Hard-walled Duct Research Articles

Duct Effects on Acoustic Source Radiation

This paper presents a detailed numerical investigation of the effects of a semi-infinite hard-walled duct on the radiation due to point dipole sources rotating around the duct axis. In particular, this study is concerned with the effect of source position relative to the duct open end and its comparison with radiation in a free field. This study was motivated by the increasing tendency of ducted propellers and fans to be located close to the open end, relative to the acoustic wavelength. In this study, the conditions under which duct effects on source radiation may be neglected are established.

Fundamental noise characterization of a ducted propeller in hover

Unmanned aerial vehicles (UAVs) are currently being used for reconnaissance missions, tactical surveillance, and infrastructure inspection. When legislation allows it, these devices will provide additional services close to inhabited areas, which could lead to noise complaints. On most UAVs, the propellers are the dominant source of noise. As a result, researchers have studied the impact of propeller shape and blade count on noise. Much of this work, however, has focused on isolated propellers. While different UAV concepts are equipped with ducts for aerodynamic and protection reasons, few studies focus on the acoustic benefit of ducts, as is achieved, for example, on turbofan aircraft. The objectives of this paper are: first, to simulate the noise radiation of a UAV propeller in static conditions based on its location in a hard wall duct; second, to analyze the contribution of the different acoustic source components (i.e., thrust, torque, and thickness); and third, to validate the approach with experiments conducted in the NASA Small Hover Anechoic Chamber over a range of propeller rotation rates. In this experiment, it is shown that the best attenuation is achieved when the propeller is centered, axially, in the duct because of interference between upstream and downstream radiated waves.

An application of the multiple-scales theory to acoustic double transition in ducts with flow

A theoretical model based on the multiple-scales technique is proposed to study the propagation of an acoustic mode that encounters multiple turning points in a hardwalled duct with slowly varying geometry and in the presence of mean flow. The case where two of such turning points arise in the duct due to radius variations is studied for both cut-off/cut-on/cut-off and cut-on/cut-off/cut-on transitions. The acoustic field through the whole duct is determined by writing the solution for a single transition around each turning point and by matching the two solutions. These two turning points provide a mechanism by which the upstream and downstream travelling waves interact and modify the overall energy balance. For the cut-off/cut-on/cut-off case, one of the possible consequences of such an interaction is that the mode is trapped by these turning points, which can lead to acoustic resonances only controlled by the geometry. As for the cut-on/cut-off/cut-on case, this interaction makes the mode not to vanish at the first transition but to tunnel through the cut-off region. Several test cases, including at least two transitions, are studied to validate the model and understand its limitations with respect to two other propagation models (finite difference and multimodal methods). The agreement between the different methods is excellent in most cases. However, when the modal scattering becomes important, the model starts to show some deviations. Nevertheless, it allows, in a preliminary design of an engine, to provide an estimate of the frequencies where an amplification occurs and thus to avoid creating resonators that can have dire implications on the sound emission.

A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions. An alternative approach

A numerical technique is devised to solve a problem of the steady laminar fluid motion in a straight plane hard-walled duct with two local axisymmetric rectangular constrictions. It uses the stream function, the vorticity and the pressure as the basic variables, has the second order of accuracy in the spatial and the first order of accuracy in the temporal coordinates, provides high stability of a solution, and needs significantly less computational time to obtain a result compared to appropriate techniques available in a scientific literature. The technique consists in: a) introducing the stream function and the vorticity, and subsequent transiting from the non-dimensional governing relations for the fluid velocity and the pressure to the corresponding non-dimensional relations for the stream function, the vorticity and the pressure; b) deriving their discrete counterparts in the nodes of the chosen space-time computational grid; c) integrating the systems of linear algebraic equations obtained after making the discretization. The discretization is based on applying appropriate differencing schemes to the terms of the equations for the basic variables. These are the two-point temporal onward difference for the unsteady term of the vorticity equation, as well as the two-point backward differences (for its convective term) and the five-point approximations (for its diffusive term and for the Poisson’s equations for the stream function and the pressure) in the axial and cross-flow coordinates. As for the velocity components, the appropriate central differences are applied to discretize their expressions. The above-mentioned systems of linear algebraic equations for the stream function and the pressure are integrated by the iterative successive over-relaxation method. The algebraic relation for the vorticity does not need application of any method to be solved, because it is a computational scheme to find this quantity based on the known magnitudes computed at the previous instant of time.

