Acoustic Modes in a Ducted Shear Flow

Abstract

The propagation of small-amplitude modes in an inviscid but sheared mean flow inside a duct is considered. For isentropic flow in a circular duct with zero swirl and constant mean flow density the pressure modes are described in terms of the eigenvalue problem for the Pridmore-Brown equation. A numerical method similar to the procedure used by Tam & Auriault is proposed for the solution of the modal equation. Since for sufficiently high Helmholtz and wavenumbers, which are of great interest for the applications, the field equation is inherently stiff, special care is taken to insure the stability of the numerical algorithm designed to tackle this problem. The accuracy of the method is checked against the well-known analytical solution for the uniform flow. The numerical method is shown to be consistent with the analytical predictions at least for the Helmholtz numbers up to 100 and the circumferential wavenumber as large as 50, typical Mach numbers being up to 0.65. In order to gain further insight into the possible structure of the modal solutions and to get an independent verification of the robustness of the numerical scheme, the asymptotic solution of the problem based on the WKB method is derived. The comparisons of the WKB solution against the exact potential flow solution show remarkably good agreement between the two. This permits us to use the asymptotic solution as a benchmark for computations with high Helmholtz numbers, where numerical solutions of other authors are not available. Numerical analysis of the problem with zero mean flow at the wall and acoustic lining shows that IngardMyers condition is recovered for vanishing boundary-layer thickness, although the boundary layer must be very thin in some cases.