Infinite Duct Research Articles

Electrostatic surface waves in a magnetized plasma propagating in channels

Dispersion relations of electrostatic surface waves propagating in magnetized plasmas contained in an infinite duct and in an infinite cylindrical column surrounded by vacuum are derived by means of a Vlasov equation and fluid equations, respectively. The kinematic boundary condition imposed on the distribution function, the specular reflection conditions on the four sides of a duct, can be satisfied by placing infinite number of fictitious surface charge sheets spaced by the duct widths. The Vlasov equation that includes these surface charge sheets is solved by summing up the contribution due to the infinite charge sheets. The method of placing appropriate fictitious surface charge sheets enables one to treat the surface waves in bounded plasmas of Cartesian structure with mathematical efficiency, kinetically. The kinetic duct dispersion relation is compared with the dispersion relation for the magnetized cylindrical plasma column. When the square duct cross-sectional area as well as the cylinder radius become infinity, both dispersion relations become the dispersion relation of the upper-hybrid wave.

Numerical solution of optimal control problems for linear systems of ordinary differential equations

Abstract An original numerical matrix algorithm aimed at solving the optimal control problems for linear systems of ordinary differential equations with constant coefficients is proposed. The work of the algorithm is demonstrated with the problem, which consists in generating a given small disturbance of the Poiseuille flow in an infinite duct by blowing and suction through the walls. The costs of creating the leading modes and optimal disturbances are compared, which is of independent interest.

Analysis of sound transmission loss in an infinite duct with three different finite linings

The study rigorously investigates the propagation of sound waves from an infinite duct lined with three different finite acoustically absorbing materials. Due to multiple discontinuity points, the mode matching method effectively solved the problem. The mathematical equations involved are highly complex due to three distinct and finite linings. To overcome this, a numerical solution is obtained, and the effects of the geometry’s parameters on the transmission loss are thoroughly examined. As the length of the finite acoustic absorbing lining increases, the transmission loss decreases. In addition, the graphs clearly illustrate the substantial influence that the choice of absorbing lining can have on transmission loss. These findings hold great relevance for absorber systems, highlighting the importance of careful selection in order to achieve optimal performance. The obtained results are compared with another study, and complete agreement is observed, indicating the study’s accuracy and reliability.

Thermoacoustic effects on the propagation of non-planar sound in a circular duct

This paper examines thermoacoustic effects on the propagation of non-planar sound in a circular duct subjected to an axial temperature gradient. Of particular concern are thermoviscous diffusive effects, which are taken into account by the boundary-layer approximation in a framework of the linear theory. For disturbances expanded into Fourier and Fourier–Bessel series in the azimuthal and radial directions, respectively, the pressure in each mode is described by a one-dimensional, dispersive wave equation, if non-diffusive propagation is assumed. When the diffusive effects are included, each radial mode is coupled to the other radial modes through the boundary layer. Focusing on a single azimuthal and radial mode only, the dispersion relation for the propagation along an infinite duct of a uniform gas is first derived. Effects of the temperature gradient are then examined by solving boundary-value problems for a duct of finite length in four typical cases. Assuming that the wall temperature increases exponentially along the duct, eigenfrequencies and decay rates in the lowest axial mode are obtained as well as axial distributions of the sound pressure and the axial velocity in the duct. The frequency and the decay rate increase as the temperature ratio at both ends becomes higher. It is found from the acoustic energy equation that the dispersion combined with the diffusion acts to reduce the damping and that the temperature gradient makes little contribution to the production of the energy. However, it is unveiled that the non-uniformity in temperature yields thermoacoustic sound confinement in the vicinity of the cold end.

Surface waves in a streaming plasma propagating in a duct: kinetic theory

The dispersion relation of a surface wave generated by a drifting plasma in an infinite duct surrounded by vacuum is derived non-relativistically by means of the Vlasov equation. The kinematic boundary condition imposed on the distribution function, the specular reflection conditions on the four sides of duct, can be satisfied by placing an infinite number of fictitious surface charge sheets spaced by the duct widths. The surface wave mode is specifically the transverse magnetic mode, often called the surface polariton, which propagates with phasor $\exp ({{\rm i}k_zz-{\rm i}\omega t})$ . The method of placing appropriate fictitious surface charge sheets enables one to treat the surface waves in semi-infinite, slab and duct plasmas simultaneously on an equal footing, kinetically. The streaming effect manifests itself through the Doppler-shifted frequency and a correction-like term ${u^2}/{c^2}$ , where u is the streaming velocity and c is the speed of light.

