Time Domain Simulations using the Modified Myers Boundary Condition

Abstract

This paper considers the simplistic 2D problem of a time-harmonic point/line mass source in uniform flow over an acoustic lining in order to investigate the relationship between time-domain numerical instabilities and “real” instabilities of flow over acoustic linings. An exact analytic solution is given for the long-time time-harmonic solution to this problem, and this is compared with a numerical time-domain solution which is carefully constructed so as not to need any selective filtering. A discrete dispersion analysis is then performed to analyse the temporal stability of this entire numerical scheme when subjected to impedance boundary conditions. The numerical instability commonly present in time-domain simulations using the Myers boundary condition is indeed shown to correspond to the illposedness of the underlying mathematical model, as previously predicted. Moreover, the modified boundary condition proposed by Brambley is numerically implemented and the stability of this boundary condition is analysed. The modified boundary condition is shown to lead to a separation in wavenumber between “real” instabilities of the underlying problem and spurious numerical instabilities, which should enable future implementations to remove the numerical instabilities while retaining any “real” instabilities. The analytic benchmark solution developed here may prove useful for numerical validation purposes, especially with regard to stability and instability.