The Acoustic Spectrum in the Sound Field of Isotropic Turbulence

Abstract

Lighthill's theory of Aerodynamic Noise uses an acoustic analogy by which the equivalent acoustic sources in a turbulent flow have first to be calculated, measured or modeled from a time-accurate evaluation of the unsteady flow. Proudman considered the application to the noise radiated from weakly compressible isotropic turbulence, by evaluating the fourth-order space-time covariance of Lighthill's stress tensor. The resulting acoustic power per unit volume of turbulence was found to follow the eighth power law of velocity and showed the efficiency of conversion of turbulent kinetic energy into sound was extremely small, being proportional to the fifth power of the turbulent Mach number. These results are of particular interest today, since, all turbulent shear flows were shown by Kolmogorov to become locally isotropic near the dissipation range. Hence the high frequency asymptotic far-field radiated noise spectrum generated from eddies in the inertial subrange of all turbulent shear flows should obey a universal frequency law as obtained from isotropic turbulence, namely f−7/ 2, a result first found by Meecham and Ford. From studies of the fluctuations in this nonlinear energy transfer cascade from large to small eddies it is possible to gain a better insight as to the physical process, and the reason for its weak efficiency, whereby sound is generated and radiated as a broadband spectrum. A key factor is shown to be the weak fluctuations in the rate of dilatation in the compressible turbulent flow, which explains the reason for the efficiency of sound conversion obtained by Lighthill.