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

Abstract

A time-domain boundary condition is derived that accounts for the acoustic impedance of a thin boundary layer over an impedance boundary, based on the asymptotic frequency-domain boundary condition of Brambley [ , AIAA J. 49(6), pp. 1272[ -1282. A finite-difference reference implementation of this condition is presented and carefully validated against both an analytic solution and a discrete dispersion analysis for a simple test case. The discrete dispersion analysis enables the distinction between real physical instabilities and artificial numerical instabilities. The cause of the latter is suggested to be a combination of the real physical instabilities present and the aliasing and artificial zero group velocity of finite-difference schemes. It is suggested that these are general properties of any numerical discretization of an unstable system. Existing numerical filters are found to be inadequate to remove these artificial instabilities as they have a too wide pass band. The properties of numerical filters required to address this issue are discussed and a number of selective filters are presented that may prove useful in general. These filters are capable of removing only the artificial numerical instabilities, allowing the reference implementation to correctly reproduce the stability properties of the analytic solution.

Acoustics in a Two-Deck Viscothermal Boundary Layer over an Impedance Surface

“…To proceed, we follow Brambley and Gabard [19] by introducing a number of new variables, the interpretation of which we leave until later. First, we appeal to the fact thatp u satisfies the inviscid linearised Euler equations in a uniform flow to make the substitution kp u /ρ 0 (ω − U 0 k) =ũ u .…”

“…Therefore the appearance of the operator ∂ 1 2 t in (69) is expected, due to the inclusion of viscous and thermal dissipation. An implementation of the proposed time-domain formulation is left for future work, although it is hypothesised that a combination of the method used in Brambley and Gabard [19] and the Grünwal-Letnikov finite difference scheme for the non-integer order derivatives [see, e.g., 36] might be profitable.…”

“…The results of such work are not always directly applicable to time-domain numerical solvers. Recent work has made progress in implementing an inviscid impedance boundary condition in the time domain to account for a finite region of shear [19]. By incorporating the modified Myers boundary condition (reformulated in the time domain) into a linearised Euler solver, the unphysical numerical instabilities associated with time-domain formulations of the ill-posed classical Ingard-Myers boundary condition can be avoided.…”

“…By incorporating the modified Myers boundary condition (reformulated in the time domain) into a linearised Euler solver, the unphysical numerical instabilities associated with time-domain formulations of the ill-posed classical Ingard-Myers boundary condition can be avoided. The physical surface wave instability is always present in the inviscid case [20], however, so the time-domain implementation in Brambley and Gabard [19] still yields instabilities. In the present work, we suggest a time-domain formulation that accounts for viscothermal effects in a thin, sheared boundary layer.…”

“…In the present work, we suggest a time-domain formulation that accounts for viscothermal effects in a thin, sheared boundary layer. Although the boundary condition is not implemented here, it is hoped that the good frequency-domain stability behaviour translates to a stable time-domain boundary condition, provided it is suitably implemented [19] and the boundary layer thickness and Reynolds number are chosen appropriately.…”