On the generalization to swirling flows of Lighthill’s eighth-power law. Part II: Laws of intensity of radiation for acoustic-vortical waves

Abstract

The Lighthill–Proudman theory of sound generation by turbulence, inhomogeneities, and dissipation in a medium at rest is extended to an axisymmetric mean flow with (i) arbitrary unidirectional shear velocity profile and (ii) arbitrary angular velocity of rotation, both depending only on the radius and also (iii) isentropic conditions. The forced acoustic-vortical wave equation through: (a) the wave operator describes the propagation of coupled acoustic-vortical waves (Part I); (b) the forcing terms specifies the compressive, shear, and swirl wave sources that model the generation of acoustic-vortical waves by turbulence, inhomogeneities, and entropy production (Part II). The energy balance is considered for acoustic-vortical waves including the energy density, flux, and production. The dimensional scaling for the wave sources and the energy flux leads to a law of intensity of radiation that generalizes the Lighthill eighth-power law of aeroacoustics by allowing for: (i) compressive, shear, and swirl wave sources; (ii) acoustic and vortical waves and their couplings; (iii) monopoles, dipoles, quadrupoles, and multipoles of any order.