Acoustic Radiation from Isotropic Turbulence

Abstract

For low Mach number turbulence, Lighthill has given a result which expresses the acoustic radiation of turbulent fluid in terms of a fourth-rank velocity correlation. Through the use of the Navier-Stokes equation this relation can be put in terms of the space-time pressure correlation. For very large Reynolds number turbulence (in the inertial subrange where Kolmogoroff's similarity principles are valid) one can write this pressure correlation in terms of a function of a single variable. Since the space-time correlation of the pressure is needed one must take into account convective effects. This is done here by introducing a new Lagrangian type of correlation which is defined in such a way that a similarity argument can be applied without difficulty from the convective effects. Such convective effects then enter only through negligible Doppler shifts. Using this similarity result and Lighthill's formulation, one obtains the acoustic self-noise power spectrum for the turbulence. The spectrum is proportional to ω−7/2M21/2 for high frequencies (ω≫c0M/L) where ω is the acoustic angular frequency, M is the turbulence Mach number, c0 is the velocity of sound in the fluid, and L is the size of the large-scale turbulent eddies. The spectrum at the high-frequency end is universal, that is it is independent of the details of the driving mechanism. Furthermore the spectrum is proportional to ω4M3 at the low-frequency end and depends on the large scale eddies there. The similarity hypothesis made here for the special Lagrangian type of space-time correlation is of interest in itself in turbulence theory. It is difficult to check this hypothesis directly; however, a measurement of the acoustic power spectrum offers an interesting indirect check on the hypothesis.