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The methodologies of Hanson and Zorumski are combined to yield a unified analysis of ducted turbomachinery noise. It is shown that the far-field broad-band and discrete noise spectral components can be expressed in terms of modal cross-spectral matrices and directivity vectors which are derivable from the duct analysis.
An internally supported, flexible duct joint for use in conducting fluids under relatively high pressures in systems where relatively large deflection angles must be accommodated is presented. The joint includes a flexible tubular bellows and an elongated base disposed within the bellows. The base is connected through radiating struts to the bellows near mid-portion and to each of the opposite end portions of the bellows through a pivotal connecting body. A motion-controlling linkage is provided for linking the connecting bodies, whereby angular displacement of the joint is controlled and uniformity in the instantaneous bend radius of the duct is achieved as deflection is imposed.
A theory is developed for evaluating the normal impedance of an orifice connecting a cylindrical cavity to a flow duct. The impedance of a piston velocity source at the orifice is calculated in terms of the cavity modes which are modified by the presence of a vortex flow. The strength of the vortex is related to the mean flow in the duct. Results are presented showing the variation of the normal impedance spectrum with the cavity vortex. The trends of these variations are confirmed by comparison with measurements.
A simplified procedure is presented for analysis of ducts with discretely nonuniform properties. The analysis uses basis functions as the generalized coordinates. The duct eigenfunctions are approximated by finite series of these functions. The emphasis is on solution of the resulting large sparse set of linear equations. Characteristics of sparse matrix algorithms are outlined and some criteria for application are established. Analogies with structural methods are used to illustrate variations which can increase efficiency in generating values for design optimization routines. The effects of basis function selection, number of eigenfunctions and identification and ordering of equations on the sparsity and solution stability are included.
The problem of acoustic wave propagation in a lined duct with non-uniform liner admittance distribution has been analyzed. Two different methods of solution are discussed. Computations are made for a simple example of symmetrically lined two-dimensional duct containing a unit pressure source distribution. The non-uniform admittance variation is assumed to be a homogeneous random function along the direction of wave propagation. The influences of the spatial scale of the admittance variation, the magnitude of the variation and their dependence on mean liner admittance on sound attenuation are presented. The predicted trends agree qualitatively with existing experimental findings.
An approach for an analysis of ducted turbomachinery noise is developed by combining features of the analysis of fan stator noise reported by Hanson (1973) with results obtained by Zorumski (1974). Hanson's analysis is briefly examined and a formula is derived for the far-field pressure due to a dipole pulse. Questions of transmission and reflection are discussed and attention is given to the radiation from ducted dipoles.
Whistler waves are launched from an electric dipole of length L in a large-volume laboratory plasma. With increasing wave amplitude, the radiation pattern narrows and finally forms a duct of diameter approximately equal to L. The ducted waves propagate nearly undamped. The observed nonlinear effects are explained by wave-particle interactions.
An acoustical field theory for axisymmetric, multisectioned lined ducts with uniform flow profiles was combined with a numerical minimization algorithm to predict optimal liner configurations having one, two, and three sections. Source models studied include a point source located on the axis of the duct and rotor/outlet-stator viscous wake interaction effects for a typical research compressor operating at an axial flow Mach number of about 0.4. For this latter source, optimal liners for equipartition-of energy, zero-phase, and least-attenuated-mode source variations were also calculated and compared with exact results. It is found that the potential benefits of liner segmentation for the attenuation of turbomachinery noise is greater than would be predicted from point source results. Furthermore, effective liner design requires precise knowledge of the circumferential and radial modal distributions.
A generalized approximate equation for duct lining sound attenuation is presented. The specification of two parameters, the maximum possible attenuation and the optimum wall acoustic impedance is shown to completely determine the sound attenuation for any acoustic mode at any selected wall impedance. The equation is based on the nearly circular shape of the constant attenuation contours in the wall acoustic impedance plane. For impedances far from the optimum, the equation reduces to Morse's approximate expression. The equation can be used for initial acoustic liner design. Not least important is the illustrative nature of the solutions which provide an understanding of the duct propagation problem usually obscured in the exact calculations. Sample calculations using the approximate attenuation equation show that the peak and the bandwidth of the sound attenuation spectrum can be represented by quite simple functions of the ratio of actual wall acoustic resistance to optimum resistance.
The propagation of plane waves and higher order acoustic modes in a circular multisectioned duct was studied. A unique source array consisting of two concentric rings of sources, providing phase and amplitude control in the radial, as well as circumferential direction, was developed to generate plane waves and both spinning and nonspinning higher order modes. Measurements of attenuation and radial mode shapes were taken with finite length liners inserted between the hard wall sections of an anechoically terminated duct. Materials tested as liners included a glass fiber material and both sintered fiber metals and perforated sheet metals with a honeycomb backing. The fundamental acoustic properties of these materials were studied with emphasis on the attenuation of sound by the liners and the determination of local versus extended reaction behavior for the boundary condition. A search technique was developed to find the complex eigenvalues for a liner under the assumption of a locally reacting boundary condition.
