An analytic Green's function for a lined circular duct containing uniform mean flow

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