Pulsating Sphere Research Articles

Harmonic acoustic pneumatic source (HAPS) to generate sound at very low-frequency

At very low frequency, the loudspeakers face technical issues (size, weight, energy consumption). An alternative to the electrodynamic loudspeaker is the Harmonic Acoustic Pneumatic Source (HAPS) demonstrated efficient in active tonal control. It comprises a high-pressure pneumatic air source, a rotating flow chopper, and an exhaust. The rotation of the flow chopper generates a pulsed flow which radiated noise out from the exhaust. An analytical model of the HAPS is presented to introduce the main challenge associated with its use at very low frequencies: having a high mean flow rate with a relatively small exhaust duct section. To overcome these challenge, a dedicated flexible tube (under light mean pressure) is added to the exhaust circuit to obtain a pulsating sphere activated by a HAPS. This configuration has been experimentally studied in a semi-anechoic room with two flexible tubes and without. The sound pressure level of the first harmonic at 1 meter range from 75 to 95 dB SPL (25 to 160 Hz, plenum pressure from 5 to 20 PSI). Thanks to fluid-structure interaction, the fundamental harmonic sound levels radiated by the compliant tube were free of jet noise. The drawback is the high harmonic distortion observed during the experiments.

A Boundary Node Method for Solving Interior and Exterior Acoustic Problems

The Boundary Node Method (BNM) is a meshfree scheme for solving Boundary Integral Equations (BIE). BNM simplifies the problem at two levels. Primarily, as BNM aims to solve the integral form of the governing equation, dimensionality of the problem is reduced by one order. Additionally, the mesh-free scheme eliminates the need for meshing, proving particularly beneficial for problems involving complex geometries. In this study, BNM is formulated using Element Free Galerkin (EFG) scheme to solve interior and exterior problems in acoustics. A 3-dimensional linear basis is used to construct moving least square shape functions. This eliminates the need to define additional local or parametric coordinate systems, making the method easily applicable to any arbitrary geometry. To illustrate the capabilities of the proposed method, four example problems are solved. An open pipe at resonance and a simple expansion muffler are analyzed to validate BNM’s performance in solving interior problems. Radiation from a pulsating sphere and radiation from a sphere with harmonic velocity excitation are analyzed as examples of exterior problems. Results from all four sample problems indicate that the BNM scheme proposed for solving acoustic problems is accurate and robust.

Comparison and analysis of different acoustic calculation methods for predicting sound radiated from a pulsating sphere

The radiated sound from a pulsating spherical source is a classical acoustic problem, which can be solved either by the classical acoustic method or by the aeroacoustics method. In this paper, an example of sound radiated by a small sphere in the air is designed, and the formulas for determining the surface pressure are derived using the classical acoustic method and the Kirchhoff method, respectively. In the derivation of Kirchhoff method, toroidal strip integrals are used. For the radiated sound field with different frequencies, sphere radius, the two methods are used to solve the problem. It is found that the results obtained by the two methods are very close to each other in the scope of actual engineering concern. Then, this paper discusses the applicability of the Kirchhoff method and points out that the main reason for the result deviation between the two ways is that the Kirchhoff method is derived from the point source Green function. For the more general equation of FW-H, the equivalent relation with the Kirchhoff method is introduced. Through the research of this paper, the characteristics and consistency of different acoustic theories in calculating the radiated sound from the pulsating sphere are explained in depth.

Analysis of an Acoustic Monopole Source in a Closed Cavity via CUF Finite Elements

The noise generated by aircraft is an important issue, which affects the external environment and the passenger’s comfort. The researches about new acoustic solutions often lead to the exploitation of innovative materials, as visco-elastic panels or acoustic metamaterials, in order to rather obtain better acoustic properties than conventional materials, in particular at low frequency. Although, there is a lack of reliable tools able to describe the complex kinematic behaviour of these new materials at low frequency. A new strong formulation, the Carrera Unified Formulation (CUF) based on the Finite Element Method (FEM), enables a wide class of refined shell models, which is able to reproduce the frequency dependent dynamic response of complex multi-layered plates. This formulation, fully developed inside the MUL2 software, is applied to vibro-acoustic analyses too, so the need to integrate new sources and boundary conditions in the software, that are essential to model the acoustic problem. A simple and powerful source is the monopole: a pulsating sphere. This source can be a first try to model the complex sources that affect the noise inside the aircraft, as the engine or the internal sources. Moreover, monopoles are widely used to estimate the transmission loss. Hence, the reason for this work: the creation inside MUL2 of a monopole boundary condition and its validation, comparing the results with those of a well-known FEM based commercial software for vibro-acoustic analyses, Actran.

