Sound generated by the interaction between shock and instability waves in supersonic round jets

TL;DR

This work develops an analytical model for sound from shock–instability interaction (SII) in supersonic round jets using a vortex-sheet base flow and Euler equations. By combining Pack’s shock model, linear instability theory, and an inhomogeneous wave equation solved via Fourier analysis and steepest descent, it yields a closed-form far-field acoustic field. The key finding is that the one-cell SII source is not monopole, and including multiple shock cells reproduces screech directivity more accurately, with Mach-wave radiation accounting for near-field and broadband noise aspects. The approach provides a physics-based framework for predicting screech and BBSAN spectra and offers a path toward incorporating installation effects in jet-noise design.

Abstract

In this paper, we develop an analytical model to investigate the sound generated by the shock-instability interactions (SII) in supersonic round jets, extending our previous two-dimensional planar study to circular configurations. The jet is represented by a vortex sheet, with its motion modeled by the Euler equations. Shock and instability waves are modeled using Pack's approach and the linear stability theory, respectively, while their interaction is calculated by solving an inhomogeneous wave equation. Using the Fourier transform and steepest descent method, we obtain a closed-form solution for the resulting acoustic field. Results due to the interaction between the instability waves and one interaction cell capture the key directivity features of screech reported in experiments and numerical simulations, indicating that the classic monopole assumption may be inadequate. In particular, the screech-tone intensity due to multiple shock cells decays rapidly as the observer angle approaches 180 degrees, which is in better agreement with the experimental data measured by Norum. We further analyze how the instability wave growth rate influences these directivity patterns and examine the sound generation efficiency of the broadband shock-associated noise. Finally, an examination of near-field pressure fluctuations due to the SII reveals that noise is produced primarily via the Mach wave radiation mechanism.

Figures (13)

• Figure 1: The schematic of the vortex-sheet flow configuration and cylindrical coordinates. The origin is fixed at the center of the nozzle while $x$ and $r$ represent the streamwise and radial coordinates, respectively.
• Figure 2: Schematic of a source term located within one single shock cell. The Prandtl-Meyer expansion waves occur at the nozzle lip and are reflected as compression waves (oblique shock waves) on the opposite jet boundary, thereby forming one shock structure shock_structure1Mehta23. The acoustic wave (with a wavelength $\lambda_c$) is generated through the interaction between shock and instablity waves. The total field is equivalent to a linear superimposition of the results from all shock cells.
• Figure 3: The branch points, branch cuts, and integral path in the complex $\lambda$ plane.
• Figure 4: Schematic of one azimuthal plane of the jet. $R$ is the distance from the observer to the starting point of one shock cell and $\psi$ represents the observer angle with respect to the downstream direction. In this plane, $r=R\sin\psi$ and $x=R\cos \psi$.
• Figure 5: The directivity patterns when three shock-cell interactions are included. The Mach number of the fully expanded jet flow is 1.19. The relative intensity of the three sources are (0.5, 1, 0.5) and (0.2, 1, 0.8) for the red solid line and the black dashed line, respectively.