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A parametric study of the broadband shock-associated noise in supersonic jets via semi-empirical modeling

Binhong Li, Benshuai Lyu

TL;DR

The paper tackles BBSAN in off-design supersonic jets by developing a semi-empirical model that couples a downstream-decaying Pack-like shock structure with a Gaussian instability-wave packet to describe SII. Far-field acoustics are obtained by solving a boundary-value Helmholtz problem with the SII source modeled on the jet lip line, enabling prediction of spectra and directivity. A comprehensive parametric study shows that non-dimensional decay rates $\epsilon_iS_0$ and $\epsilon_sS_0$, spacing decay $\sigmaS_0$, and offset $X_m/S_0$ govern spectral peak location, bandwidth, and lobes, while PSE-based parameterization ensures physical consistency. Validation against multiple experimental datasets demonstrates improved accuracy when incorporating shock amplitude/spacing decays, reproducing key features such as the spectral peak and upstream lobes and their frequency/angle dependencies. The approach provides a practical framework for understanding and potentially controlling BBSAN in realistic jet configurations, with future work extending the model to nozzle and wing scattering effects.

Abstract

A semi-empirical model is developed in this paper to predict the broadband shock-associated noise (BBSAN) generated by shock-instability interaction (SII) in imperfectly expanded supersonic jets. The model makes use of a semi-empirically modified Pack's model that accounts for the decay in both shock amplitude and shock spacing and a Gaussian wave-packet model for the instability waves. The near-field pressure perturbation due to the SII is treated as a boundary value for the Helmholtz equation, which is subsequently solved to predict the far-field acoustic spectra and directivity patterns. A comprehensive parametric study is conducted to reveal the effects of the key parameters on the acoustic spectral and directivity features. It is found that decreasing the instability-wave decay rate narrows the spectral bandwidth and the major lobes in directivity patterns, while variations in shock spacing shift the spectral peak frequency and the major radiation angle. Mechanisms of such changes are discussed based on the model. Further validation against multiple experimental datasets demonstrates that incorporating more realistic parameters in the model-particularly those accounting for the shock spacing and amplitude decays-considerably improves its prediction accuracy and physical consistency. The improved model successfully reproduces several key spectral features observed in experiments; these include, for example, the peak frequency and the tendency of bandwidth contradiction as the observer angle increases. Moreover, the predicted directivity patterns closely match the experiments outside the shallow-angle region dominated by jet mixing noise. In particular, it captures the major radiation lobes and their frequency-dependent amplitude and shape variations.

A parametric study of the broadband shock-associated noise in supersonic jets via semi-empirical modeling

TL;DR

The paper tackles BBSAN in off-design supersonic jets by developing a semi-empirical model that couples a downstream-decaying Pack-like shock structure with a Gaussian instability-wave packet to describe SII. Far-field acoustics are obtained by solving a boundary-value Helmholtz problem with the SII source modeled on the jet lip line, enabling prediction of spectra and directivity. A comprehensive parametric study shows that non-dimensional decay rates and , spacing decay , and offset govern spectral peak location, bandwidth, and lobes, while PSE-based parameterization ensures physical consistency. Validation against multiple experimental datasets demonstrates improved accuracy when incorporating shock amplitude/spacing decays, reproducing key features such as the spectral peak and upstream lobes and their frequency/angle dependencies. The approach provides a practical framework for understanding and potentially controlling BBSAN in realistic jet configurations, with future work extending the model to nozzle and wing scattering effects.

Abstract

A semi-empirical model is developed in this paper to predict the broadband shock-associated noise (BBSAN) generated by shock-instability interaction (SII) in imperfectly expanded supersonic jets. The model makes use of a semi-empirically modified Pack's model that accounts for the decay in both shock amplitude and shock spacing and a Gaussian wave-packet model for the instability waves. The near-field pressure perturbation due to the SII is treated as a boundary value for the Helmholtz equation, which is subsequently solved to predict the far-field acoustic spectra and directivity patterns. A comprehensive parametric study is conducted to reveal the effects of the key parameters on the acoustic spectral and directivity features. It is found that decreasing the instability-wave decay rate narrows the spectral bandwidth and the major lobes in directivity patterns, while variations in shock spacing shift the spectral peak frequency and the major radiation angle. Mechanisms of such changes are discussed based on the model. Further validation against multiple experimental datasets demonstrates that incorporating more realistic parameters in the model-particularly those accounting for the shock spacing and amplitude decays-considerably improves its prediction accuracy and physical consistency. The improved model successfully reproduces several key spectral features observed in experiments; these include, for example, the peak frequency and the tendency of bandwidth contradiction as the observer angle increases. Moreover, the predicted directivity patterns closely match the experiments outside the shallow-angle region dominated by jet mixing noise. In particular, it captures the major radiation lobes and their frequency-dependent amplitude and shape variations.
Paper Structure (9 sections, 28 equations, 17 figures, 3 tables)

This paper contains 9 sections, 28 equations, 17 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model formulation
  3. The parametric model
  4. Determination of model parameters
  5. Results
  6. Typical directivity patterns and frequency spectra
  7. Effects of the non-dimensional parameters
  8. Comparison with experimental data
  9. Conclusion

Figures (17)

  • Figure 1: Schematic of the jet flow in a cylindrical coordinate frame. The origin is fixed on the jet center line, while $x$, $r$, and $\theta$ represent the streamwise, radial, and azimuthal coordinates, respectively. Note that the instability wave reaches its maximum intensity at $x=0$ while the nozzle is located at $x=-X_m$.
  • Figure 2: The obtained shock spacing from 1982NormSeiner. The designed Mach number of the nozzle is $M_d=1$ and the Mach number of the fully expanded jet is calculated via $M_{-}=\sqrt{\beta^2+1}$. (a) $\beta=0.4$; (b) $\beta=0.6$; (c) $\beta=0.8$; (d) $\beta=1$.
  • Figure 3: The decay rate of the shock spacing. $\Diamond$: 1982NormSeiner; $\Box$: shock_spacing_panda; $\vartriangleright$: 2014edgington; $\circ$: edgington_shockleakage. (a) $M_d=1$; (b) $M_d=1.5$.
  • Figure 4: The obtained shock intensities from 1982NormSeiner, which are normalized by the intensity of the first shock cell. The designed Mach number of the nozzle is $M_d=1$ and the Mach number of the fully expanded jet is calculated via $M_{-}=\sqrt{\beta^2+1}$. (a) $\beta=0.4$; (b) $\beta=0.6$; (c) $\beta=0.8$; (d) $\beta=1$.
  • Figure 5: The decay rate of the shock amplitude. $\Diamond$: 1982NormSeiner; $\Box$: shock_spacing_panda; $\vartriangleright$: 2014edgington; $\circ$: edgington_shockleakage. (a) $M_d=1$; (b) $M_d=1.5$.
  • ...and 12 more figures