Cut-on Modes Research Articles

Generation of a single cylindrical duct mode using a mode synthesiser

The paper presents a method to generate a single user-selected waveguide cut-on mode by means of a mode synthesiser consisting of a set of point sources, each of them modelled by a thin pipe outlet and driven by a separate loudspeaker. Analysis of sound propagation in duct-like cylindrical facilities frequently needs considering the presence of higher cut-on modes together with mode reflection/coupling phenomena occurring on the duct outlet, impedance variations, etc. With increasing reduced frequency (Helmholtz number), description of the field becomes more and more complex. To avoid this complexity, at least in the case of tests carried out in laboratory conditions, the simplest approach seems to be able to generate a single cut-on mode by means of a properly constructed system, because to analyse the multi-mode wave, the contribution to the overall field from individual higher order modes must be established in advance. Based on in-duct field properties, the authors designed a mode synthesiser composed of N⩽13 point sources which was constructed and integrated into the measurement duct. Two methods were used to approach the problem of generating a single higher mode. In the first one, the amplitudes of the individual point sources were calculated by means of the in-duct Green’s function. In the second method, amplitudes of individual sources were calibrated based on sound pressure measurement data taken on the duct cross-section with only one source operating at a time. These measurements, fully automated at a given cross-section, were carried out for some reduced frequencies allowing for propagation of at least four cut-on modes. The advantages and disadvantages of the two methods and the causes of discrepancies between theoretical predictions and the results obtained were analysed.

Cascade trailing-edge noise modeling using a mode-matching technique and the edge-dipole theory

An original analytical approach is proposed to model the broadband trailing-edge noise produced by high-solidity outlet guide vanes in an axial turbomachine. The model is formulated in the frequency domain and first in two dimensions for a preliminary assessment of the method. In a first step the trailing-edge noise sources of a single vane are shown to be equivalent to the onset of a so-called edge dipole, the direct field of which is expanded in a series of plane-wave modes. A criterion for the distance of the dipole to the trailing-edge and a scaling of its amplitude is defined to yield a robust model. In a second step the diffraction of each plane-wave mode is derived considering the cascade as an array of bifurcated waveguides and using a mode-matching technique. The cascade response is finally synthesized by summing the diffracted fields of all cut-on modes to yield upstream and downstream sound power spectral densities. The obtained spectral shapes are physically consistent and the present results show that upstream radiation is typically 3dB higher than downstream radiation, which has been experimentally observed previously. Even though the trailing-edge noise sources are not vane-to-vane correlated their radiation is strongly determined by a cascade effect that consequently must be accounted for. The interest of the approach is that it can be extended to a three-dimensional annular configuration without resorting to a strip theory approach. As such it is a promising and versatile alternative to previously published methods.

A Couple of Savonius Wind Mill and Centrifugal Reaction Pump as a Wind Energy Water Pump System

Wind energy is one of renewable energy which became the center of attention and grew rapidly. Especially for remote area, wind energy is one of alternative dependable energy sources which can be used for water lifting. Savonius wind mill can be a solution for decentralized power generation, with low cost and reduced environmental impacts. This study observed a couple of Savonius wind mill and a centrifugal reaction pump which used as a wind energy water pump system (WEWPS). The Savonius wind mill has 0.8 m diameter, 1.0 m height, 2 stages, 2 buckets in every stage and 0.1 m width of the buckets spacing. The centrifugal reaction pump with a T-junction has 0.9 m diameter, 18.5 mm (0.5 inch) nominal inner diameter for both vertical and horizontal pipes. Both arms of T-junction have similar dimension and functioned as impeller. The pump, which is suitable for low shaft speed, is modified by replacing the couple of fixed orifices and sliding orifice with double U pipe configuration to restrict the air entering the pipe channel, either while stopped or rotated. The transmission used to connect the shafts of both devices is the couple of belt and pulleys with transmission ratio 7:4. WEWPS started pumping the water at 4.5 m/s wind speed and total head 1.5 m. The wind speed produced low shaft speed 120 rpm, shaft power 13 Watt and through the transmission driving the pump into cut-on mode.

