A full discrete dispersion analysis of time-domain simulations of acoustic liners with flow

Abstract

The effect of flow over an acoustic liner is generally described by the Myers impedance condition. The use of this impedance condition in time-domain numerical simulations has been plagued by stability issues, and various ad hoc techniques based on artificial damping or filtering have been used to stabilise the solution. The theoretical issue leading to the ill-posedness of this impedance condition in the time domain is now well understood. For computational models, some trends have been identified, but no detailed investigation of the cause of the instabilities in numerical simulations has been undertaken to date. This paper presents a dispersion analysis of the complete numerical model, based on finite-difference approximations, for a twodimensional model of a uniform flow above an impedance surface. It provides insight into the properties of the instability in the numerical model and clarifies the parameters that influence its presence. The dispersion analysis is also used to give useful information about the accuracy of the acoustic solution.Comparison between the dispersion analysis and numerical simulations shows that the instability associated with the Myers condition can be identified in the numerical model, but its properties differ significantly from that of the continuous model. The unbounded growth of the instability in the continuous model is not present in the numerical model due to the wavenumber aliasing inherent to numerical approximations. Instead, the numerical instability includes a wavenumber component behaving as an absolute instability. The trend previously reported that the instability is more likely to appear with fine grids is explained. While the instability in the numerical model is heavily dependent on the spatial resolution, it is well resolved in time and is not sensitive to the time step. In addition filtering techniques to stabilise the solutions are considered and it is found that, while they can reduce the instability in some cases, they do not represent a systematic or robust solution in general.

Time-domain implementation of an impedance boundary condition with boundary layer correction

“…For further details the reader is referred to Ref. 19. Once the DDA is validated against the full numerical time-domain solution (and the agreement will be seen to be excellent), the DDA can be used to investigate the effects of varying parameters, introducing or removing damping, and changing the way the boundary condition is discretized and implemented, without the noise inherent in post-processing the output of the full time-domain solution.…”

“…20 for further details. Modified BC [39] DDA [19] Figure 6: Exponential growth rate as a function of wavenumber for case B, grid 4. The time-domain computational result is obtained from a discrete Fourier transform of the growth between times t = 3.2 and t = 4.5.…”

“…The time-domain computational result is obtained from a discrete Fourier transform of the growth between times t = 3.2 and t = 4.5. Also plotted are growth rates given by the discrete dispersion analysis [19], the continuous analytic result using the modified boundary condition [39], and the dispersion analysis from solving the Pridmore-Brown equation resolving the boundary layer.…”

“…However, high-frequency numerical instabilities are invariably present in time-domain simulations [4,17,18] when such impedances are used with slipping mean flow using the Myers boundary condition (1). These numerical instabilities are different from the instabilities due to the use of low-dissipation numerical schemes mentioned above [19]. Stabilizing the boundary condition requires the use of strong, wide-band filters to indiscriminately remove any form of instability.…”

“…One other aim of this paper is therefore to describe general numerical difficulties that arise when simulating unstable linear systems. In particular, careful distinction is needed between genuine instabilities of the system being simulated and artificial instabilities introduced by the numerical discretization [19]. While there have been previous investigations of special cases, such as the careful numerical treatment of the continuous spectrum by Marx [24], here we propose for general simulations a general class of artificial instabilities caused by a combination of instability of the underlying physical system and the finite resolution of the numerical discretization.…”