Sound generated by instability wave/shock-cell interaction in supersonic jets.

*(English)*Zbl 1141.76451Summary: Broadband shock-associated noise is an important component of the overall noise generated by modern airplanes. In this study, sound generated by the weakly nonlinear interaction between linear instability waves and the shock-cell structure in supersonic jets is investigated numerically in order to gain insight into the broadband shock-noise problem. The model formulation decomposes the overall flow into a mean flow, linear instability waves, the shock-cell structure and shock-noise. The mean flow is obtained by solving RANS equations with a \(k-\epsilon \) model. Locally parallel stability equations are solved for the shock structure, and linear parabolized stability equations are solved for the instability waves. Then, source terms representing the instability wave/shock-cell interaction are assembled and the inhomogeneous linearized Euler equations are solved for the shock-noise.Three cases are considered, a cold under-expanded \(M_{j}\) = 1.22 jet, a hot under-expanded \(M_{j}\) = 1.22 jet, and a cold over-expanded \(M_{j}\) = 1.36 jet.

Shock-noise computations are used to identify and understand significant trends in peak sound amplitudes and radiation angles. The peak sound radiation angles are explained well with the Mach wave model of C.K.W. Tam and H.K. Tanna [J. Sound Vib. 81, 337–358 (1982; Zbl 0488.76071)]. The observed reduction of peak sound amplitudes with frequency correlates well with the corresponding reduction of instability wave growth with frequency. However, in order to account for variation of sound amplitude for different azimuthal modes, the radial structure of the instability waves must be considered in addition to streamwise growth. The effect of heating on the \(M_{j} = 1.22\) jet is shown to enhance the sound radiated due to the axisymmetric instability waves while the other modes are relatively unaffected. Solutions to a Lilley-Goldstein equation show that sound generated by ‘thermodynamic’ source terms is small relative to sound from ‘momentum’ sources though heating does increase the relative importance of the thermodynamic source. Furthermore, heating preferentially amplifies sound associated with the axisymmetric modes owing to constructive interference between sound from the momentum and thermodynamic sources. However, higher modes show destructive interference between these two sources and are relatively unaffected by heating.

Shock-noise computations are used to identify and understand significant trends in peak sound amplitudes and radiation angles. The peak sound radiation angles are explained well with the Mach wave model of C.K.W. Tam and H.K. Tanna [J. Sound Vib. 81, 337–358 (1982; Zbl 0488.76071)]. The observed reduction of peak sound amplitudes with frequency correlates well with the corresponding reduction of instability wave growth with frequency. However, in order to account for variation of sound amplitude for different azimuthal modes, the radial structure of the instability waves must be considered in addition to streamwise growth. The effect of heating on the \(M_{j} = 1.22\) jet is shown to enhance the sound radiated due to the axisymmetric instability waves while the other modes are relatively unaffected. Solutions to a Lilley-Goldstein equation show that sound generated by ‘thermodynamic’ source terms is small relative to sound from ‘momentum’ sources though heating does increase the relative importance of the thermodynamic source. Furthermore, heating preferentially amplifies sound associated with the axisymmetric modes owing to constructive interference between sound from the momentum and thermodynamic sources. However, higher modes show destructive interference between these two sources and are relatively unaffected by heating.

##### MSC:

76Q05 | Hydro- and aero-acoustics |

76J20 | Supersonic flows |

76L05 | Shock waves and blast waves in fluid mechanics |

76E99 | Hydrodynamic stability |

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\textit{P. K. Ray} and \textit{S. K. Lele}, J. Fluid Mech. 587, 173--215 (2007; Zbl 1141.76451)

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