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

An analytic Green's function for a lined circular duct containing uniform mean flow
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. 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 et al. in the Cargill splice scattering model. The explicit form of the Green's function for the annular duct is new.

A more comprehensive causality analysis suggests the possibility of certain upstream modes being really downstream instabilities. As their growth rates are usually exceptionally large, including these modes as instabilities is both not practical and in disagreement with most (not all) experiments. Therefore, we outline the possibility but do not include them in the presented examples. We follow the “modelling assumption” that all modes decay in their respective direction of propagation.

To illustrate the advantages of our analytic result compared to the matrix inversion technique of Alonso et al., we compute the mode amplitudes from both methods for a typical aircraft engine intake condition. The comparisons 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, enough to include any important surface waves, the numerically obtained modal amplitudes of Alonso et al. 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. It also may have application to post-processing of phased array measurements inside lined ducts.
0022-460X
994-1016
Rienstra, Sjoerd W.
74822d94-5be4-4f91-9c68-300ef07ee36a
Tester, Brian J.
1bd4a793-131b-4173-93cc-3eca70b2d116
Rienstra, Sjoerd W.
74822d94-5be4-4f91-9c68-300ef07ee36a
Tester, Brian J.
1bd4a793-131b-4173-93cc-3eca70b2d116

Rienstra, Sjoerd W. and Tester, Brian J. (2008) An analytic Green's function for a lined circular duct containing uniform mean flow. Journal of Sound and Vibration, 317 (3-5), 994-1016. (doi:10.1016/j.jsv.2008.03.048).

Record type: Article

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. 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 et al. in the Cargill splice scattering model. The explicit form of the Green's function for the annular duct is new.

A more comprehensive causality analysis suggests the possibility of certain upstream modes being really downstream instabilities. As their growth rates are usually exceptionally large, including these modes as instabilities is both not practical and in disagreement with most (not all) experiments. Therefore, we outline the possibility but do not include them in the presented examples. We follow the “modelling assumption” that all modes decay in their respective direction of propagation.

To illustrate the advantages of our analytic result compared to the matrix inversion technique of Alonso et al., we compute the mode amplitudes from both methods for a typical aircraft engine intake condition. The comparisons 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, enough to include any important surface waves, the numerically obtained modal amplitudes of Alonso et al. 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. It also may have application to post-processing of phased array measurements inside lined ducts.

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More information

e-pub ahead of print date: 9 June 2008
Published date: 11 November 2008
Organisations: Fluid Dynamics & Acoustics Group

Identifiers

Local EPrints ID: 65189
URI: https://eprints.soton.ac.uk/id/eprint/65189
ISSN: 0022-460X
PURE UUID: 33cc5505-cc85-4c8c-bbad-0c0c62c7da9a

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Date deposited: 05 Feb 2009
Last modified: 05 Dec 2018 17:33

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