On the flow-acoustic coupling of fan blades with over-the-rotor liner

Over-the-rotor liner exhibits the potential to further attenuate turbofan noise, but the physics involved remain to be explored. In this paper, a three-dimensional coupled singularity method is proposed to investigate the flow-acoustic coupling effects of axially overlapping annular rotor and finite-length liner in subsonic flow. The formulation adopts the orthogonal basis expansion of the generated disturbances in terms of the hard-walled duct modes. The sound scatterings at the rotor and the liner are then characterized, respectively, by the underdetermined dipole and monopole distributions. We derive a simultaneous solution to the coupled unsteady rotor and liner responses, which ensures that the resultant perturbed field satisfies both the impermeable boundary condition on the blade surfaces and the impedance boundary condition on the lined wall. The effect of a perforated porous-material liner on the wake–rotor interaction tones is investigated. The analysis reveals that for the sound field of varying mode and frequency characteristics, moving the inlet liner to the over-the-rotor location generally leads to limited loss or even an increase of upstream sound absorption, along with additional acoustic benefits in the aft duct. The flow-acoustic coupling between the axially overlapping rotor and liner is shown to alleviate significantly the unsteady blade loading and meanwhile intensify the fluid particle oscillation through the acoustically treated wall. Sound source reduction and sound dissipation enhancement are thus identified as the governing noise attenuation mechanisms. Finally, we extend the analysis to provide insights into the effectiveness of over-the-rotor acoustic treatment with shortened axial length.

A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions

A second-order numerical technique is developed to study the steady laminar fluid motion in a straight two-dimensional hard-walled duct with two axisymmetric rectangular constrictions. In this technique, the governing relations are solved via deriving their integral analogs, performing a discretization of these analogs, simplifying the obtained (after making the discretization) coupled nonlinear algebraic equations, and the final solution of the resulting (after making the simplification) uncoupled linear ones. The discretization consists of the spatial and temporal parts. The first of them is performed with the use of the TVD-scheme and a two-point scheme of discretization of the spatial derivatives, whereas the second one is made on the basis of the implicit three-point asymmetric backward differencing scheme. The above-noted uncoupled linear algebraic equations are solved by an appropriate iterative method, which uses the deferred correction implementation technique and the technique of conjugate gradients, as well as the solvers ICCG and Bi-CGSTAB.

Flow modelling in a straight rigid-walled duct with two rectangular axisymmetric narrowings. Part 2. An alternative approach

A second-order technique is suggested to study fluid motion in a two-dimensional hard-walled duct with two abrupt constrictions. In this technique, the governing relationships are integrated via their rewriting in a non-dimensional form, deriving their integral analogues, performing a discretization of the derived integral relationships, simplifying the obtained (after making the discretization) coupled non-linear algebraic equations, and final solving the resulting (after making the simplification) uncoupled linear ones. The discretization consists of the spatial and temporal parts. The first of them is performed with the use of the total variation diminishing scheme and the two-point scheme of discretization of the spatial derivatives, whereas the second one is made on the basis of the implicit three-point non-symmetric backward differencing scheme. The noted uncoupled linear algebraic equations are solved by an appropriate iterative method.

Flow modelling in a straight hard-walled duct with two rectangular axisymmetric narrowings

A method for modelling the flow in a rigid-walled duct with two narrowings has been developed. It has the second order of accuracy in the spatial and the first order of accuracy in the temporal coordinates, provides high stability of the solution, and compared to the similar methods requires much less computational time to obtain a result. According to the method, the stream function and the vorticity are introduced initially, and consequently the transition from the governing equations, as well as the initial and boundary conditions to the proper relationships for the introduced variables is performed. The obtained relationships are rewritten in a non-dimensional form. After that a computational domain and a uniform computational mesh are chosen, and the corresponding discretization of the non-dimensional relationships is performed. Finally, the linear algebraic equations obtained as a result of the discretization are solved.

