The Source of Aerodynamic Noise

Abstract

This paper is a tribute to Alan Powell's achievements and his extensive publications in hydro and aeroacoustics.¹ The theory of Aerodynamic Noise was established by Sir James Lighthill in 1952. The beauty of Lighthill's treatment was that he based this theory on the exact Navier-Stokes equations and showed, by their rearrangement, how the source of aerodynamic noise could be obtained from exact time-accurate calculations or experiment. Lighthill used the emission or propagation theory whereby an observer in a uniform medium at rest receives acoustic radiation from a distribution of moving sources of sound. Their properties are found using an acoustic analogy. The relevant fluctuations in a turbulent fluid flow can be expressed in terms of Lighthill's stress tensor T_ij, which is used to define a distribution of equivalent acoustic sources, which move through an otherwise uniform stationary fluid. An alternative procedure is to concentrate on the acoustic generation and to regard the sources of sound at rest or in motion in a uniform medium moving at a constant speed. (The approach can be extended to consider any arbitrary mean fluid motion.) The advantage of the present approach, involving the convective wave equation is that flow-acoustic interaction becomes part of the solution. In Lighthill's theory, flow-acoustic interaction is either ignored or at best is included as an equivalent source. The purpose of the present paper is to show there is no unique source of aerodynamic noise for it depends on the flow quantity used to describe the radiated sound. The convective wave equation is introduced and shown to involve similar sources to those found by Lighthill for the wave equation in a medium at rest. The source function found for the convective wave equation for a turbulent flow is shown to involve a modified Lighthill's stress tensor, which is non-linear in velocity and temperature fluctuations. It is further shown when the rate of dilatation covariance is examined, which can be derived from Lighthill's solution, that this small quantity, which Lighthill so carefully treated in retaining the exact properties of the compressible flow, is itself directly responsible for the rate of change of volume in the fluid and the creation of the noise which is radiated from the turbulent flow.