Fundamental problems with the model of uniform flow over acoustic linings

Abstract

The walls of ducts containing flow are often acoustically lined in order to reduce sound. Many simple models of acoustic linings assume the lining to be linear and locally reacting; examples considered here include the three-parameter, mass–spring–damper, Helmholtz resonator and enhanced Helmholtz resonator models. All of these models have been found to have stability issues with uniform mean grazing flow, and there has been some confusion over the existence of hydrodynamic instability surface waves over such linings. Mathematically, the standard proven Briggs–Bers stability analysis is not applicable. Computationally, the hydrodynamic modes are routinely ignored (in the frequency domain) and instabilities filtered out (in the time domain). This confusion also causes significant problems for mode-matching, Green functions, and scattering analyses. In this paper, it is shown that any situation not capable of being analysed using the Briggs–Bers criterion is illposed, and that this is the root cause of the confusion over hydrodynamic instabilities. A large class of lining models, including all those mentioned above, are shown to be illposed with uniform mean flow. An explanation is given of the effects of this illposedness in practice, and it is argued that illposed models should not be used.An alternative stability criterion to Briggs–Bers has been occasionally used in the literature. This alternative criterion, involving analytic continuation to purely imaginary frequencies, was recently christened the “Crighton–Leppington” stability criterion. However, this stability criterion is incorrect, and several counter-examples are given here.