Sound generated aerodynamically revisited: large-scale structures in a turbulent jet as a source of sound

Lighthill’s formulation of the aerodynamic sound problem (Lighthill, Proc. R. Soc. Lond. A 211, 564 (1952)) is here considered as fundamental to the sound generated by real turbulent jets. For convenience, the aerodynamic sound integral is recast, via Michalke & Fuchs [J.Fluid Mech. 70, 179 (1975)), into a form involving the pressure fluctuations. It is first conjectured that the large-scale coherent structures in the turbulent jet, whose existence is now well recognized, would be responsible for the spectrally dependent highly oriented radiation patterns in the aerodynamic sound field. Accordingly, only contributions that arise from the coherent structures are retained in the aerodynamic sound integral. The neglected fine-grained turbulence as far as the sound field is concerned is thought otherwise to contribute to the broadband, nearly isotropic radiation. The present source description follows Mankbadi & Liu [Phil.Trans. R. Soc. Lond. A 298, 541 (1981)), but suitably modified to include an ensemble of n = 0 axisymmetric and n = 1 spiral modes in the relevant Strouhal number range. The coherent structures interact with the mean flow and the finegrained turbulence as an ensemble through energy exchanges dictated by rates according to their individual spectral characteristics. Because such coherent structures are relatively ‘weak’ in a real, developing turbulent jet, their mutual interactions are neglected as a first approximation. The sound sources, in a stationary coordinate system and evaluated at the appropriate retarded time, give rise to an equivalent streamwise distribution of line radiators after performance of the azimuthal and radial integrations in the aerodynamic sound integral. The streamwise oscillation of the equivalent sources is determined by an axial interference function strongly influenced by the wavenumber of each individual mode whereas the streamwise growth and decay of the source envelope is determined primarily by the coherent structure amplitude whose spectral dependence is also strong. The streamwise net imbalance of the source contribution, reflected by the axial integration in the aerodynamic sound integral, gives rise to the far sound field. It is found that in general, the radiation is primarily in the direction of the jet exhaust; the radiation patterns of the n = 0 modes resembling those of longitudinal quadrupoles and those of the n = 1 modes resembling those of lateral quadrupoles. However, the n = 0 modes tend to peak at Strouhal numbers less than those of the n = 1 modes. The superposition gives a directional-spectral behaviour that strikingly resembles that of observations: lower frequency sound radiates preferentially in the forward direction and as the frequency increases, the peak radiation moves towards the lateral directions; it is also found that contributions to the high-frequency sound come from coherent structures that peak nearer the nozzle lip, whereas contributions to the low-frequency sound come from such structures that peak further downstream in the jet. The calculated spectral shapes are narrower than observations by typically a deficit of 4—7 dB per octave on both the high and low frequency sides and this is most likely attributable to the nearly isotropic radiation caused by the broad-band fine-grained turbulence whose direct contribution to the sound field is not accounted for. For the same reason, the calculated aerodynamic sound field has a large deficit compared with observations in the vicinity of the 90- degree region. The dominant contributions to the radiation come from the so-called shear noise in the forward arc, whereas both the shear and self-noise of the coherent structures become equally insignificant to the same order in the 90-degree region. Although the source distribution within the jet is calculated for an identically incompressible fluid, it is used in a limited sense to study the effect of jet exit velocity on the peak radiation frequency in the forward direction: it is found that the peak value of fd/a0, where f is the frequency, d the jet nozzle diameter and a0 the ambient sound speed, take on a value of about 0.30 independently of the jet velocity and this compares favourably with an observational value of about 0.20. In general, the angular distribution of the peak frequency due to coherent structures radiation compared favourably with observations. Compressibility effects that somewhat limit the amplification of coherent structures, as well as the effects of higher azimuthal modes whose radiation would peak at higher frequencies and larger lateral directions, warrant further study in the light of the present considerations. The present work, however, has already shown that the consequences of Lighthill's formulation of the aerodynamic sound problem agree with major features of observations and that this is brought about by taking into account as sources the growing and decaying largescale coherent eddies whose development within the turbulent jet and whose radiational properties are all strongly dependent upon their spectral contents.