A moving embedded boundary approach for the compressible Navier-Stokes equations in a block-structured adaptive refinement framework

A computational technique has been developed to perform compressible flow simulations involving moving boundaries using an embedded boundary approach within the block-structured adaptive mesh refinement (SAMR) framework of AMReX [1,91,92]. We leverage the SAMR capability to obtain quantitatively accurate results whilst using robust, second-order finite volume schemes. A conservative, unsplit, cut-cell approach is utilized and a ghost-cell approach is developed for computing the flux on the moving, embedded boundary faces. A third-order least-squares formulation has been developed to compute the wall velocity gradients, and was found to significantly improve the performance of the solver in terms of the quantitative comparison of surface quantities such as the skin friction coefficient. Various test cases are performed to validate the method, and compared with analytical, experimental, and other numerical results in literature. Inviscid and viscous test cases are performed that span a wide regime of flow speeds – acoustic (harmonically pulsating sphere), smooth flows (expansion fan created by a receding piston) and flows with shocks (shock-cylinder interaction, shock-wedge interaction, pitching NACA 0012 airfoil and shock-cone interaction). A closed system with moving boundaries – an oscillating piston in a cylinder, showed that the percentage error in mass within the system decreases with refinement, demonstrating that the numerical scheme is conservative with grid refinement, but is not discretely conservative. Viscous test cases involve that of a horizontally moving cylinder at Re=40, an inline oscillating cylinder at Re=100, and a transversely oscillating cylinder at Re=185. The judicious use of adaptive mesh refinement with appropriate refinement criteria to capture the regions of interest leads to well-resolved flow features, and good quantitative comparison is observed with the results available in literature.

Acoustic radiation from a sphere pulsating near an impedance plane using a boundary integral equation method

In this study, a boundary integral equation method is proposed for investigating acoustic pressure radiation from a sphere pulsating near a free surface or an impedance plane. The half-space and free-space problems are investigated for the acoustic radiation of pulsating sphere. The effects of free surface and impedance boundaries are introduced into the mathematical model by employing three different half-space Green’s functions, respectively. These Green’s functions are derived, respectively, using the single image-source method, multiple equivalent-source method, and complex equivalent-source method. Green’s functions are implemented into the boundary element (BE) formulation. The surface of the pulsating sphere is discretized with linear and quadratic BEs, and the Combined Helmholtz Integral Equation Formulation (CHIEF) is employed to overcome the non-uniqueness problem. Four different case studies are considered for the sphere pulsating near a free surface or an impedance plane. The first case study involves the sphere pulsating near a free surface (perfectly reflective) and the single image-source method is used in the boundary element method (BEM) formulation. In the second case study, the sphere is assumed as pulsating near a perfectly reflecting and perfectly absorbing impedance planes, respectively. The multiple equivalent-source method is employed for the perfectly reflecting plane, but the multiple equivalent-source method and complex equivalent-source methods for the perfectly absorbing plane. The third case study involves a general impedance plane, and all the methods are employed, respectively, in the BE formulation. The final case study assumes a general impedance plane forming a perpendicular incidence and the complex equivalent-source method is used in this particular case. It is observed that there is a very good comparison between the results obtained from all these methods.

A Semi-Analytical Method for the 3D Elastic Structural-Acoustic Radiation in Shallow Water

A hybrid computational method is proposed for three-dimensional structural-acoustic radiation in shallow water, which combines the wave superposition method (WSM) with the adjustable Green function and the multi-physical-coupled finite element method (FEM). A low-frequency acoustic model in the near field is established by the FEM to discretize the continuous and arbitrary-shaped radiator, and the normal vibration velocity on the wetted surface is used as the input in the WSM to calculate the strengths of simple sources enclosed within the radiator. Finally, the sound propagation in the far field of the elastic structure is efficiently computed by superimposing the sound field of each simple source. The numerical examples of the pulsating sphere and elastic spherical shell in the shallow water are developed to verify the high accuracy and efficiency of the presented method, respectively. Moreover, the regularization algorithms are employed as the complementary approaches to improve the stability of this method in engineering practices, it is found that the algorithms can significantly hold the calculation accuracy in case the velocity data contain noise, and the least-squares QR is an efficient and accurate method with a few iteration numbers and better stability effect as compared with other algorithms.