A simple approach for evaluation of cut-on condition of higher order modes in inhomogeneous uniform acoustic waveguides

Uniform acoustic waveguides in initially quiescent inhomogeneous media are governed by Bergmann’s wave equation. The transverse modes are independent of the ambient gradient, but there is a void of papers on the cut-on conditions of the higher order axial modes. This paper presents a sufficient cut-on condition and discusses its application to hard-walled ducts in which the mean temperature varies linearly and exponentially along the duct axis.

Experimental investigation of a new two-microphone method for the determination of broadband noise radiation from ducts

This paper experimentally investigates a new technique for measuring the modal amplitude distribution, sound power transmission and radiation, and far field directivity of the broadband noise from hard walled ducts. The innovative aspect of this method is that it only requires the measurements of the two-point complex coherence function between the acoustic pressures at two closely spaced points on the duct wall. This method is therefore very useful when direct measurements of sound power and directivity are not possible. This paper describes detailed measurements of the sound power spectrum and coherence function from a hard walled circular duct excited at one end by a diffuse sound field. The other open end is terminated within an anechoic chamber with which to measure the radiated sound field at 11 microphones distributed over a polar arc. Measurements of the complex coherence were made at the duct and used to infer the sound power spectrum and far field directivity. This paper demonstrates generally good agreement between direct measurements of sound power and directivity and those inferred from the coherence function. The method is restricted to broadband noise in large ducts in the frequency range where many modes are able to propagate and the modal amplitudes are mutually uncorrelated.

A High-Order Finite Element Method for the Linearised Euler Equations

Sound propagation in complex non-uniform mean flows is an important feature of turbofan exhaust noise radiation. The Linearised Euler Equations are able to represent the strong shear layer refraction effects on the sound field, as well as multiple length scales. Frequency domain solvers are suitable for tonal noise and considered a way to avoid linear instabilities, which may occur with time domain solvers. However, the classical Finite Element Method suffers from dispersion error and high memory requirements. These shortcomings are particularly critical for high frequencies and for the Linearised Euler Equations, which involve up to five unknowns. In this paper, a high-order Finite Element Method is used to solve the Linearised Euler Equations in the frequency domain in order to overcome those issues. The model involves high-order polynomial shape functions, unstructured triangular meshes, numerical stabilisation and Perfectly Matched Layers. The acoustic radiation from a straight circular semi-infinite hard-wall duct with several mean flow configurations is computed. Comparisons with analytic solutions demonstrate the method accuracy. The acoustic and vorticity waves are well represented, as well as the refraction of the sound field across the jet shear layer. The high-order approach allows to use coarse meshes, while maintaining a sufficient accuracy. The benefits in terms of memory requirements are significant when compared to standard low-order Finite Element Method.

Multi-Modal Acoustic Flow Decomposition Examined in a Hard Walled Cylindrical Duct

Abstract Flow fields could be of great interest in the study of sound propagation in aeroengines. For ducts with rigid boundaries, the fluid-resonant category may contribute significantly to unwanted noise. An understanding of the multi-modal propagation of acoustic waves in ducts is of practical interest for use in the control of noise in, for example, aero-engines, automotive exhaust and heating or ventilation systems. The purpose of our experiments was to test the acoustic energy transmission of duct modes based on studies carried out by the sound intensity technique. Sound intensity patterns in circular duct are discussed of modal energy analysis with particular reference to proper orthogonal decomposition and dynamic mode decomposition. The authors try to justify some advantages of the sound intensity experimental research in this area. In the paper, the wide-band sound signal propagated from source approximated with loudspeaker in hard-walled duct is imaged using a sound intensity - based approach. For a simple duct geometry, the sound intensity field is examined visually and by performing a modal decomposition greater insight into the acoustic structures is obtained. The image of sound intensity fields below and above “cut-off” frequency region are found to compare acoustic modes which might resonate in duct.