Radiation of sound waves from a coaxial duct with perforated screen

Abstract Radiation of sound waves by a coaxial rigid duct with perforated screen is investigated by using the Mode Matching technique in conjunction with the Jones’ Method. The geometry of the problem consists semi-infinite outer duct and infinite inner duct. It is assumed that the duct walls are fully rigid. The solution of current study contains an infinite sets of coefficients satisfying an infinite systems of linear algebraic equations. These systems are truncated at a certain number and then solved numerically. The effects of open and perforated case, frequency and porosity on the radiation phenomenon are shown graphically. In the present study, perforated screen makes the problem more interesting when it is compared with the unperforated screen. In this context, solution of the problem is compered numerically with similar studies, which are used different method to obtain Wiener–Hopf equation, existing in the literature. As a result, it is observed that in the absence of a perforated screen, there is a perfect agreement between the two results.

Design and experimental validation of a high level electropneumatic source

The study and characterization of non-linear phenomena in silencers and microperforated panels under high sound pressure level and mean flow require a dedicated experimental bench with adequate acoustic sources. For this purpose, electropneumatic sources are excellent candidates to reach the expected operating conditions in the laboratory. The challenge is to design a source that can generate a high tonal sound level (> 160 dB SPL) under a significant mean flow (M > 0.1) with a low harmonic distortion. The objective is to focus on mastering the design process of such a tonal source based on an analytical modeling of the physical phenomenon of tonal sound generation and flow as a function of geometrical and pneumatic supply parameters. Based on this validated analytical modeling, a new original source has been developed with the objective of generating a pure harmonic sound in the frequency range [100; 800] Hz, that can reach 180 dB in an infinite duct, combined with a flow up to Mach 0.3. For this purpose, a flow chopper system with a digitally optimized pattern has been specifically developed to minimize harmonic distortion. Considering the need to experimentally evaluate the rate of harmonic distortion with a high degree of accuracy, a complex envelope analysis by Vold-Kalman was implemented in post-processing. Finally, an experimental test campaign was set up on a dedicated bench to test with an open ended duct, for different supply pressures going from 2.07 105 to 3.45 105 Pa, the performances of the new acoustic source. The results of this test campaign show that the device can generate a tonal sound level with a distortion of less than 40 % for the second order and less than 20 % for the higher orders, with a sound pressure level that can reach 186 dB at the resonances in presence of an average flow reaching M = 0.24. These observations are consistent with the predictions of both analytical and numerical models.

Vibroacoustic performance assessment of aircraft panels in low, mid and high frequency regimes

In the present work, vibroacoustic (VA) characteristics, namely sound transmission loss (STL), overall sound pressure levels (OASPLs) of aircraft panels made up of aluminum, composites and fiber metal laminates (FML) are studied. The investigation involves modeling of aircraft panels using finite element method (FEM) for low frequency, Boundary Element Method (BEM) for mid-frequency and statistical energy analysis (SEA) in high-frequency bands. To obtain the VA characteristics of the panels, twin chambers, namely source and receiver are numerically modeled, and the panels are placed in between them. The VA performances of the considered panels under low, mid and high-frequency sound environments are numerically simulated using an infinite duct model. The sound source in low, mid and high frequency is modeled in the source chamber as sound pressure, plane wave and diffusive acoustic field, respectively and the receiver chamber is idealized as an anechoic chamber at the ends. This numerical study helps in understanding the VA behavior of aircraft materials and also minimizes the cost and time involved in conducting experiments. The results are presented in the form of STL and OASPL. From this study, it was found that the aluminum and fiber metal laminates have good VA characteristics when compared to composites in all the three frequency regimes and composite panels behaving better in the mid-frequency region of 200 Hz to 300 Hz than aluminum and fiber metal laminates. An optimization approach for aircraft panels is also presented.

Mutual Effect Between Swept-and-Leaned Vanes and Acoustic Liners on Fan Interaction-Noise Reduction

The swept-and-leaned vanes and acoustic liners can both be used to reduce the turbofan noise. In this work, an analytical model was proposed under an idealized condition, in which an annular stator cascade and a locally reacting liner exist in an infinite duct at the same time, to study the mutual effect between these two techniques. Based on the transfer element method, the system matrix of multi-elements can be established based on the continuity of disturbance pressure and velocity at the interfaces. The acoustic elements for a stator and a liner were established, respectively, by employing the three-dimensional lifting-surface method and the equivalent surface source method, which are both the so-called methods of singularity. The interface matching pattern between adjacent elements was improved in this paper with more physics included, which can also be combined with the numerical methods easier. Because the technologies for fan-noise reduction, which are the swept-and-leaned vanes and acoustic liners in this paper, are both closely related to the modal acoustic field in the duct, the coupling effect can be strong in some conditions. The results showed that the noise reduction gained by these two techniques cannot be superposed simply. They should not be considered isolatedly. Further discussions on the transmission and reflection between different kinds of acoustic elements should be made to achieve the system optimization in the nacelle.

