Shear-layer pressure fluctuations and superdirective acoustic sources.

*(English)*Zbl 0708.76116Summary: We consider a sequence of boundary-value problems for the acoustic wave equation, with the pressure specified on the boundary as a function of space and time, and simulating features of the pressure field measured just outside a turbulent shear layer supporting large-scale coherent structures. The boundary pressure field has the form of a travelling subsonic plane wave, modulated by a large-scale envelope function. Three models for the envelope distribution are studied in detail, and the particular features which they exhibit are shown to be representative of large classes of amplitude functions.

We start by looking at the hydrodynamic near field of the boundary pressure fluctuations, over spatial regions throughout which the motion can be taken as incompressible. Very close to the boundary, the pressure fluctuations decay exponentially with transverse distance, while at sufficiently large distances from the whole wave packet on the boundary, the pressure fluctuations have a dipole algebraic decay. We investigate the transition from exponential to algebraic decay, and find that it is effected through quite a complicated multilayer tructure which depends crucially on the detailed form of the envelope.

Acoustic fields are then determined both from exact solutions to the wave equation, and from matching arguments. In some cases, where the boundary source is compact, the distant acoustic fields have a simple compressible dipole type of behaviour. In other cases, however, when the boundary source is non-compact, the acoustic field has a superdirective character, the angular variation being described by exponentials of cosines of the angle with the streamwise direction. It is shown how the superdirective acoustic sources are completely compatible with the features of the inner incompressible field, and a criterion for the occurrence of the superdirective acoustic fields will be given. Superdirective fields of this kind have been observed in measurements by J. Laufer and T. C. Yen [ibid. 134, 1-32 (1983)] on a low-speed round jet of Mach number 0.1, and the general relation of our results to those experiments is explained.

We start by looking at the hydrodynamic near field of the boundary pressure fluctuations, over spatial regions throughout which the motion can be taken as incompressible. Very close to the boundary, the pressure fluctuations decay exponentially with transverse distance, while at sufficiently large distances from the whole wave packet on the boundary, the pressure fluctuations have a dipole algebraic decay. We investigate the transition from exponential to algebraic decay, and find that it is effected through quite a complicated multilayer tructure which depends crucially on the detailed form of the envelope.

Acoustic fields are then determined both from exact solutions to the wave equation, and from matching arguments. In some cases, where the boundary source is compact, the distant acoustic fields have a simple compressible dipole type of behaviour. In other cases, however, when the boundary source is non-compact, the acoustic field has a superdirective character, the angular variation being described by exponentials of cosines of the angle with the streamwise direction. It is shown how the superdirective acoustic sources are completely compatible with the features of the inner incompressible field, and a criterion for the occurrence of the superdirective acoustic fields will be given. Superdirective fields of this kind have been observed in measurements by J. Laufer and T. C. Yen [ibid. 134, 1-32 (1983)] on a low-speed round jet of Mach number 0.1, and the general relation of our results to those experiments is explained.

##### MSC:

76Q05 | Hydro- and aero-acoustics |

##### Keywords:

sequence of boundary-value problems; acoustic wave equation; large-scale coherent structures; travelling subsonic plane wave
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\textit{D. G. Crighton} and \textit{P. Huerre}, J. Fluid Mech. 220, 355--368 (1990; Zbl 0708.76116)

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##### References:

[1] | Lighthill, Proc. R. Soc. Lond. 211 pp 564– (1952) |

[2] | DOI: 10.1017/S0022112083003195 |

[3] | Gutmark, AIAA J. 23 pp 354– (1985) |

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