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Broadband Sound Generation

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Broadband sound generation by confined turbulent jetsZhaoyan Zhang, Luc Mongeau,a)and Steven H. FrankelSchool of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907-1077共Received 9 November 2001; revised 14 May 2002; accepted 17 May 2002兲Sound generation by confined stationary jets is of interest to the study of voice and speechproduction, among other applications. The generation of sound by low Mach number, confined,stationary circular jets was investigated. Experiments were performed using a quiet flow supply,muffler-terminated rigid uniform tubes, and acrylic orifice plates. A spectral decomposition methodbased on a linear source-filter model was used to decompose radiated nondimensional soundpressure spectra measured for various gas mixtures and mean flow velocities into the product of 共1兲a source spectral distribution function; 共2兲 a function accounting for near field effects and radiationefficiency; and 共3兲 an acoustic frequency response function. The acoustic frequency responsefunction agreed, as expected, with the transfer function between the radiated acoustic pressure at onefixed location and the strength of an equivalent velocity source located at the orifice. The radiationefficiency function indicated a radiation efficiency of the order (kD)2over the planar wavefrequency range and (kD)4at higher frequencies, where k is the wavenumber and D is the tube crosssectional dimension. This is consistent with theoretical predictions for the planar wave radiationefficiency of quadrupole sources in uniform rigid anechoic tubes. The effects of the Reynoldsnumber, Re, on the source spectral distribution function were found to be insignificant over therange 2000⬍Re⬍20000. The source spectral distribution function approximately obeyed a St⫺ 3power law for Strouhal number values St⬍0.9, and a St⫺ 5power law for St⬎2.5. The influence ofa reflective open tube termination on the source function spectral distribution was found to beinsignificant, confirming the absence of a feedback mechanism. © 2002 Acoustical Society ofAmerica. 关DOI: 10.1121/1.1492817兴PACS numbers: 43.70.Aj, 43.28.Ra 关MSH兴LIST OF SYMBOLSa sphere diameter 共m兲c speed of sound 共m/s兲d orifice diameter 共m兲D tube cross section dimension 共m兲f frequency 共Hz兲E nondimensional sound pressure spectrumF source spectral distribution functiong Green’s functionG acoustic frequency response functionGexpmeasured system transfer functionHe Helmholtz number, fD/ck wave number 共m⫺1兲L distance from the orifice 共m兲M radiation efficiency functionP sound powerp total pressure 共Pa兲p⬘acoustic pressure 共Pa兲r distance between source and observer 共m兲R reflection coefficientRe Reynolds number, Ud/vSppsound pressure spectral densitySt Strouhal number, fD/UTijLighthill stress tensorU jet centerline velocity 共m/s兲vcvelocity of the sphere center 共m/s兲x observer positiony source position⌬p mean pressure drop across the orifice 共Pa兲␳0ambient density 共kg/m3兲␳⬘acoustic density 共kg/m3兲␻angular frequency 共rad/s兲␯kinematic viscosity 共m/s2兲␴ijviscous stress tensor 共Pa兲␶source time coordinate␴standard deviation␾sound source strengthI. INTRODUCTIONSound generation by jet flows has been the object ofmany previous investigations 共Blake, 1986兲. Most of thesestudies have considered the problem of sound production byfree high-speed jets because of its importance for aircraftengine noise 共Hubbard, 1994兲. There has been relatively lessinterest in noise from low-speed confined jets. The motiva-tion for the study described in this paper is the potentialsignificance of this problem for speech production. Soundgeneration by confined low-speed jet flows is always in-volved in the generation of fricative consonants, an impor-tant component of speech. It may also contribute to voiceproduction, since the jet plume downstream of the glottis isa兲Author to whom correspondence should be addressed. Electronic mail:[email protected]. Acoust. Soc. Am. 112 (2), August 2002 0001-4966/2002/112(2)/677/13/$19.00 © 2002 Acoustical Society of Americamost likely turbulent over a large fraction of one glottis os-cillation. Turbulence may be responsible for a random com-ponent in the sound produced by the air flow through thelarynx, which may play a significant role in defining voicequality. A better understanding of this contribution couldhelp develop better models for speech synthesis, and assist inthe diagnosis of pathological voice conditions.Jet noise theory is usually based on the work of Lighthill共1952兲, and subsequent refinements by many investigators, inwhich the basic equations for the fluid motion are formulatedin the form of an inhomogeneous wave equation. This ap-proach, termed the acoustic analogy, allows presumed soundsource terms to be identified. In particular, for nonheatedsubsonic free jet flows, the source term is the double diver-gence of Lighthill’s stress tensor, a quantity dominated bythe instantaneous, time varying Reynolds stress tensor. In afree field, the source distribution is of a quadrupole-type,known to be an inefficient sound generation mechanism rela-tive to monopole or dipole contributions. The radiated soundpressure may be obtained from the convolution of the sourceterm and a free space Green’s function over the entire regionwhere the instantaneous Reynolds stress is significant.The confinement of the source in a channel modifies thegeneration and propagation of sound waves significantly. Thecharacteristics of the turbulent jet flow are also different dueto the effects of confinement. For example, the strength ofthe recirculation region is often increased. Additional sourcesmay arise from the impingement of the jet plume on the rigidwalls. Finally, the one-dimensional nature of the radiatedsound waves at low frequencies changes the basic characterof the sound sources from quadrupole- to dipole-, ormonopole-type sources. This generally enhances sound pro-duction efficiency.The theory of sound production by turbulence in a rigidpipe has been discussed by Davies and Ffowcs Williams共1968兲. Lighthill’s acoustic analogy was used, with a theo-retical expression for the Green’s function satisfying rigidboundary conditions on the walls of a rectangular pipe. Theradiated sound pressure spectral density was expressed as thesummation of acoustic modes within the pipe. A


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