How Loud Can you go? Physical and Physiological Constraints to Producing High Sound Pressures in Animal Vocalizations

Sound is vital for communication and navigation across the animal kingdom and sound communication is unrivaled in accuracy and information richness over long distances both in air and water. The source level (SL) of the sound is a key factor in determining the range at which animals can communicate and the range at which echolocators can operate their biosonar. Here we compile, standardize and compare measurements of the loudest animals both in air and water. In air we find a remarkable similarity in the highest SLs produced across the different taxa. Within all taxa we find species that produce sound above 100 dBpeak re 20 μPa at 1 m, and a few bird and mammal species have SLs as high as 125 dBpeak re 20 μPa at 1 m. We next used pulsating sphere and piston models to estimate the maximum sound pressures generated in the radiated sound field. These data suggest that the loudest species within all taxa converge upon maximum pressures of 140–150 dBpeak re 20 μPa in air. In water, the toothed whales produce by far the loudest SLs up to 240 dBpeak re 1 μPa at 1 m. We discuss possible physical limitations to the production, radiation and propagation of high sound pressures. Furthermore, we discuss physiological limitations to the wide variety of sound generating mechanisms that have evolved in air and water of which many are still not well-understood or even unknown. We propose that in air, non-linear sound propagation forms a limit to producing louder sounds. While non-linear sound propagation may play a role in water as well, both sperm whale and pistol shrimp reach another physical limit of sound production, the cavitation limit in water. Taken together, our data suggests that both in air and water, animals evolved that produce sound so loud that they are pushing against physical rather than physiological limits of sound production, radiation and propagation.

A fast multipole boundary element method for three‐dimensional acoustic problems in a subsonic uniform flow

AbstractA fast multipole boundary element method (FMBEM) in a subsonic uniform flow is presented. It is based on the boundary integral equation (BIE) in a subsonic uniform flow. The convected Green's function complicates its multipole expansion as well as the implementation of the computer code. Although the Lorentz transformation allows the Helmholtz equation in the uniform flow to be reduced to the standard Helmholtz equation, the deformation of the domain complicates the boundary conditions and may cause the elements' distortion. In this work, the analytical evaluations of singular integrals are achieved. Then a nonsingular BIE in a subsonic uniform flow is obtained and is incorporated in building FMBEM with the plane wave multipole expansion of Green's function directly. Details on the implementation of the algorithm are described. Numerical examples including a pulsating sphere radiation problem, a multibody scattering problem and an aircraft model are performed to validate the accuracy and efficiency of the proposed method. Results show that FMBEM solutions are in good agreement with analytical solutions. The difference between the analytical moments and numerical moments is also investigated carefully in the implementation of the fast multipole method. Dramatical improvements on solution efficiency are observed by comparing the developed algorithm with the CBEM.

Boundary integral equations for sound radiation from a harmonically vibrating body moving uniformly in a free space.

In this high-speed transit era, the prediction of sound radiation from a moving and vibrating object is an important research topic. A typical example is a high-speed train wheel that not only moves fast in the railway direction but, due to wheel/rail interaction, also vibrates at high frequency, generating a complex sound field. By making use of sound spectra generated by moving harmonic compact sources, three-dimensional (3D) boundary integral equations are established in this paper for sound radiation from a harmonically vibrating body moving uniformly in a free space. The 3D boundary integral equations are reduced to 2D ones for a special case in which the body is axisymmetric and moves in its axial direction. The 2D boundary element method is applied to solve the 2D boundary integral equations and to produce results for a pulsating and moving sphere. Results show that the moving speed of the pulsating sphere has a significant effect on the generated sound field. This means that, for sounds radiated from a high-speed train wheel vibro-acoustically, the motion of the wheel has to be taken into account.

The sound of a pulsating sphere in a rarefied gas: continuum breakdown at short length and time scales

The pressure field of a pulsating sphere is a canonical problem in classical acoustics, used to illustrate the acoustic efficiency of a monopole source at continuum conditions. We consider the counterpart vibroacoustic and thermoacoustic problems in a rarefied gas, to investigate the effect of continuum breakdown on monopole radiation. Focusing on small-amplitude normal-to-boundary mechanical and heat-flux excitations, the perturbation field is analysed in the entire range of gas rarefaction and input frequencies. Numerical calculations are carried out via the direct simulation Monte Carlo method, and are used to validate analytical predictions in the free-molecular and near-continuum regimes. In the latter, the regularized thirteen moments model (R13) is applied, to capture the system response at states where the Navier–Stokes–Fourier (NSF) description breaks down. Comparing with the continuum inviscid solution, the results quantitate the dampening effect of gas rarefaction on source point-wise strength and acoustic power. At near-continuum conditions, the acoustic field is composed of exponentially decaying ‘compression’, ‘thermal’ and ‘Knudsen-layer’ modes, reflecting thermoviscous and higher-order rarefaction effects. With reducing rarefaction, the contributions of the latter two modes vanish, and the former degenerates into the ideal-flow inverse-to-distance decaying wave. Stronger attenuation is obtained with increasing rarefaction, where boundary sphericity results in a ‘geometric reduction’ of the molecular layer affected by the source. Notably, while the R13 model at low frequencies appears valid up to moderate gas rarefaction rates, both the NSF and R13 descriptions break down at common low Knudsen numbers at higher frequencies. Further study should therefore be carried out to extend the applicability of moment models to unsteady flows with short time scales.

