An Analytic Green's Function for a Lined Circular Duct Containing Uniform Mean Flow

Abstract

An analytic Green's function is derived for a lined circular duct, both hollow and annular, containing uniform mean flow, from first principles by Fourier transformation. The derived result takes the form of a common mode series. All modes are assumed to decay in their respective direction of propagation. A more comprehensive causality analysis suggests the possibility of upstream modes being really downstream instabilities. As their growth rates are usually exceptionally large, this possibility is not considered in the present study.We show that the analytic Green's function for a lined hollow circular duct, containing uniform mean flow, is essentially identical to that used by Tester e.a. in the Cargill splice scattering model. The Green's function for the annular duct is new.Comparisons between the numerically obtained modal amplitudes of Alonso e.a. and the present analytic results for a lined, hollow circular duct show good agreement without flow, irrespective of how many modes are included in the matrix inversion for the numerical mode amplitudes. With flow, the mode amplitudes do not agree but as the number of modes included in the matrix inversion is increased the numerically obtained modal amplitudes of Alonso e.a. appear to be converging to the present analytical result.In practical applications our closed form analytic Green's function will be computationally more efficient, especially at high frequencies of practical interest to aero-engine applications, and the analytic form for the mode amplitudes could permit future modelling advances not possible from the numerical equivalent.

Numerical study of acoustic modes in ducted shear flow

“…These appear to be much more difficult than one might think on the face of it and require special investigation, Rienstra & Tester [14], Rienstra [15,16], Brambley & Peake [17]. Therefore, we decided to leave these issues outside the scope of the present work.…”

“…The relevant numerical procedure, which has also been used to integrate the generic problem (11,13,14), is described in the next Section. However, prior to proceeding to the numerical analysis, we briefly summarize here the main analytic results for the eigenvalue problem (15,16,17) reported in Vilenski & Rienstra [12].…”

“…In the eigenvalue problem (11,13,14) the excitation frequency ω and the circumferential wavenumber m are given parameters while the axial wavenumber k is the unknown spectral variable.…”

“…The eigenvalue problem (11,13,14) was solved numerically using the following method. The field equation (11) was first rewritten in the form of a system of two first-order ordinary differential equations…”

Spatial Instability of Boundary Layer Along Impedance Wall

“…The aero-acoustics community followed them and now the Ingard-Myers boundary condition is the state-of-the-art for this kind of mean flows. However, more refined analyses [3][4][5][6] as well as numerical time-domain results [7][8][9] indicate that the wall vortex sheet along the lined wall may be spatially unstable. This instability is to be interpreted as related to the Kelvin-Helmholtz instability [10] inherent to vortex sheets separating two inviscid fluids of different velocity.…”