Scalar and vector viscous wave equations of fluid bounded by solid boundaries

Abstract

Quadrupole sources in the acoustic analogies developed by Lighthill, Curle, and Ffowcs Williams and Hawkings, include the effect of fluid viscosity on both sound generation and propagation. However, all or partial quadrupole sources are usually ignored to save the computational cost, and this simplification also ignores the effect of fluid viscosity on sound propagation. In this paper, viscous scalar and vector wave equations, and the corresponding acoustic pressure, acoustic velocity and vortical velocity integral formulations are derived analytically to separate the effect of fluid viscosity on sound generation and propagation. An acoustic energy balance equation for the viscous fluid is derived analytically. The developed equation reveals two types of acoustic dissipation mechanisms, namely, the viscous dissipation and the acoustic-vortical interaction. The viscous dissipation occurs during the entire sound-propagation process but the acoustic-vortical interaction occurs mainly in the near field around sources because the vortical wave usually decays rapidly away from the sources. Using the developed analytical formulations, the acoustic power loss caused by fluid viscosity is calculated and analyzed.