The generation of noise by isotropic turbulence

Abstract

A finite region, with fixed boundaries, of an infinite expanse of compressible fluid is in turbulent motion. This motion generates noise and radiates it into the surrounding fluid. The acoustic properties of the system are studied in the special case in which the turbulent region consists of decaying isotropic turbulence. It is assumed that the Reynolds number of the turbulence is large, and that the Mach number is small. The noise appears to be generated mainly by those eddies of the turbulence whose contribution to the rate of dissipation of kinetic energy by viscosity is negligible. It is shown that the intensity of sound at large distances from the turbulence is the same as that due to a volume distribution of simple acoustic sources occupying the turbulent region. In this analogy, the whole fluid is to be regarded as a stationary and uniform acoustic medium. The local value of the acoustic power output P per mass of turbulent fluid is given approximately by the formula P = -$\frac{3}{2}\alpha \frac{\text{d}\overline{u^{2}}}{\text{d}t}\left(\frac{\overline{u^{2}}}{c^{2}}\right)^{\frac{5}{2}}$, where $\alpha $ is a numerical constant, $\overline{u^{2}}$ is the mean-square velocity fluctuation, t is the time, and c is the velocity of sound in the fluid. The constant $\alpha $ is expressed in terms of the well-known velocity correlation function f(r) by assuming the joint probability distribution of the turbulent velocities and their first two time-derivatives at two points in space to be Gaussian. The numerical value $\alpha \sim $ 38 is then obtained by substituting the form of f(r) corresponding to Heisenberg's theoretical spectrum of isotropic turbulence. It is found that the effects of decay make only a small contribution to the value of $\alpha $, and that the order of magnitude of $\alpha $ is not changed when widely differing forms of the function f(r) are used.