Acoustically Consistent Investigation of Combustion Instabilities in a Dump Combustor

An acoustically consistent, linear modal analysis-based analytical method is presented to predict the longitudinal and transverse combustion instabilities in a two-dimensional Cartesian dump combustor. At first, rigorous acoustical analysis (without combustion) is performed of two duct configurations with one and two discontinuities in the cross-sectional area. Novel, acoustically consistent jump or matching conditions are developed and applied at the duct cross-sectional interface(s), with distinct forms for the purely axial and nonaxial modes. The effects of uniform and nonuniform mean flows, cross-sectional area ratio, as well as of different types of boundary conditions on the duct acoustic modes are investigated. Acoustic modal frequency predictions are in excellent agreement with the analytical and numerical results of Meissner (“Effect of Cross-Sectional Area Discontinuities in Closed Hard-Walled Ducts on Frequency of Longitudinal Modes,” Archives of Acoustics, Vol. 35, No. 3, 2010, pp. 421–435). In the second part, combustion instabilities of a two-dimensional Cartesian dump combustor are investigated. The instability analysis employs the developed acoustically consistent jump conditions instead of the conventional mass, momentum, and energy balance-based conditions. Effects of the fluctuating heat-release source term in the acoustic wave equation are incorporated directly into the longitudinal wave number, obviating the need for a separate energy matching condition across the flame. A detailed investigation of the parametric space and boundary conditions affecting combustion instabilities is undertaken, and the consistency of the modal analysis with the Rayleigh criterion is explicitly demonstrated. Further, the present approach enables the consideration of arbitrary mean flame shapes in determining the unstable modes. Instabilities are demonstrated for the fundamental longitudinal mode and its harmonics, as well as for the fundamental transverse mode. The effects of the cross-sectional area ratio and flow Mach number on the unstable-mode growth rates are also presented.

Acoustics flow analysis in circular duct using sound intensity and dynamic mode decomposition

Sound intensity generation in hard-walled duct with acoustic flow (no mean-flow) is treated experimentally and shown graphically. In paper, numerous methods of visualization illustrating the vortex flow (2D, 3D) can graphically explain diffraction and scattering phenomena occurring inside the duct and around open end area. Sound intensity investigation in annular duct gives a physical picture of sound waves in any duct mode. In the paper, modal energy analysis are discussed with particular reference to acoustics acoustic orthogonal decomposition (AOD). The image of sound intensity fields before and above "cut-off" frequency region are found to compare acoustic modes which might resonate in duct. The experimental results show also the effects of axial and swirling flow. However acoustic field is extremely complicated, because pressures in non-propagating (cut-off) modes cooperate with the particle velocities in propagating modes, and vice versa. Measurement in cylindrical duct demonstrates also the cut-off phenomenon and the effect of reflection from open end. The aim of experimental study was to obtain information on low Mach number flows in ducts in order to improve physical understanding and validate theoretical CFD and CAA models that still may be improved.

Acoustic Response of a Sinusoidally Perturbed Hard-Walled Duct

Acoustic wave propagation in hard-walled ducts is of interest in many fields including vehicle design, musical instruments acoustics, and architectural and environmental noise-control. For the case of small sinusoidal perturbation of the cross-section, it is possible to derive simple though approximate analytical formulas of its plane wave acoustic reflection and transmission spectral response that resembles the optical situation of uniform Bragg gratings. The proof is given here, starting from the “horn equation” and then exploiting the coupled-modes theory. Examples of the results obtained with these analytical formulas are shown for some sinusoidally perturbed ducts and compared to results obtained through a numerical method, revealing a very good agreement.

Flow and geometry induced scattering of high frequency acoustic duct modes

Cut-on cut-off transition of acoustic modes in hard-walled ducts with irrotational mean flow is well understood for Helmholtz numbers of order unity. Previous finite-element simulations of this phenomenon, however, appear to indicate the possibility of energy scattering into neighbouring modes at moderately large Helmholtz numbers. In this paper, such scattering phenomena are explained and predicted in slowly varying aeroengine ducts using a multiple-scales approach. It is found that, for sufficiently high frequencies, two mechanisms exist whereby energy can be scattered into neighbouring modes by an incident propagating mode. One mechanism occurs only when there is a mean flow inside the duct and induces scattering at significantly lower frequencies than the other mechanism which remains present without mean flow. A coupled system of ordinary differential equations is derived and then solved numerically for a number of example cases to obtain the corresponding transmitted and reflected amplitudes of the scattered modes as well as the overall acoustic pressure field. The theory appears to demonstrate that some exchange of energy between the acoustic and mean flow fields occurs during scattering.

