On the precise implications of acoustic analogies for aerodynamic noise at low Mach numbers

Abstract

We seek a clear statement of the scaling which may be expected with rigour for transportation or other noise at low Mach numbers M, based on Lighthill's and Curle's theories of 1952 and 1955. In the presence of compact solid bodies, the leading term in the acoustic intensity is of order M6. Contrary to the belief held since that time that it is of order M8, the contribution of quadrupoles, in the presence of dipoles, is of order only M7. Retarded-time-difference effects are also of order M7. Curle's widely used approximation based on unsteady forces neglects both effects. Its order of accuracy is thus lower than was thought, and the common estimates of the value of M below which it applies appear precarious. The M6 leading term is modified by powers up to the fourth of (1−Mr), where Mr is the relative Mach number between source and observer; at speeds of interest the effect is several dB. However, this is only one of the corrections of order M7, which makes its value debatable. The same applies to the difference between emission distance and reception distance. The scaling with M6 is theoretically correct to leading order, but this prediction may be so convincing, like the M8 scaling for jet noise, that some authors rush to confirm it when their measurements are in conflict with it. We survey experimental studies of landing-gear noise, and argue that the observed power of M is often well below 6. We also object to comparisons across Mach numbers at fixed frequency; they should be made at fixed Strouhal number St instead. Finally, the compact-source argument does not only require M⪡1; it requires MSt⪡1. This is more restrictive if the relevant St is well above 1, a situation which can be caused by interference with a boundary or by wake impingement, among other effects. The best length scales to define St for this purpose are discussed.