Absorption of sound by homogeneous turbulence

Abstract

The following problem is treated: given a plane acoustic wave propagating through an unbounded field of turbulence, calculate the amount of acoustic energy converted into turbulent kinetic energy. The fluid velocities due to the acoustic waves and the turbulence are assumed to be small compared with the speed of sound. Thus the sound-turbulence interaction is weak and the turbulent field may be considered to be incompressible. The analysis is based on the interaction of two opposite effects: the acoustic distortion of the turbulence (producing anisotropic Reynolds stresses) and the redistribution of the kinetic energy among components (tendency towards isotropy) and among wavenumbers (energy cascade and dissipation). These phenomena are described using semi-empirical turbulence arguments. It is seen that the simplest model for the redistribution among components is not sufficient for unsteady flows. A more complete model is used which is modified to agree with the exact instantaneous distortion analysis of Ribner [map ] Tucker to first order. Owing to the two redistribution effects, the Reynolds stress behaves inelastically and is out of phase with the acoustic field. Thus there is an average production of turbulent energy corresponding to the absorption of acoustic energy and attenuation of the incident wave. For nearly isotropic turbulence, the attenuation coefficient is found to be proportional to the rate of viscous dissipation and independent of the frequency.In order to compare the theory with experiment several constants involved in the semi-empirical model of the turbulence must be found. Owing to the lack of better information these constants are estimated here by order-of-magnitude considerations. No existing experiments correspond to the homogeneous turbulence assumed by the theory. Comparison with the few reasonably applicable experiments shows qualitative agreement though the importance of the turbulent absorption is generally of nearly the same order as the measurement error. Several discrepancies between jet noise experiments and aerodynamic noise predictions may be roughly explained using the above analysis.