On low frequency sound propagation across closely coupled narrow cavities along an infinite duct and the similarity in stopband cut-on frequencies

The propagation of sound across closely coupled narrow rectangular cavities in an infinitely long duct is studied numerically using finite-element simulation and the method of mode matching in the present investigation. Sound frequencies below the first cut-off frequency of the main duct are the focuses. Results show that the middle region of the coupled cavity section plays the most important role in affecting the spectral characteristics of sound transmission across the coupled cavities. It is also found that there exists a critical frequency above which a coupled cavity section can offer reasonably good sound transmission loss. In addition, similarity in this stopband cut-on frequency across coupled cavity sections of different dimensions is established empirically after the introduction of a new length scale. The corresponding frequencies of the symmetrical non-offset coupled cavity sections are also a vital parameter in the scaling. Two frameworks are developed to predict the stopband cut-on frequencies. Both of them are able to give predictions with accuracy to within engineering tolerance.

Vibroacoustic study of a point-constrained plate mounted in a duct

The vibroacoustic study of the interaction of sound with a point-constrained, simply-supported square plate is considered in this paper. The plate is mounted flush on one of the walls of an infinite duct of rectangular cross section and is backed by a cavity. The plate response and the acoustic field is predicted by solving the coupled governing equations using modal expansion with the relevant eigenmodes of the plate dynamics and acoustic fields in the duct and cavity. By varying the location of the point constraint, the frequency characteristics of the transmission loss in the duct can be tuned. The point constraint can also alter the amplitude and spectral characteristics of the plate's response. Interestingly, some new peaks are observed in the response because of the excitation of unsymmetric modes which are otherwise dormant. Mode-localization phenomenon, which is the localization of vibration in specific regions of the plate, is observed for selected constrained points.

An improved time-dependent Boundary Element Method for two-dimensional acoustic problems in a subsonic uniform flow

In this paper, we have developed an improved formulation of two-dimensional Convected Boundary Element Method (CBEM) for radiation and propagation acoustic problems in a uniform mean flow with arbitrary orientation. The improved CBEM approach is derived from an advanced form of time-space two-dimensional Boundary Integral Equation (BIE) according to new Sommerfeld Radiation Conditions (SRC) with arbitrary mean flow. The acoustic variables of these formulations are expressed only in terms of the acoustic field as well as its normal and tangential derivatives. The multiplication operators are based explicitly on the two-dimensional Green's function and its convected normal derivative kernel. The proposed terms significantly reduce the presence of flow quantities incorporated in the classical integral formulations. Precisely, the convected kernel only requires the evaluation of two terms instead of several terms in conventional formulations due to the flow effects in the temporal and spatial derivatives. Also, for the singular integrations, the kernels containing logarithmic and weak singularities are converted to regular forms and evaluated partially analytically and numerically. The formulation is derived to be easy to implement as a numerical tool for computational codes of acoustic mediums with arbitrary mean flow. The accuracy and robustness of this technique is assessed through several examples such as the two-dimensional monopole, dipole and quadrupole sources in a uniform mean flow. An application of the improved formulation coupled with Particular Dirichlet-to-Neumann operators (PDtN) has been presented to describe the acoustic field inside two-dimensional infinite ducts in a uniform mean flow. The numerical results are compared to analytical, conventional BEM and Finite Element Method (FEM) formulations.

Reflection coefficient of a dominant mode in a pentafurcated duct

This paper presents the diffraction of acoustic plane wave through a semi-infinite soft duct. The soft plates are enclosed symmetrically inside an infinite duct with hard boundary conditions. We fully emphasize the analysis of the behavior of the reflected field amplitude for the pentafurcated duct by applying the straightforward mode-matching technique. We also depict some graphical representations by determining the reflected field amplitude for various dimensions of the pentafurcated duct. We also present comparisons of results with the existing results of a hard pentafurcated duct problem. This research work will assist the future researchers in reducing noise effects in complicated devices and exhaust systems.

