Equation for Aeroacoustics in a Quiescent Environment

TL;DR

This paper derives a comprehensive spectral equation for aeroacoustic perturbations in a dissipative, quiescent environment, revealing the dispersion relation and classifying the PDE type based on viscosity and scale parameters.

Contribution

It introduces a general spectral PDE for aeroacoustics in dissipative media without assumptions, analyzing dispersion and PDE classification across different viscosities and scales.

Findings

Dispersion relation links wavenumber and frequency in dissipative media.

Classification of PDE as parabolic or hyperbolic based on spectral analysis.

Identification of a critical wavenumber depending on space-time scales.

Abstract

The perturbation equation for aeroacoustics has been derived in a dissipative medium from the linearized compressible Navier-Stokes equation without any assumption, by expressing the same in spectral plane as in Continuum perturbation field in quiescent ambience: Common foundation of flows and acoustics Sengupta et al., Phys. Fluids,35, 056111 (2023). The governing partial differential equation (PDE) for the free-field propagation of the disturbances in the spectral plane provides the dispersion relation between wavenumber and circular frequency in the dissipative medium, as characterized by a nondimensional diffusion number. Here, the implications of the dispersion relation of the perturbation field in the quiescent medium are probed for different orders of magnitude of the generalized kinematic viscosity, across large ranges of the wavenumber and the circular frequency. The adopted global spectral analysis helps not only classify the PDE into parabolic and hyperbolic types, but also explain the existence of a critical wavenumber depending on space-time scales.