On Sound Generated Aerodynamically. II. Turbulence as a Source of Sound

M. J. Lighthill

Abstract

The theory of sound generated aerodynamically is extended by taking into account the statistical properties of turbulent airflows, from which the sound radiated (without the help of solid boundaries) is called aerodynamic noise. The theory is developed with special reference to the noise of jets, for which a detailed comparison with experiment is made (section 7 for subsonic jets, section 8 for supersonic ones). The quadrupole distribution of part I (Lighthill 1952) is shown to behave (see section 3) as if it were concentrated into independent point quadrupoles, one in each 'average eddy volume'. The sound field of each of these is distorted, in favour of downstream emission, by the general downstream motion of the eddy, in accordance with the quadrupole convection theory of part I. This explains, for jet noise, the marked preference for downstream emission, and its increase with jet velocity. For jet velocities considerably greater than the atmospheric speed of sound, the 'Mach number of convection' M$_{c}$ may exceed 1 in parts of the jet, and then the directional maximum for emission from these parts of the jet is at an angle of sec$^{-1}$ (M$_{c}$) to the axis (section 8). Although turbulence without any mean flow has an acoustic power output, which was calculated to a rough approximation from the expressions of part I by Proudman (1952) (see also section 4 below), nevertheless, turbulence of given intensity can generate more sound in the presence of a large mean shear (section 5). This sound has a directional maximum at 45 degrees (or slightly less, due to the quadrupole convection effect) to the shear layer. These results follow from the fact that the most important term in the rate of change of momentum flux is the product of the pressure and the rate of strain (see figure 2). The higher frequency sound from the heavily sheared mixing region close to the orifice of a jet is found to be of this character. But the lower frequency sound from the fully turbulent core of the jet, farther downstream, can be estimated satisfactorily (section 7) from Proudman's results, which are here reinterpreted (section 5) in terms of sound generated from combined fluctuations of pressure and rate of shear in the turbulence. The acoustic efficiency of the jet is of the order of magnitude 10$^{-4}$ M$^{5}$, where M is the orifice Mach number. However, the good agreement, as regards total acoustic power output, with the dimensional considerations of part I, is partly fortuitous. The quadrupole convection effect should produce an increase in the dependence of acoustic power on the jet velocity above the predicted U$^{8}$ law. The experiments show that (largely cancelling this) some other dependence on velocity is present, tending to reduce the intensity, at the stations where the convection effect would be absent, below the U$^{8}$ law. At these stations (at 90 degrees to the jet) proportionality to about U$^{6\cdot 5}$ is more common. A suggested explanation of this, compatible with the existing evidence, is that at higher Mach numbers there may be less turbulence (especially for larger values of nd/U, where n is frequency and d diameter), because in the mixing region, where the turbulence builds up, it is losing energy by sound radiation. This would explain also the slow rate of spread of supersonic mixing regions, and, indeed, is not incompatible with existing rough explanations of that phenomenon. A consideration (section 6) of whether the terms other than momentum flux in the quadrupole strength density might become important in heated jets indicates that they should hardly ever be dominant. Accordingly, the physical explanation (part I) of aerodynamic sound generation still stands. It is re-emphasized, however, that whenever there is a fluctuating force between the fluid and a solid boundary, a dipole radiation will result which may be more efficient than the quadrupole radiation, at least at low Mach numbers.