 5, and 7% when the elevation angle is within 0–30°, 30–60°, 60–90°, respectively.

Experimental and Numerical Investigation into the Propagation of Entropy Waves

Entropy waves are an important source of indirect combustion noise and potentially contribute to the generation of thermoacoustic instabilities in gas-turbine combustors. Entropy fluctuations generated by unsteady combustion are known to disperse and diffuse as they convect toward the combustor exit. In this work, the propagation of entropy waves is investigated by means of experiments in a newly developed entropy rig and numerical simulations based on the large-eddy simulation approach. Both experimental and numerical results demonstrate that the amplitude of entropy fluctuations decays as a function of wave parameters and propagation distance, and it scales well with a local Helmholtz number . A new theoretical model for the computation of the entropy transfer function suitable for inclusion in low-order models for combustion instabilities is proposed. Assessment against numerical and experimental results shows the capability of the model to give a proper representation of the decay of entropy waves in terms of both magnitude and phase of the entropy transfer function. Furthermore, by comparison with the large-eddy simulation results, it is shown that, at low Helmholtz numbers, the contribution of the differential convection to the decay of entropy waves is dominant; whereas for high values of the Helmholtz number, the turbulent mixing and diffusion also become important.

Modeling the damping properties of perforated screens traversed by a bias flow and backed by a cavity at low Strouhal number

Acoustic absorption properties of perforated screens traversed by a bias flow and backed by a resonant cavity are examined at a new absorption regime. The reflection coefficient of these dampers can be canceled at specific frequencies for low Strouhal numbers St ⪡ 1 and Helmholtz numbers of order unity He ∼ 1 . This regime differs from the classical one where resonance is sought at low Helmholtz numbers He ⪡ 1 and large Strouhal numbers St ⪢ 1 . A theoretical analysis is carried out and analytical expressions are derived to determine conditions for optimal absorption in this regime. It is shown that the optimal bias flow velocity and back cavity length can be fixed separately. The former is determined by the plate porosity and the latter controls the peak absorption frequency. This operating regime at low Strouhal numbers enables to increase sound absorption at low frequencies over a larger frequency bandwidth around the peak absorption frequency than the one obtained with a damping system working at high Strouhal numbers. This is confirmed by experiments carried out on a set of dampers in the frequency range 100–1000 Hz. Predictions of the reflection coefficient, including effects of bias flow velocity and cavity depth, show good agreement with experimental data for small pressure perturbation levels. These elements can contribute to the development of robust dampers to control low frequency thermoacoustic instabilities in gas turbines and jet engines.

Mechanism of F− Elimination from Zeolitic D4R Units: A Periodic B3LYP Study on the Octadecasil Zeolite

The mechanism of elimination of F− trapped inside the double four-membered ring (D4R) cages of octadecasil has been determined by means of periodic calculations at the B3LYP/VDZP//VTZP level of theory with the CRYSTAL06 program. Finite temperature and entropic effects are taken into account to study the efficiency of the elimination reaction, which is exothermic, requires acid media, and is catalyzed by cationic polar molecules with accessible protons, from which a hydronium ion has been considered in the present study. The mechanism consists of three steps that schematically correspond to hydrolysis of one SiOSi bridge in the D4R cage, extraction of the F− from the unit, and recondensation of the bridge with formation of slightly adsorbed HF·H2O in the large cavity that can be easily eliminated by diffusion. The limiting step is the first one and involves a quite low Helmholtz free energy barrier even at 100 °C (51.6 kJ mol−1) that slowly decreases with increasing temperature.

A boundary element extension of Curle's analogy for non-compact geometries at low-Mach numbers

The aeroacoustic analogy derived by Curle for the prediction of the sound resulting from turbulence/body interactions has proved quite powerful for low Helmholtz numbers, i.e. when the interaction region is acoustically compact. In such case, incompressible flow modeling can be used to obtain the source field used as the input of the analogy. It was, however, shown in a previous paper that Curle's analogy can yield erroneous results for non-compact cases, when an incompressible flow model is adopted. Yet, at low-Mach numbers, incompressible flow modeling can be substantially more efficient than compressible simulations, due to the stiffness issues faced by the latter. The present work focuses on the derivation of a method combining Curle's analogy with a boundary element method (BEM), in order to compensate for the weaknesses of the traditional approach at high Helmholtz numbers. The validation of this method is performed by application to a generic test case: the spinning of two vortex filaments in an infinite two-dimensional duct. This flow model is amenable to a nearly exact derivation by an incompressible vortex model. Moreover, the acoustic field can also be obtained very accurately using the tailored Green's function based on the duct modes, providing a reference solution to validate our numerical approach. The sound field predicted using the BEM/Curle approach shows excellent agreement with the reference solution based on the tailored Green's function, thereby validating the general principle of BEM/Curle method and its numerical implementation.