Comparisons of regularization methods and regularization parameter selection methods in sound source identification using inverse boundary element method

In noise source identification based on the inverse boundary element method (IBEM), the boundary vibration velocity is predicted based on the field pressure through a transfer matrix of the vibration velocity and field pressure established on the Helmholtz integral equation. Because the matrix is often ill-posed, it needs to be regularized before reconstructing the vibration velocity. Two regularization methods and two methods of selecting the regularization parameter are investigated through the simulation analysis of a pulsating sphere. The result of transfer matrix regularization is further verified through the reconstruction of the vibration of an aluminum plate. Additionally, to reduce the large errors at some frequencies in the reconstruction result, increasing the number of measuring points is more effective than reducing the distance between the measurement plane and the sound source.

Numerical Method of Radiation Impedance Calculation in Enclosed Space

A numerical method for calculating the radiation impedance of sound sources in enclosed space is established in this paper. The correctness of the proposed method is verified by analytical results. Primarily, the radiation impedance of a pulsating sphere sound source in the free field and the enclosed space is analyzed using analytical calculation method. Furthermore, the radiation impedance of a pulsating ball sound source is calculated by acoustic finite element software Actran using the proposed numerical method. The research shows that the method of calculating the sound source radiation impedance by Actran is suitable for the analysis of the acoustic impedance characteristics of underwater sound sources. The numerical method established in this paper provides a useful numerical method to analyze the radiation impedance characteristics of underwater sound sources with complex geometric structure.

A Boundary Element Method Based on the Hierarchical Matrices and Multipole Expansion Theory for Acoustic Problems

The coefficient matrices of conventional boundary element method (CBEM) are dense and fully populated. Special techniques such as hierarchical matrices (H-matrices) format are required to extent its ability of handling large-scale problems. Adaptive cross approximation (ACA) algorithm is a widely adopted algorithm to obtain the H-matrices. However, the accuracy of the ACA boundary element method (ACABEM) cannot be adjusted by changing the tolerance [Formula: see text] when it exceeds a certain value. In this paper, the degenerate kernel approximation idea for the low-rank matrices is developed to build a fast BEM for acoustic problems by exploring the multipole expansion of the kernel, which is referred as the multipole expansion H-matrices boundary element method (ME-H-BEM). The newly developed algorithm compresses the far-field submatrices into low rank submatrices with the expansion terms of Green’s function. The obtained H-matrices are applied in conjunction with the generalized minimal residual method (GMRES) to solve acoustic problems. Numerical examples are carefully set up to compare the accuracy, efficiency as well as memory consumption of the CBEM, ACABEM, fast multipole boundary element method (FMBEM) and ME-H-BEM. The results of a pulsating sphere indicate that the ME-H-BEM keeps both storage and operation logarithmic-linear complexity of the H-matrices format as the ACABEM does. Moreover, the ME-H-BEM can achieve better convergence and higher accuracy than the ACABEM. For the analyzed complicated large-scale model, the ME-H-BEM with appropriate number of expansion terms has an advantage in terms of efficiency as compared with the ACABEM. Compared with the FMBEM, the ME-H-BEM is easier to be implemented.

A New Method for Determining Optimal Regularization Parameter in Near‐Field Acoustic Holography

Tikhonov regularization method is effective in stabilizing reconstruction process of the near‐field acoustic holography (NAH) based on the equivalent source method (ESM), and the selection of the optimal regularization parameter is a key problem that determines the regularization effect. In this work, a new method for determining the optimal regularization parameter is proposed. The transfer matrix relating the source strengths of the equivalent sources to the measured pressures on the hologram surface is augmented by adding a fictitious point source with zero strength. The minimization of the norm of this fictitious point source strength is as the criterion for choosing the optimal regularization parameter since the reconstructed value should tend to zero. The original inverse problem in calculating the source strengths is converted into a univariate optimization problem which is solved by a one‐dimensional search technique. Two numerical simulations with a point driven simply supported plate and a pulsating sphere are investigated to validate the performance of the proposed method by comparison with the L‐curve method. The results demonstrate that the proposed method can determine the regularization parameter correctly and effectively for the reconstruction in NAH.