Effect of Cross-Sectional Area Discontinuities in Closed Hard-Walled Ducts on Frequency of Longitudinal Modes

A simple analytical method is developed to estimate frequencies of longitudinal modes in closed hard-walled ducts with discontinuities in a cross-sectional area. The approach adopted is based on a general expression for the acoustic impedance for a plane wave motion in a duct and conditions of impedance continuity at duct discontinuities. Formulae for mode frequencies in a form of transcendental equations were found for one, two and three discontinuities in a duct cross-section. An accuracy of the method was checked by a comparison of analytic predictions with calculation data obtained by use of numerical implementation based on the forced oscillator method with a finite difference algorithm.

Elaboration of a sacttering matrix measurement procedure using the p‐v probe

The scattering matrix which relates traveling waves amplitudes as state variables has been shown to be more attractive than transfer or mobility matrices since it reflects the fundamental duct nature: it gives a more complete description of the transmission, reflection, and conversion properties of the duct. In the University of Technology of Compiègne, an experimental procedure was developed to measure this matrix: a p‐p probe mounted on a setup designed during DUCAT project is used to measure pressures at two cross sections on the both side of the test lined duct, then by using a modal decomposition and separation techniques, the scattering matrix is deduced. In this paper, a method to measure the multimodal scattering matrix based on the use of a p‐v probe getting simultaneously the acoustic pressure and velocity at one section on the both side of the test duct is developed. A comparison of some acoustics values (scattering matrix coefficients, acoustic powersœ) of a hard wall duct straight duct obtained by each technique with the theory is presented to evaluate its advantages and limitations.

Wide forbidden band induced by the interference of different transverse acoustic standing-wave modes

A non-Bragg nature forbidden band is experimentally observed in an axially symmetric hard-walled duct with a periodically varying cross section. Unlike the familiar Bragg ones, the observed bandgap is found to result from the interference of sound wave modes having different transverse standing-wave profiles, the so-called non-Bragg resonance. The experiments also show that the non-Bragg band can be comparably wider than the Bragg one; furthermore, the sound transmission loss within the band can be much more effective, exhibiting the great significance of the non-Bragg resonance in wave propagation in periodic waveguides.

Effects of Cross-Sectional Partitioning on Active Noise Control in Round Ducts

Active noise control (ANC) is particularly useful in hard-walled ducts where plane waves propagate. Higher order mode waves are much more difficult to control. Basic acoustic principles dictate that the cut-on frequency at which higher order modes will first begin to eclipse simple plane waves in a duct will be determined by the cross-sectional diameter of the duct. The lowest frequency for higher order modes will increase as duct diameter decreases. Therefore, the range of frequencies where plane waves dominate will be greater, and effective control using ANC will be better as duct diameter decreases. The result is that somewhat higher frequencies can be controlled with ANC for smaller diameters. If smaller diameters have broader frequency ranges that can be controlled with ANC, perhaps one could extend the frequency range for a large cross section by partitioning it into smaller cross sections using axial vane splitters. This hypothesis was tested by two methods of cross-sectional partitioning. Partitioning was achieved in one design by inserting a smaller duct inside a large duct. In a second design, a cross-shaped splitter was inserted inside the large duct. Summed ANC insertion loss (IL) at low frequencies (≤250 Hz) was at least 16 dB and at least 14 dB at middle frequencies (≥315 Hz). ANC IL results were 1.7 to 2 dB better for the large duct partitioned by a smaller inner duct than the large duct alone (p = 0.0146 for low frequency and p = 0.0333 for middle frequency). ANC insertion loss was 5.6 dB better for the large duct partitioned by a cross-shaped splitter at high frequencies than the large duct alone (p = 0.0003). However, the cross-shaped partition system was 5.8 dB less effective at low frequencies than the large duct ANC IL alone (p < 0.0001).