Numerical study of acoustic modes in ducted shear flow

Abstract

The propagation of small-amplitude modes in an inviscid but sheared mean flow inside a duct is studied numerically. 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. 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, comparison to the asymptotic solution of the problem based on the WKB method is performed. The asymptotic solution is also used as a benchmark for computations with high Helmholtz numbers, where numerical solutions of other authors are not available.The bulk of the analysis concentrates on the influence of the wall lining. The proposed numerical procedure is adapted in order to include Ingard-Myers boundary conditions. In parallel with this, the WKB solution is used to check the numerical predictions of the typical behaviour of the axial wavenumber in the complex plane, when the wall impedance varies in the complex plane.Numerical analysis of the problem with zero mean flow at the wall and acoustic lining shows that the use of Ingard-Myers condition in combination with an appropriate slipstream approximation instead of the actual no-slip mean flow profile gives valid results in the limit of vanishing boundary-layer thickness, although the boundary layer must be very thin in some cases.

The critical layer in sheared flow

“…A relation similar to (2) regarding the existence of the critical layer is mentioned in Ref. [19] for uniform density.…”

Spatial Instability of Boundary Layer Along Impedance Wall

“…The foregoing eigenvalue problem in k was solved numerically by a combination of an implicit finite-difference (backward Euler) scheme for the differential equation and a modified Newton's method to iterate to eigenvalue k as described in [18,19]. To the authors' knowledge, this is the only numerical method currently available in the literature which, apart from the acoustic spectrum, allows to accurately and reliably resolve the continuous (i.e., hydrodynamic) part of the spectrum, where the Pridmore-Brown equation is singular.…”

“…In contrast to the much more general theory of [18,19], the configuration considered here is taken as simple as possible in order obtain canonical and generic results. In particular, the mean flow variables are uniform everywhere except for a ∼ tanh-profile of the axial velocity (see Figure 1) The resulting system of equations (note slight notation differences with [18,19])…”

“…In particular, the mean flow variables are uniform everywhere except for a ∼ tanh-profile of the axial velocity (see Figure 1) The resulting system of equations (note slight notation differences with [18,19])…”

“…Using the Pridmore-Brown model for duct modes of given frequency in shear flow along a lined wall, developed earlier [18,19], we were able to analyse the spatial modes in more detail, in particular as a function of complex frequency. When the axial wave number k i (ω) ∈ C can be traced to the other complex half plane as we let ω ∈ C vary far enough into the complex plane, arguments of causality applied to the time-Fourier transformed field tell us that k i really is an instability [20][21][22][23].…”