The noise from turbulence convected at high speed

Abstract

The theory initiated by Lighthill (1952) for the purpose of estimating the sound radiated from a turbulent fluid flow is extended to deal with both the transonic and supersonic ranges of eddy convection speed. The sound is that which would be produced by a distribution of convected acoustic quadrupoles whose instantaneous strength per unit volume is given by a turbulence stress tensor, T$_{ij}$. At low subsonic speeds the radiated intensity increases with the eighth power of velocity although quadrupole convection augments this basic dependence by a factor $|$1 - M cos $\theta|^{-5}$, where M is the eddy convection Mach number and $\theta $ the angular position of an observation point measured from the direction of eddy motion. At supersonic speeds the augmentation factor becomes singular whenever the eddy approaches the observation point at sonic velocity, M cos $\theta $ = 1. At that condition a quadrupole degenerates into its constituent simple sources, for each quadrupole element moves with the acoustic wave front it generates and cancelling contributions from opposing sources, so essential in determining quadrupole behaviour, cannot combine but are heard independently. This simple-source radiation is likened to a type of eddy Mach wave whose strength increases with the cube of a typical flow velocity. When quadrupoles approach the observer with supersonic speed sound is heard in reverse time, but is once again of a quadrupole nature and the general low-speed result is again applicable. The limiting high-speed form of the convection augmentation factor is $|$M cos $\theta|^{-5}$ which combines with the basic eighth power velocity law to yield the result that radiation intensity increases only as the cube of velocity at high supersonic speed. The mathematical theory is developed in some detail and supported by more physical arguments, and the paper is concluded by a section where some relevant experimental evidence is discussed.