On the Directivity and Spectra of Noise from Unheated and Heated Jets

Abstract

The present paper considers the transmission of sound from a source distribution inside a jet to a receiver or observer outside; the source can be represented by a combination of multipoles moving relative to the jet. The shear layer has a random shape with a plane mean position, and entrains a region of turbulence. The transmission across the plane shear layer involves: (i) scattering by the convected, irregular and unsteady interface across which occurs the transition from the jet to the ambient medium; (ii) refraction in the region of low Mach number turbulence entrained by the shear layer. If the jet Mach number does not exceed two the local turbulence can be taken as incompressible. The transmission involves: (i) a deterministic amplitude factor due to the difference in sound speed, mass density and velocity between the jet and ambient medium; (ii) a random phase shift due to the scattering of sound by the irregular interface and the convection of sound by the turbulence entrained with the shear layer. Since there is a random phase shift for the acoustic pressure, the acoustic power, which is quadratic in the acoustic pressure, involves the statistics of interference between two waves. The latter specify the spectral directivity, defined as the acoustic power radiated per unit solid angle and unit frequency band, i.e. either the spectrum received in each direction or the directivity of sound received outside the jet at each frequency. The statistics of sound scattering by irregular interfaces and convection by turbulence are subject to assumptions and constraints which affect critically the directivity and spectra. The evaluation of radiation integrals without low-or high-frequency approximations is also essential to reproduce experimental results on the noise of cold or hot jets. An accuracy of 3 dB can be expected for directivities and 6 dB for spectra, for directions not too far away from the vertical, (i.e perpendicular to the jet). ACOUSTIC MODEL OF THE SHEAR LAYER The present model (Figure 1) assumes that the mean position of the shear layer is the plane x 3 = 0, and the coordinates x ជ * = (x 1 , x 2) lie on this plane. Thus the model applies to the sides of a rectangular jet; it applies locally to the tangent plane of a circular jet if the wavelength is small compared with the radius. There are four relevant velocities of which the first three are horizontal: (i) the velocity u ជ 0 of the sound source; (ii) the velocit v ជ a of the Dr. Dennis K. McLaughlin served as guest editor for this paper jet; (iii) the velocity of convection of the shear layer v ជ 0 = αv ជ a , which is typically a fraction α ∼ 0.6 of the jet velocity; (iv) the velocity of turbulence in the shear layer u ជ, whose magnitude is typically a fraction β ∼ 0.15 of the jet velocity |u ជ| ~ βv a. The coordinates convected by the shear layer in the plane of the mean position of the shear layer are y ជ = x ជ * − v ជ 0 t; the turbulent velocity u ជ(y ជ,x 3 ,t) is a random function of the convected coordinate in the plane of the shear layer...

On Physical Aeroacoustics with Some Implications for Low-Noise Aircraft Design and Airport Operations

“…The review of aeroacoustics research (Section 2) has focused on some aspects not widely known but that could have considerable impact such as: (i) the existence of two alternative theories of aerodynamic noise [1][2][3][4][5][6][7][8][9] and [62][63][64][65][66][67][68]; (ii) the coupling of sound with vortical and entropy modes in jet engines . The account on sound transmission from the aircraft noise sources to the interior of the near airport resident (Section 3) has included: (i) the spectral and directional broadening of noise [255][256][257][258][259][260][261]; (ii) the psychoacoustic distinctions between noise components. The noise mitigation measures (Section 4) mentioned include: (i) optimization of non-uniform acoustic liners [406][407][408][409][410][411][412][413][414][415][416] and use of partial chevron nozzle [417]; (ii) low noise operating procedures [432,433] consistent with flight safety and air traffic management rules [444][445][446][447]…”

“…The edge diffraction changes the directivity of sound radiated out the nozzle [234][235][236][237][238][239][240][241][242][243][244][245][246][247][248][249][250] including the effect of the vortex sheet [253,254]. In reality the shear layer includes [255,256]: (i) an irregular interface across which velocity, density and sound speed change between the hot jet and the cold atmosphere, implying wave reflection and transmission; (ii) the shear layer entrains turbulence, that is itself a source of noise, and causes refraction of sound waves travelling from the jet to the exterior. In addition both effects are random, that is the shape of the irregular interface and the velocity field of turbulence lead to spectral and directional broadening, that is the scattering and refraction by the turbulent and irregular shear layer distributes the acoustic energy over a wider range of directions and a broader frequency spectrum thus providing significant attenuation [6,257].…”

“…The comparison of theory [256][257][258][259] with experimental data [260,261] shows ( Figure 6) that jet noise is dominated by spectral and directional broadening: (i) a monopole inside a jet emits isotropic monochromatic radiation; (ii) an observer outside receives a spectrum consisting of an attenuated spike at the source frequency, with the remaining acoustic energy scattered into a broadband; (iii) the spike is more attenuated and the broadband wider in directions away from the vertical; (iv) the broadband is nearly symmetrical about the tone for transmission normal to the shear layer; (v) the broadband becomes unsymmetrical for oblique propagation, extending more towards high-frequencies downstream and to lower frequencies upstream.…”

“…If the shear layer were made more irregular and turbulent the spikes would be attenuated into the broadband (top) and the directivity (circles at the bottom) shows even greater noise reduction. This demonstrates the effect of spectral and directional broadening of sound due to transmission across a turbulent and irregular shear layer [255,256].…”

On magnetoacoustic-gravity-inertial (MAGI) waves - I. Generation, propagation, dissipation and radiation

“…The theory of aerodynamic sound has seen several extensions to account for (i) the presence of fixed (Curle 1955) or moving (Ffowcs‐Williams & Hawkins 1968) solid boundaries; (ii) two‐phase flow (Crighton & Ffowcs‐Williams 1969), vorticity (Powell 1968) and fluid inhomogeneities (Howe 1975), including ionization (Campos 1978); (iii) shear flows, viz. unidirectional (Lilley 1974; Goldstein 1976, 2001; Mani 1976; Mohring, Muller & Obermeier 1983; Musafir 1993; Colonius, Lele & Moin 1997; Campos & Serrão 1998; Campos & Kobayaski 2000, 2007, 2010; Campos 2007b); (iv) swirling flows (Howe 1977; Tam & Auriault 1998; Campos 2007c). Several alternative forms of the acoustic analogy exist (Philips 1960; Campos 1983a, 1986; Doak 1998; Goldstein 2003), and in principle each of the 60 forms of the acoustic wave equation (Campos 2007b,c) could lead to an acoustic analogy.…”