An improved axisymmetric convected boundary element method formulation with uniform flow

This paper presents an improved form of the convected Boundary Element Method (BEM) for axisymmetric problems in a subsonic uniform flow. The proposed formulation based on the axisymmetric Convected Helmholtz Equation (CHE) and its fundamental solution that describes the sound radiation from a monopole source. The variables in the new axisymmetric boundary integral formulation can be expressed explicitly in terms of the acoustic pressure and its particular normal derivative. Also, the constant coefficients are expressed only in terms of the axisymmetric convected Green's function and its convected normal derivative. The particular and convected derivatives reduce the flow effects of the normal and the flow direction derivatives incorporated in the conventional convected boundary integral formulas. The advanced form of the axisymmetric boundary integral representation with flow is a similar form of the axisymmetric boundary element method without flow. Precisely, the two new operators significantly reduce the computational burden of the classical BEM and then becomes the CPU time of BEM without flow. The formula is verified comparing to both analytical and Finite Element Methods (FEM) of an axisymmetric infinite rigid duct in a subsonic uniform flow.

Scattering matrices in non-uniformly lined ducts

Sudden area expansion and sudden area contraction in an infinitely long duct with discontinuous locally reacting lining are defined by respective mixed boundary value problems. In the absence of a sudden area change, a separate problem with an infinite duct having bifid lining on its wall is described. Introducing Fourier transform along the duct axis boundary value problems is solved by the well-known Wiener–Hopf technique, and then, corresponding scattering matrices are constructed. To show the proper use of scattering matrices in the case of several discontinuities and also validation and comparison purposes, transmitted field in a duct with an inserted expansion chamber whose walls are treated by acoustically absorbent material is derived by the help of the relevant scattering matrices. A perfect agreement is observed when the transmitted fields are compared numerically with a similar work exists in the literature.

Wave scattering by soft–hard three spaced waveguide

In this work we consider the sound radiation of a fundamental plane wave mode from a semi-infinite soft–hard duct. This duct is symmetrically located inside an infinite duct. This infinite waveguide consist of soft and hard plates. The whole system constitutes a three spaced waveguide. A closed form solution is obtained by using eigenfunction expansion matching method. This particular problem has been solved previously by Rawlins in closed form but without numerical work. Here the numerical results for reflection coefficient are given when the lowest mode propagates out from the semi-infinite duct. At the end we give comparison to both methods.

Discontinuous Galerkin method for the computation of acoustic modes in lined flow ducts with rigid splices

A discontinuous Galerkin method for the discretisation of the linearised Euler equations is derived for the computation of acoustic modes in an infinite duct of arbitrary geometric shape in the presence of axial sheared flow. Validations are performed with analytical solutions in the case of a uniformly lined duct and comparisons are done with the literature in the case of a lined cylindrical duct with rigid splices. We take advantage of the discontinuous formulation to study the behaviour of the acoustic eigenfunctions in the vicinity of the liner-splice transition where singular behaviours of the acoustic perturbation are found.

Stability of contact discontinuity for steady Euler system in infinite duct

In this paper, we prove structural stability of contact discontinuities for full Euler system.

Reflection coefficient of a dominant mode in a trifurcated duct of soft walls in the presence of mean flow

The problem of sound radiation from a soft semi-infinite duct is analyzed. This duct is symmetrically placed inside another soft but infinite duct. The problem is solved when the whole fluid region inside these ducts is in motion with a constant fluid velocity. Mathematical formulation is made. The resulting problem is solved by employing integral transform and the Jones’ method based on the Wiener-Hopf technique. The kernel functions for the problem have been factorized. Graphs are sketched and analyzed. It is noticed that through specific values of the involved parameters in the solution, the unwanted noise can be abated.

CAA-based optimisation of a sensor array to characterize aft and forward fan noise in a 3D nacelle

In the context of the on-going European project JTI-SFWA, the modal acoustic content inside a realistic fan duct, designed by Dassault-Aviation, must be characterized experimentally. The objective is to use Onera's CAA solver sAbrinA-V0 to design a non-intrusive sensor distribution inside the nacelle, upstream/downstream the fan plane, accounting for the heterogeneous mean flow and the complex internal geometry. By computing the propagation of several interacting modes in the duct, a region in which the acoustic pattern is close to the infinite duct analytical solution was characterized. Then several 3D distributions of pressure sensors were tested in this zone. Radial and azimuthal sensor arrays were located at the nacelle wall, whereas a radial sensor array was placed along the bifurcation leading edge, without additional intrusivity. The response of the 3D arrays to a given acoustic field, either analytical or propagated through CAA, was projected on an analytical modal basis. For several array configurations, this projection clearly showed the emergence of the modal injected content. From these simulations, the most efficient sensor distribution will be used to equip an experimental fan noise simulator, and the modal projection process will be applied on measurements to evaluate the actual modal content in the duct.