On Certain Practical Issues Relating to Construction of the In-duct Single Mode Synthesizer

Abstract It is convenient to have a device and a method of generating single cut-on modes in cylindrical hard-walled waveguides or at least in laboratory models of such systems. This allows to examine, among other things, properties of various active and/or passive elements inserted in a cylindrical duct by testing them in conditions when the incident (input) wave comprises only one cut-on mode and determining the reflection and transmission coefficients for single selected incident modes. As it has been already demonstrated by the present authors, it is possible to generate single cut-on modes in a circular duct using a small (although increasing with mode order) number of acoustic monopoles arranged properly on a duct cross-section and driven with appropriate acoustic volume amplitudes and phases. Laboratory models of such sources are proposed in this paper and results of tests verifying their directional properties are presented. The other technical issue relating to practical utilization of the proposed method is the possible error introduced by the apparatus used for scanning the acoustic field inside the duct model. It is shown that insertion of the measuring probe changes the total energy radiated into the free space only by a fraction of a decibel.

Noise Sources for Duct Acoustics Simulations: Broadband Noise and Tones

Although simulating acoustic tones propagating in ducts or waveguides is relatively well established, broadband noise is composed of a wide range of frequencies, and at each frequency, the acoustic energy is generally distributed over all the cut-on modes of the duct. Simulating the propagation and absorption of broadband noise is further complicated by the random nature of the sound field. This paper presents a computational method to generate truly broadband, multimode noise in a time-domain simulation. This method is equally applicable to the generation of deterministic tones. For broadband noise, we devise a stochastic source that is tailored to radiate a sound field characterized by the cross-correlation between mode amplitudes. This stochastic source allows to compute in a single time-domain simulation the complete spectrum of the broadband multimode sound field. Unlike other methods proposed for predicting broadband noise, there is no need to perform several simulations for different realizations. Results are presented for broadband noise alone as well as for a combination of tones and broadband noise.

Broadband Wave Propagation from an Aeroengine Duct

Accurate and efficient computation of multimode/broadband noise propagating into and radiating out of a duct has both fundamental and practical interests, particularly for broadband turbofan noise problems. In this work, a new complex formulation of the linearized Euler equations is proposed in the time domain for accurate and efficient multimode noise computations via a single-mode summation approach. The approach achieves incoherent results by summation of individual predictions from excitation modes at all discrete frequencies. The solutions are obtained using high-order computational aeroacoustics algorithms. An advantage of the proposed formulation is that the number of computations is equal to the cuton mode number at the highest frequency regardless of the frequency sampling rate. Furthermore, the computation cost only increases marginally despite a doubling of the governing equations in real space. Tests show that the method could potentially achieve significant computational cost savings. This approach is compared with a time-domain random phase approach with which incoherent results are obtained by averaging the results from repeated simulations with random phases added to the wave inputs in each computation. Examples including a generic bypass engine duct are included to illustrate the reliability and efficiency of the method.

Comparative study on mode-identification algorithms using a phased-array system in a rectangular duct

To identify multiple acoustic duct modes, conventional beam-forming, CLEAN as well as L2 (i.e. pseudo-inverse) and L1 generalized-inverse beam-forming are applied to phased-array pressure data. A tone signal of a prescribed mode or broadband signal is generated upstream of a curved rectangular duct, and acoustic fields formed in both upstream and downstream stations of the test section are measured with identical wall-mounted microphone arrays. Sound-power distributions of several horizontal and vertical modes including upstream- and downstream-propagating waves can be identified with phased-array techniques, and the results are compared among the four approaches. The comparisons using synthetic data demonstrate that the L2 generalized-inverse algorithm can sufficiently suppress undesirable noise levels and detect amplitude distributions accurately in over-determined cases (i.e. the number of microphones is more than the number of cut-on modes) with minimum computational cost. As the number of cut-on modes exceeds the number of microphones (i.e. under-determined problems), the L1 algorithm is necessary to retain better accuracy. The comparison using test data acquired in the curved duct test rig (CDTR) at NASA Langley Research Center suggests that the L1/L2 generalized-inverse approach as well as CLEAN can improve the dynamic range of the detected mode by as much as 10dB relative to conventional beam-forming even with mean flow of M=0.5.