A computer code, based upon a finite element solution algorithm, was developed to solve the governing equations for three-dimensional, reacting boundary region, and constant area ducted flow fields. Effective diffusion coefficients are employed to allow analyses of turbulent, transitional or laminar flows. The code was used to investigate mixing and reacting hydrogen jets injected from multiple orifices, transverse and parallel to a supersonic air stream. Computational results provide a three-dimensional description of velocity, temperature, and species-concentration fields downstream of injection. Experimental data for eight cases covering different injection conditions and geometries were modeled using mixing length theory (MLT). These results were used as a baseline for examining the relative merits of other mixing models. Calculations were made using a two-equation turbulence model (k+d) and comparisons were made between experiment and mixing length theory predictions. The k+d model shows only a slight improvement in predictive capability over MLT. Results of an examination of the effect of tensorial transport coefficients on mass and momentum field distribution are also presented. Solutions demonstrating the ability of the code to model ducted flows and parallel strut injection are presented and discussed.
Recent research which contributes to the prediction of the acoustic impedance of duct liners is reviewed. This review includes the linear and nonlinear properties of sheet and bulk type materials and methods for the measurement of these properties. It also includes the effect of grazing flow on the acoustic properties of materials. Methods for predicting the properties of single or multilayered, point reacting or extended reaction, and flat or curved liners are discussed. Based on this review, methods for predicting the properties of the duct liners which are typically used in aircraft engines are recommended. Some areas of needed research are discussed briefly.
A generalized approximate equation for duct lining sound attenuation is presented. The specification of two parameters, the maximum possible attenuation and the optimum wall acoustic impedance is shown to completely determine the sound attenuation for any acoustic mode at any selected wall impedance. The equation is based on the nearly circular shape of the constant attenuation contours in the wall acoustic impedance plane. For impedances far from the optimum, the equation reduces to Morse's approximate expression. The equation can be used for initial acoustic liner design. Not least important is the illustrative nature of the solutions which provide an understanding of the duct propagation problem usually obscured in the exact calculations. Sample calculations using the approximate attenuation equation show that the peak and the bandwidth of the sound attenuation spectrum can be represented by quite simple functions of the ratio of actual wall acoustic resistance to optimum resistance.
Analytical and experimental investigations of phased linings for a duct containing a uniform mean flow were conducted as part of a program concerned with the design of more efficient acoustic liners for the reduction of noise emission. An analytical technique was developed which predicts the propagation of sound waves in flow ducts with segmented liners. The calculations for implementing the derived algorithms were conducted with the aid of FORTRAN IV computer programs. Attention is also given to details regarding the experimental program which was designed to supplement the theoretical studies.
Methods of determining the transmission and attenuation of sound propagating in nonuniform ducts with and without mean flows are discussed. The approaches reviewed include purely numerical techniques, quasi-one-dimensional approximations, solutions for slowly varying cross sections, solutions for weak wall undulations, approximation of the duct by a series of stepped uniform cross sections, variational methods and solutions for the mode envelopes.
The propagation of sound in a converging-diverging duct containing a quasi-one-dimensional steady flow with a high subsonic throat Mach number was studied. The behavior of linearized acoustic theory at the throat of the duct was shown to be singular. This singularity implies that linearized acoustic theory is invalid. The explicit singular behavior was determined and used to sketch the development (by the method of matched asymptotic expansions) of a nonlinear theory for sound propagation in a sonic throat region.
The nature of the solution to the convected acoustic wave equation and associated boundary conditions for rectangular ducts containing uniform mean flow is examined in terms of the complex mapping between the wall admittance and characteristic mode eigenvalues. It is shown that the Cremer optimum suppression criteria must be modified to account for the effects of flow below certain critical values of the nondimensional frequency parameter of duct height divided by sound wavelength. The implications of these results on the design of low frequency suppressors are considered.
Nonlinear acoustic interactions at the second order (so-called weak interactions) of the perturbation scheme have been recognized for a long time. This paper examines the possibility of stronger interactions at the first order which are found in rigid rectangular ducts and to a lesser degree in slightly soft rectangular ducts. The strong interactions occur if any two modes have identical phase speeds, the case of multimode and multifrequency input for example. An important consequence of the strong interaction is the self-generation of subharmonics. Even a small amount of subharmonic can, given the right phase conditions, grow through a strong interaction with a higher harmonic.