Lighthill's acoustic analogy

The sound generated by turbulent flow is called aerodynamic sound. Most unsteady flows of technological interest are of high Reynolds number and turbulent, and the acoustic radiation is a very small by-product of the motion. The turbulence is usually produced by fluid motion over a solid boundary or by flow instability. Lighthill (1952) investigated aerodynamic sound by transforming the Navier–Stokes and continuity equations to form an exact, inhomogeneous wave equation whose source terms are important only within the turbulent region. He argued that sound is a very small component of the whole motion and that, once generated, its ‘back-reaction’ on the main flow can usually be ignored. The properties of the unsteady flow in the source region may then be determined by neglecting the production and propagation of the sound, a reasonable approximation if the Mach number M is small, and there are many important flows where the hypothesis is obviously correct, and where the theory leads to unambiguous predictions of the sound.

Lighthill was initially interested in solving the problem illustrated in Figure 4.1.1a, of the sound produced by a turbulent nozzle flow. However, his original theory actually applies to the simpler situation shown in Figure 4.1.1b, in which the sound is imagined to be generated by a finite region of ‘rotational’ flow in an unbounded fluid. This avoids complications caused by the presence of the nozzle. The fluid is assumed to be at rest at infinity, where the mean pressure, density and sound speed are respectively equal to po, ρo, co. Lighthill compared the equations for the production of acoustic density fluctuations in the real flow with those in an ideal, linear acoustic medium that coincides with the real fluid at large distances from the sources.