The propagation of acoustic waves in a slowly varying duct with radially sheared axial mean flow

We consider the propagation of acoustic waves along a cylindrical duct carrying radially sheared axial mean flow, in which the duct radius is allowed to vary slowly along the axis. In previous work [A.J. Cooper & N. Peake, Journal of Fluid Mechanics 445 (2001) 207–234.] radially sheared axial mean flow with nonzero swirl in a slowly varying duct was considered, but in this paper we set the swirl to zero, thereby allowing simplification of the calculations of both the mean and unsteady flows. In this approach the acoustic wavenumber and corresponding eigenfunction are determined locally, while the wave amplitude is found by solving an evolution equation along the duct. Sample results are presented, including a case in which, perhaps surprisingly, the number of cut-on modes increases as the duct radius decreases.

Numerical investigation of flutter stability in subsonic space turbine blisk with emphasis on cut-on/cut-off modes and interblade phase angles

The so-called blisks, i.e. integrally bladed disks, are characterized by very low viscous material damping and make the flutter prediction much more critical. In that framework, a two-dimensional numerical study of a space turbine blisk featuring complex deformation of blades and high eigenfrequency (>40kHz) is performed. The simulations are based on unsteady Reynolds Averaged Navier Stokes computations linearized in the frequency domain and consist in the superposition of an unsteady linear (in time) pressure field, generated by a harmonic perturbation, upon a steady nonlinear (in space) flow. The aerodynamic damping coefficient is calculated over a range of nodal diameters, and the blades are predicted aeroelastically stable. However, violent changes occur and are rather critical since sudden and large deviations in stability appear. In that context, the nature of the waves propagating from the cascade are evaluated. Such an approach provides fundamental knowledge about the perturbations which can either propagate to the far-field (cut-on mode) or decay (cut-off mode). It is expected that the ability of the flow to damp or to amplify the blade motion is strongly affected by the way unsteady perturbations are transferred from the cascade to the far-field. The nature of the waves are first assessed from the aforementioned linearized results, then they are evaluated analytically and finally compared. A good agreement is found despite the strong assumptions of the analytical model. The results show a clear correlation between the cut-on/cut-off conditions and stability. The least stable configuration corresponds to cut-off mode at the inlet and no wave at the outlet. Without outgoing waves from the cascade, the blade is prone to be less stable: the energy from the blades vibration is necessarily dissipated or sent out by the cascade.

Open End Correction for Arbitrary Mode Propagating in a Cylindrical Acoustic Waveguide

The paper presents a method of theoretical derivation and numerical calculation of the open-end correction coefficient for an arbitrary cut-on mode propagating in acoustic waveguide. Actually, the so-called open-end correction coefficient of acoustic tube, frequently discussed in literature, refers to specific conditions, when the wave heading the outlet is the plane wave. It follows from the fact that the plane wave is a commonly applied approximation when considering phenomena in duct-like devices or systems (tubes, musical instruments, heating or ventilation systems). The aim of the paper is to extend the concept of the open-end correction on the so-called higher Bessel modes, that under some conditions can also propagate in a duct. Theoretical results, forming the basis for numerical calculations, were obtained by considering diffraction at the duct end and applying the Wiener–Hopf factorization method. As a result, the formula for the acoustic field inside the duct was derived. For each Bessel mode present in the incident wave the reflected wave is composed of all cut-on modes of the same circumferential order. Each mode present in the reflected wave is characterized by the complex reflection/coupling coefficient, argument of which describes phase change at the duct end and therefore the open-end correction coefficient can be attributed to each coupled pair of modes.

INVESTIGATION OF MODE SCATTERING BY HARD STRIPS IN LINED DUCTS USING COLLOCATION

When mounting acoustic liners inside the inlet duct of aircraft jet engines, hard strips exist in-between liner sections in order to hold them in place. It has been generally acknowledged, from experiments, that the existence of hard strips inside lined ducts affects the attenuation behaviour of the duct. The acoustic energy is scattered and rearranged among different modes so that it might be transferred into modes which are less attenuated by the liner. Moreover, cut-off modes may scatter into cut-on modes. In this paper, flow is included in the locally reacting liner case, which is more interesting to the aeronautics applications. Moreover, the duct is made finite and connected to two semi-infinite hard inlet and outlet ducts. By using mode matching, it is possible to input any mode at the inlet side and to study the modes on the outlet side. Comparisons are made between different hard strip cases and the splice-less liner case.

