Reflection of an acoustic line source by an impedance surface with uniform flow

Abstract

An exact analytic solution is derived for the 2D acoustic pressure field generated by a time-harmonic line mass source located above an impedance surface with uniform grazing flow. Closed-form asymptotic solutions in the far-field are also provided. The analysis is valid for both locally-reacting and nonlocally-reacting impedances, as is demonstrated by analysing a nonlocally reacting effective impedance representing the presence of a thin boundary layer over the surface. The analytic solution may be written in a form suggesting a generalization of the method of images to account for the impedance surface. The line source is found to excite surface waves on the impedance surface, some of which may be leaky waves which contradict the assumption of decay away from the surface predicted in previous analyses of surface waves with flow. The surface waves may be treated either (correctly) as unstable waves or (artificially) as stable waves, enabling comparison with previous numerical or mathematical studies which make either of these assumptions.The computer code for evaluating the analytic solution and far-field asymptotics is provided in the supplementary material. It is hoped this work will provide a useful benchmark solution for validating 2D numerical acoustic codes.

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

“…A complete analytical solution for this benchmark problem is provided in reference [8] and it includes a surface wave which can be treated as an instability. The analytical solution is shown in figure 2.3a, and this unstable surface wave can be seen to emanate from a point on the liner located downstream of the source.…”

“…This condition assumes an infinitely thin boundary layer, and effectively uses a vortex sheet model where the continuity of pressure and normal displacement is applied [16,39,15]. In the limit of an infinitely small source (w → 0) the present test case is equivalent to the benchmark problem proposed in reference [8] and for which a complete analytical solution has been made available.…”

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

“…We achieve this here by considering as a test case a simple two-dimensional situation that has previously been investigated analytically [39], sketched in figure 1. A fluid with constant density ρ 0 ≡ 1 and constant sound speed c 0 ≡ 1 occupies y < 0 and flows with uniform Mach number M in the positive x-direction past a wall at y = 0, with the Euler equations therefore implying the steady pressure p 0 is constant.…”

“…The first is the analytic modal analysis and long-time limit of Ref. 39, calculated using the code in the supplementary material of that reference. Since this is an analytic solution of the LEE in an unbounded half-space, effects due to discretization and the finite computational domain size are absent in this solution.…”

“…In order to verify that the numerical scheme developed here accurately reproduces the correct results and stability of the underlying equations, we will here consider a simple situation of a time-harmonic line source in a uniform flow past an acoustic liner which admits an analytic solution [39]. Following a description of the general method in §2, this simple situation is described in §3, together with the numerical scheme used and a comparison of numerical results with the analytic solution.…”