The Noise Generation Mechanism

It is now well established that broadband shock-associated noise is gen­erated by the interaction of the large-scale structures that propagate down­stream and the quasi-periodic shock cell structure. The point-source array model proposed by Harper-Bourne and Fisher Harper-Bourne and Fisher (1974) was successful in explaining many of the observed noise character­istics. Tam and Tanna (1982) proposed an alternate theory based on the observed properties of the large-scale turbulence structures, which possess the important characteristics of being coherent and spatially quasi-periodic over several jet diameters. Thus the large-scale structures are wave-like when viewed as a whole. As these structures propagate downstream, they interact with the shock cell system established in the jet plume of an im­perfectly expanded supersonic jet. Tam and Tanna (1982) proposed a simple analytical model to explain the noise generation mechanism. They first expressed the pressure perturbation associated with the shock cells as a summation of the waveguide modes. Such a first order shock solution had been developed by Pack (1950), based on the work of Prandtl (1904).

In a cylindrical polar coordinate system centered at the nozzle exit, any perturbation associated with the shock cells can be expressed as,

OO 1 OO

Us = ^ Апфп (r) cos (knx) = – ^ Лпфп (r) (eiknx + e~lknx), (58)

n= 1 2 n=1

where Лп, фп(r) and kn are the amplitude, the eigenfunction, and the axial wavenumber of the n-th mode, respectively. The fundamental shock cell spacing L1 = 2n/k1.

The large-scale turbulent structures can be represented by a linear su­perposition of the normal wave modes of the flow with random amplitude functions. For a frequency f = ш/2п, the corresponding disturbance quan­tity can be expressed as

ut = R{B (x) ф (r) exp [i (kx – шЬ + тф)]} , (59)

where B(x), Ф(г), k, and m are the amplitude, the eigenfunction or radial distribution, the axial wavenumber, and the azimuthal mode number of the traveling instability wave, respectively. The wavenumber and frequency are related by uc = ш/k, where uc is the convection or phase velocity of the large-scale turbulent structure or instability wave.

The perturbations created by a weak interaction between the instability waves or large scale structures and the shock cell structure are given by the product of the Eqns. (58) and (59). The expression for the shock cell structure involves two summations corresponding to the different signs of the exponent. Tam and Tanna (1982) noted that the phase velocities of the terms associated with the shock cell component with the positive exponent, given by ш/^ + k„,), are less than those of the instability wave alone. They are usually subsonic relative to the ambient speed of sound and do not radiate. For the term involving the product with the component with the negative exponent, the interaction quantity is given by

R {2B (x) ф (r) Лпфп (r) exp [i(k – k^x – ішЬ]} . (60)

This expression represents a traveling wave with wavenumber (k – kn) and phase velocity equal to ш/^ – kn), if any amplitude variation B(x) is ig­nored. If kn is slightly larger than k, then the phase velocity is negative. This phase velocity could be supersonic relative to the ambient speed of sound even if the convection velocity of the large-scale structures them­selves, given by ш/k, is subsonic. These supersonic components would gen­erate Mach wave radiation mainly in the upstream direction. The direction

of radiation can be related to the phase velocity and ambient speed of sound

Подпись: ao The Noise Generation Mechanism Подпись: (61)

by

Подпись: n Подпись: uc Ln (1 + Mc cos ф) ’ Подпись: n = 1, 2, 3,... Подпись: (62)

where ф is measured from the jet inlet direction. This relation can be rewritten as an expression for the frequency as a function of angle as,

where Mc = uc/a is the convection Mach number of the large scale structures relative to the ambient speed of sound and Ln is the wavelength of the n­th Fourier mode of the shock cell structure. Li is the fundamental shock cell spacing. The peak frequency for a given angle of radiation would be close to but not exactly that given by Eqn. (62). The axial variation of the instability wave amplitude B(x), broadens the wavenumber spectrum, as shown by Tam and Morris (1980) and discussed in Section 3.2. This broadening produces a band of components with different supersonic phase velocities at frequency f. These components radiate at different angles, producing the observed directivity pattern at this frequency.

Since the shock cell system is composed of several waveguide modes, with different wavelengths, the interaction effects of the different waveguide modes are different. The principal direction of radiation and the spectral content of the noise are different for each mode. Thus, the far field noise that is made up of the superposition of the contributions from all the modes should exhibit multiple peaks and directional dependencies. These are pre­cisely the characteristics observed experimentally as shown in Figure 10. Since the amplitude of the broadband shock noise is directly proportional to the amplitude of the waveguide modes, and the amplitudes decrease rapidly with mode number, the spectral levels associated with the higher order modes are smaller than that of the fundamental. Even though multi­ple peaks are possible, they may not be easily observed. This explains why only a single dominant peak is often observed in the measured spectra.

Tam and Tanna (1982) also developed an expression for the intensity of broadband shock associated noise for jets operated at slightly off-design con­ditions |Mj – Mj| < 1, where Md and Mj are the design and fully-expanded Mach numbers of the jet respectively. The intensity is given by,

Is MMj – Mj)2. (63)

This expression, which is valid for both convergent and C-D nozzles, was shown to provide excellent agreement with measured data for both cold and
hot jets, and for over – and under-expanded modes of operation for the C-D nozzle. This expression is not valid when a strong shock, such as a Mach disc, is present in the plume.

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