An indirect method for the characterization of locally reacting liners

An indirect technique for educing the homogenized acoustic impedance of a liner mounted on the wall of a barrel is presented. It is based on measurements and computational simulations of the multimodal scattering matrix of this lined duct. Measurements are performed with a multisource method and the use of an anechoic duct termination. The numerical computation of the scattering matrix relies on a finite element model, and assume that the duct is axisymmetric and uniformly covered by a locally reacting material. The impedance is educed by minimizing the difference between the theoretical and experimental acoustic power dissipations, which are deduced from the scattering matrix. The source is an incoming pressure field generated at one end of the duct only and composed of all cut-on modes. This technique was tested on a cylindrical barrel whose wall was partially lined with a realistic, locally reacting material made of honeycomb cells and a perforated facing sheet. Results for the acoustic impedance are compared with those deduced from semiempirical models and the experimental two microphone method. The best agreement with the indirect method is found with the semiempirical impedance results when the difference between the acoustic power dissipated by the actual lined barrel and the reference barrel is chosen as the cost function of the minimizing procedure.

Multimode radiation from an unflanged, semi-infinite circular duct with uniform flow

Multimode sound radiation from an unflanged, semi-infinite, rigid-walled circular duct with uniform subsonic mean flow everywhere is investigated theoretically. The multimode directivity depends on the amplitude and directivity function of each individual cut-on mode. The amplitude of each mode is expressed as a function of cut-on ratio for a uniform distribution of incoherent monopoles, a uniform distribution of incoherent axial dipoles, and for equal power per mode. The directivity function of each mode is obtained by applying a Lorentz transformation to the zero-flow directivity function, which is given by a Wiener-Hopf solution. This exact numerical result is compared to an analytic solution, valid in the high-frequency limit, for multimode directivity with uniform flow. The high-frequency asymptotic solution is derived assuming total transmission of power at the open end of the duct, and gives the multimode directivity function with flow in the forward arc for a general family of mode amplitude distribution functions. At high frequencies the agreement between the exact and asymptotic solutions is shown to be excellent.

Time-domain computation of multimode propagation in an aeroengine duct

Accurate computation of multimode propagation in and radiation from a turbofan duct is important for aircraft noise prediction. A formulation of linearised Euler equations (LEE) is proposed for efficient computation. Its solutions are sought in time-domain using high-order computational aeroacoustic (CAA) algorithms. An advantage of the proposed method is that the number of computation is the cut-on mode number at the highest frequency regardless frequency sample interval. Furthermore the computing cost only increases marginally despite a doubling of the governing equations in real space. Tests show that the method could potentially achieve computing cost saving. Examples including a generic bypass engine duct are given to illustrate the reliability and efficiency of the method.

Reprint of: Time-Domain Computation of Multimode Propagation in an Aeroengine Duct

Accurate computation of multimode propagation in and radiation from a turbofan duct is important for aircraft noise prediction. A formulation of linearised Euler equations (LEE) is proposed for efficient computation. Its solutions are sought in time-domain using high-order computational aeroacoustic (CAA) algorithms. An advantage of the proposed method is that the number of computation is the cut-on mode number at the highest frequency regardless frequency sample interval. Furthermore the computing cost only increases marginally despite a doubling of the governing equations in real space. Tests show that the method could potentially achieve computing cost saving. Examples including a generic bypass engine duct are given to illustrate the reliability and efficiency of the method.

Scattering of acoustic duct modes by axial liner splices

Recent engine test data and results of computational analysis show that the engine inlet acoustic liner splices have a significant impact on aircraft flight noise certification and cabin noise levels. The phenomenon of scattering of acoustic duct modes by axial liner splices is investigated. Previous studies, invariably, follow the frequency-domain approach. The present study, however, uses the time-domain approach. It is demonstrated that time-domain computation yields results that are in close agreement with frequency-domain results. The scattering phenomenon under consideration is very complex. This study concentrates on the effects of four parameters. They are the width of the splices, the frequency of the incident duct mode, the number of splices and the length of splices. Based on the computed results, the conditions under which scattered wave modes would significantly increase the intensity of transmitted waves are identified. It is also found that surface scattering by liner splices has the tendency to distribute energy equally to all the cut-on scattered azimuthal modes. On the other hand, for each scattered azimuthal mode, the high-order cut-on radial mode, generally, has the highest intensity. Moreover, scattering by liner splices is a local phenomenon. It is confined primarily to an area of the duct adjacent to the junction between the hard wall near the fan face and the spliced liner.

