A classification of duct modes based on surface waves

For the relatively high frequencies relevant in a turbofan engine duct the modes of a lined section may be classified in two categories: genuine acoustic 3D duct modes resulting from the finiteness of the duct geometry, and 2D surface waves that exist only near the wall surface in a way essentially independent of the rest of the duct. Per frequency and circumferential order there are at most 4 surface waves. They occur in two kinds: 2 acoustic surface waves that exist with and without mean flow, and 2 hydrodynamic surface waves that exist only with mean flow. The number and location of the surface waves depends on the wall impedance Z and mean flow Mach number. When Z is varied, an acoustic mode may change via small transition zones into a surface waves and vice versa. Compared to the acoustic modes, the surface waves behave-for example as a function of the wall impedance-rather differently as they have their own dynamics. They are therefore more difficult to find. A method is described to trace all modes by continuation in Z from the hard-wall values, by starting in an area of the complex Z plane without surface waves. Keywords: Acoustic surface waves; Duct geometries; Duct modes; High frequency HF; Mean flow; Transition zones; Wall impedance; Wall surface

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

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