A new mechanism for size separation of particles immersed in a fluid

Work presented deals with a new mechanism for particle size-separation based on differential grouping induced by a controlled oscillating flow field. This separation by size may be used for coagulation of like-sized particles. Flow oscillations are induced by a pulsating sphere. The dynamics of particles immersed in such a flow field and their trajectories are analysed mathematically based on perturbation analysis. It is shown that in proximity to the sphere, particle trajectories display strong dependence on particle size (through particle Stokes number), far more so than on initial position. Grouping behaviour is observed under specific operating conditions, and this suggests feasibility of applications in filtration systems, as well as for size-based separation.

Sound Radiation of a Pulsating Sphere in the Outlet of a Hard/Soft Semi-Spherical Cavity in a Flat Screen

Abstract A rigorous analysis of sound radiation by a pulsating sphere forming a resonator together with a semi-spherical cavity is presented. Both hard and soft boundaries are considered, as well as mixed. The problem is solved by dividing the entire region into two subregions, one surrounding the sphere and containing the cavity and the other for the remaining half-space. The continuity conditions are applied to obtain the acoustic pressure. Then the acoustic radiation resistance is calculated both in the near- and far-field. The acoustic radiation reactance is calculated in the impedance approach. The resonance frequencies are determined, for which a significant growth of the sound pressure level is observed as well as the sound field directivity. The accuracy and convergence of these rigorous results has been examined empirically.

Cepheid distances from the SpectroPhoto-Interferometry of Pulsating Stars (SPIPS)

The parallax of pulsation, and its implementations such as the Baade-Wesselink method and the infrared surface bright- ness technique, is an elegant method to determine distances of pulsating stars in a quasi-geometrical way. However, these classical implementations in general only use a subset of the available observational data. Freedman & Madore (2010) suggested a more physical approach in the implementation of the parallax of pulsation in order to treat all available data. We present a global and model-based parallax-of-pulsation method that enables including any type of observational data in a consistent model fit, the SpectroPhoto-Interferometric modeling of Pulsating Stars (SPIPS). We implemented a simple model consisting of a pulsating sphere with a varying effective temperature and a combina- tion of atmospheric model grids to globally fit radial velocities, spectroscopic data, and interferometric angular diameters. We also parametrized (and adjusted) the reddening and the contribution of the circumstellar envelopes in the near-infrared photometric and interferometric measurements. We show the successful application of the method to two stars: delta Cep and eta Aql. The agreement of all data fitted by a single model confirms the validity of the method. Derived parameters are compatible with publish values, but with a higher level of confidence. The SPIPS algorithm combines all the available observables (radial velocimetry, interferometry, and photometry) to estimate the physical parameters of the star (ratio distance/ p-factor, Teff, presence of infrared excess, color excess, etc). The statistical precision is improved (compared to other methods) thanks to the large number of data taken into account, the accuracy is improved by using consistent physical modeling and the reliability of the derived parameters is strengthened thanks to the redundancy in the data.

Acoustic radiation analysis for a control domain based on Green's function

The acoustic radiation problem in the control domain with complex boundaries is investigated. Acoustic Green's function for acoustic point sources in the control domain was obtained by using conformal transformation theory. The expression of the acoustic radiation function for the control domain was derived. The acoustic radiation characteristics of a pulsating sphere in a quarter-infinite domain and numerical simulations were compared to illustrate the effects of the boundary characteristic, location, and radiation frequency on acoustic radiation power and acoustic directivity. This study provides a new method to analyze the acoustic radiation problem with complex boundaries.

Is the Burton–Miller formulation really free of fictitious eigenfrequencies?

This paper is concerned with the fictitious eigenfrequency problem of the boundary integral equation methods when solving exterior acoustic problems. A contour integral method is used to convert the nonlinear eigenproblems caused by the boundary element method into ordinary eigenproblems. Since both real and complex eigenvalues can be extracted by using the contour integral method, it enables us to investigate the fictitious eigenfrequency problem in a new way rather than comparing the accuracy of numerical solutions or the condition numbers of boundary element coefficient matrices. The interior and exterior acoustic fields of a sphere with both Dirichlet and Neumann boundary conditions are taken as numerical examples. The pulsating sphere example is studied and all fictitious eigenfrequencies corresponding to the related interior problem are observed. The reasons are given for the usual absence of many fictitious eigenfrequencies in the literature. Fictitious eigenfrequency phenomena of the Kirchhoff–Helmholtz boundary integral equation, its normal derivative formulation and the Burton–Miller formulation are investigated through the eigenvalue analysis. The actual effect of the Burton–Miller formulation on fictitious eigenfrequencies is revealed and the optimal choice of the coupling parameter is confirmed.