Mode detection in turbofan inlets from near field sensor arrays

Knowledge of the modal content of the sound field radiated from a turbofan inlet is important for source characterization and for helping to determine noise generation mechanisms in the engine. An inverse technique for determining the mode amplitudes at the duct outlet is proposed using pressure measurements made in the near field. The radiated sound pressure from a duct is modeled by directivity patterns of cut-on modes in the near field using a model based on the Kirchhoff approximation for flanged ducts with no flow. The resulting system of equations is ill posed and it is shown that the presence of modes with eigenvalues close to a cutoff frequency results in a poorly conditioned directivity matrix. An analysis of the conditioning of this directivity matrix is carried out to assess the inversion robustness and accuracy. A physical interpretation of the singular value decomposition is given and allows us to understand the issues of ill conditioning as well as the detection performance of the radiated sound field by a given sensor array.

Multiload procedure to measure the acoustic scattering matrix of a duct discontinuity for higher order mode propagation conditions

A procedure for measuring the acoustic scattering matrix coefficients of a duct discontinuity for higher order acoustic duct mode propagation conditions is described and tested. The technique requires measurement of pressure waves per mode coming in and out of the discontinuity. Assuming N cut-on modes, the (2N)2 scattering matrix coefficients are determined after repeating the experiment for N linearly independent pressure distributions for at least two load configurations. Experiments were conducted for a straight duct and a reactive chamber. A good agreement was found between experiment and theory except near cut-off frequencies. The overdetermination method based on four loads was shown to improve the results. An analytical simulation of the experiment was developed to compute the influence on the [S] calculation of an error in temperature and total modal pressure assumed to be representative of a real measurement situation. This simulation with a discussion explains the discrepancies between experiment and theory. The test with the chamber shows that the load method fails as expected in determining the coefficients associated to the wave coming in the discontinuity from the open end side because of the property of the middle duct to filter modes making the results very sensitive to uncertainties.

Approximations for the scattered field potential from higher mode transmission in rectangular apertures

This paper considers the wave fields that result when a plane wave impinges at an arbitrary angle on a rectangular aperture in a rigid, thick wall. A nondimensional form of a prior Fourier transform solution of this problem is derived, from which it is more easy to appreciate the relationship between the physical attributes of the aperture and incoming wave and the resultant acoustic fields. The scattered field from the aperture is examined in detail, in particular the modal contributions to the driving function for the amplitude of the velocity potential. Although the full scattered field contains both modal sum and modal coupling effects, it is shown that neglect of the modal coupling effects introduces minimal error to the solution in certain situations. An approximate analytical solution to the uncoupled analysis is then developed, which is accurate provided that the aperture is acoustically large, such that there are several cut-on modes within the aperture. The full nature of the scattered field can be easily interpreted from the nondimensional, analytical solution. In particular, an error analysis is given from which one can determine the required number of modes in a solution for given accuracy.

Numerical Simulations of Fan Interaction Noise Using a Hybrid Approach

A source-to-far-field computation procedure aimed at predicting the noise generated by rotor-stator fan interactions is presented here. Interaction noise sources are assessed using a three-dimensional Reynolds-averaged Navier-Stokes (RANS) code. A modal expansion is applied to the computational fluid dynamics (CFD) data, and a source model is used to link the CFD to an Euler code simulating the acoustic propagation. Far-field noise is classically obtained by means of a Kirchhoff integral. The source model is detailed and validated over a selected benchmark. Then the hybrid method is applied to a turbofan model, tested in the German-Dutch Wind Tunnel. Simulations are compared to boundary element method computations previously performed and used here to validate the no-flow solution. Interacting cut-on mode amplitude deduced from CFD output postprocessing is adjusted to fit in-duct measurements on the outer wall. The predicted radiation field is related to the directivity patterns measured at several axial positions upstream of the fan inlet and fairly good agreement is found using the RANS/Euler/Kirchhoff procedure.