Computer simulation on the void formation in neutron-irradiated Cu and Ni at high temperature

Atomistic processes of evolution of damage structure in neutron-irradiated Cu and Ni at high temperature are investigated on the linkage of experimental results and computer-simulation. Interstitials and their clusters move to form a grouping of interstitial clusters which subsequently evolve to dislocation at high temperature. Vacancies aggregate to stacking fault tetrahedron (SFT) and void. At high temperature gas atoms do not make a significant contribution to the nucleation of voids at low fluence of neutron irradiation such as 10 18 n/cm 2. SFT relax at a high temperature to a string of vacancy clusters, in which vacancies are connected one-dimensionally. The break-up of SFT occurs by jumping of an atom into a SFT. A loosely bound vacancy cluster of a string shape moves with low activation energy of 0.2 eV. Voids are formed by the coalescence of moving vacancy clusters of string shape at high temperature. This is due to low Helmholtz free energy of void-containing crystal than that of SFT-containing crystal. The role of gas atoms in the formation of voids is to degrade the mobility of movable vacancy clusters when captured by the clusters.

Numerical computation of underwater sound scattering from a complex elastic object

There are numerous analytical approaches for the calculation of sound scattering from cylinderlike structures. However, limitations exist in the complexity of objects, which can be handed that way. In this case numerical methods are necessary to compute the acoustical response of the model. The investigated structure consists of two cylinders of different radii connected by a conical section. The ends are covered by hemispherical caps. An axisymmetric finite-element code (SARA2d) has been used to calculate the response of the target, due to an incident plane wave. Semi-infinite elements are implemented at the boundary of the acoustic fluid to take care of the Sommerfeld radiation condition. In the covered frequency range, which comprised Helmholtz numbers up to ka∼13, specular and elastic properties of the target contribute to the scattered signal. The calculations were performed during a one year research stay at the Naval Undersea Warfare Center (NUWC) in Newport, Rhode Island, USA. Additionally an attempt has been made to compare parts of the results with boundary element methods (SYSNOISE) and coupled FEM/BEM (ANSYS/SYSNOISE). For the coupled calculations only three-dimensional methods have been available, hence those results are restricted to relatively low Helmholtz numbers.

Damping and reflection coefficient measurements for an open pipe at low Mach and low Helmholtz numbers

The propagation of plane acoustic waves in smooth pipes and their reflection at open pipe terminations have been studied experimentally. The accuracy of the measurements is determined by comparison of experimental data with results of linear theory for the propagation of acoustic waves in a pipe with a quiescent fluid. The damping and the reflection at an unflanged pipe termination are compared.In the presence of a fully developed turbulent mean flow the measurements of the damping confirm the results of Ronneberger &amp; Ahrens (1977). In the high-frequency limit the quasi-laminar theory of Ronneberger (1975) predicts accurately the convective effects on the damping of acoustic waves. For low frequencies a simple theory combining the rigid-plate model of Ronneberger &amp; Ahrens (1977) with the theoretical approach of Howe (1984) yields a fair prediction of the influence of turbulence on the shear stress. The finite response time of the turbulence near the wall to the acoustic perturbations has to be taken into account in order to explain the experimental data. The model yields a quasi-stationary limit of the damping which does not take into account the fundamental difference between the viscous and thermal dissipation observed for low frequencies.Measurements of the nonlinear behaviour of the reflection properties for unflanged pipe terminations with thin and thick walls in the absence of a mean flow confirm the theory of Disselhorst &amp; van Wijngaarden (1980), for the low-frequency limit. It appears however that a two-dimensional theory such as proposed by Disselhorst &amp; van Wijngaarden (1980) for the high-frequency limit underestimates the acoustical energy absorption by vortex shedding by a factor 2.5.The measured influence of wall thickness on the reflection properties of an open pipe end confirms the linear theory of Ando (1969). In the presence of a mean flow the end correction δ of an unflanged pipe end varies from the value at the high-Strouhal-number limit of δ/a= 0.61, withathe pipe radius, which is close to the value in the absence of a mean flow given by Levine &amp; Schwinger (1948) of δ/a= 0.6133, to a value of δ/a= 0.19 in the low-Strouhal-number limit which is close to the value predicted by Rienstra (1983) of δ/a= 0.26.The pressure reflection coefficient is found to agree with the theoretical predictions by Munt (1977, 1990) and Cargill (1982b) in which a full Kutta condition is included. The accuracy of the theory is fascinating in view of the dramatic simplifications introduced in the theory. For a thick-walled pipe end and a pipe terminated by a horn the end correction behaviour is similar. It is surprising that the nonlinear behaviour at low frequencies and high acoustic amplitudes in the absence of mean flow does not influence the end correction significantly.The aero-acoustic behaviour of the pipe end is dramatially influenced by the presence of a horn. In the presence of a mean flow the horn is a source of sound for a critical range of the Strouhal number.The high accuracy of the experimental data suggests that acoustic measurements can be used for a systematic study of turbulence in unsteady flow and of unsteady flow separation.