The critical layer in linear-shear boundary layers over acoustic linings

E.J. Brambley, M. Darau, S.W. Rienstra

Research output: Contribution to journalArticleAcademicpeer-review

36 Citations (Scopus)

Abstract

Acoustics within mean flows are governed by the linearized Euler equations, which possess a singularity wherever the local mean flow velocity is equal to the phase speed of the disturbance. Such locations are termed critical layers, and are usually ignored when estimating the sound field, with their contribution assumed to be negligible. This paper studies fully both numerically and analytically a simple yet typical sheared ducted flow in order to investigate the influence of the critical layer, and shows that the neglect of critical layers is sometimes, but certainly not always, justified. The model is that of a linear-then-constant velocity profile with uniform density in a cylindrical duct, allowing exact Green’s function solutions in terms of Bessel functions and Frobenius expansions. For sources outside the sheared flow, the contribution of the critical layer is found to decay algebraically along the duct as O(1/x^4), where x is the distance downstream of the source. For sources within the sheared ¿ow, the contribution from the critical layer is found to consist of a nonmodal disturbance of constant amplitude representing the hydrodynamic trailing vorticity of the source, and a disturbance decaying algebraically as O(1/x^5). For thin boundary layers, these disturbances trigger the inherent convective instability of the flow. Extra care is required for high frequencies as the critical layer can be neglected only in combination with a particular downstream pole. The advantages of Frobenius expansions over direct numerical calculation are also demonstrated, especially with regard to spurious modes around the critical layer.
Original languageEnglish
Pages (from-to)545-568
JournalJournal of Fluid Mechanics
Volume710
DOIs
Publication statusPublished - 2012

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linings
Linings
Ducts
boundary layers
Boundary layers
Acoustics
shear
Bessel functions
acoustics
Euler equations
Acoustic fields
Vorticity
Green's function
Flow velocity
disturbances
Poles
Hydrodynamics
ducts
ducted flow
expansion

Cite this

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title = "The critical layer in linear-shear boundary layers over acoustic linings",
abstract = "Acoustics within mean flows are governed by the linearized Euler equations, which possess a singularity wherever the local mean flow velocity is equal to the phase speed of the disturbance. Such locations are termed critical layers, and are usually ignored when estimating the sound field, with their contribution assumed to be negligible. This paper studies fully both numerically and analytically a simple yet typical sheared ducted flow in order to investigate the influence of the critical layer, and shows that the neglect of critical layers is sometimes, but certainly not always, justified. The model is that of a linear-then-constant velocity profile with uniform density in a cylindrical duct, allowing exact Green’s function solutions in terms of Bessel functions and Frobenius expansions. For sources outside the sheared flow, the contribution of the critical layer is found to decay algebraically along the duct as O(1/x^4), where x is the distance downstream of the source. For sources within the sheared ¿ow, the contribution from the critical layer is found to consist of a nonmodal disturbance of constant amplitude representing the hydrodynamic trailing vorticity of the source, and a disturbance decaying algebraically as O(1/x^5). For thin boundary layers, these disturbances trigger the inherent convective instability of the flow. Extra care is required for high frequencies as the critical layer can be neglected only in combination with a particular downstream pole. The advantages of Frobenius expansions over direct numerical calculation are also demonstrated, especially with regard to spurious modes around the critical layer.",
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pages = "545--568",
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The critical layer in linear-shear boundary layers over acoustic linings. / Brambley, E.J.; Darau, M.; Rienstra, S.W.

In: Journal of Fluid Mechanics, Vol. 710, 2012, p. 545-568.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

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AU - Brambley, E.J.

AU - Darau, M.

AU - Rienstra, S.W.

PY - 2012

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N2 - Acoustics within mean flows are governed by the linearized Euler equations, which possess a singularity wherever the local mean flow velocity is equal to the phase speed of the disturbance. Such locations are termed critical layers, and are usually ignored when estimating the sound field, with their contribution assumed to be negligible. This paper studies fully both numerically and analytically a simple yet typical sheared ducted flow in order to investigate the influence of the critical layer, and shows that the neglect of critical layers is sometimes, but certainly not always, justified. The model is that of a linear-then-constant velocity profile with uniform density in a cylindrical duct, allowing exact Green’s function solutions in terms of Bessel functions and Frobenius expansions. For sources outside the sheared flow, the contribution of the critical layer is found to decay algebraically along the duct as O(1/x^4), where x is the distance downstream of the source. For sources within the sheared ¿ow, the contribution from the critical layer is found to consist of a nonmodal disturbance of constant amplitude representing the hydrodynamic trailing vorticity of the source, and a disturbance decaying algebraically as O(1/x^5). For thin boundary layers, these disturbances trigger the inherent convective instability of the flow. Extra care is required for high frequencies as the critical layer can be neglected only in combination with a particular downstream pole. The advantages of Frobenius expansions over direct numerical calculation are also demonstrated, especially with regard to spurious modes around the critical layer.

AB - Acoustics within mean flows are governed by the linearized Euler equations, which possess a singularity wherever the local mean flow velocity is equal to the phase speed of the disturbance. Such locations are termed critical layers, and are usually ignored when estimating the sound field, with their contribution assumed to be negligible. This paper studies fully both numerically and analytically a simple yet typical sheared ducted flow in order to investigate the influence of the critical layer, and shows that the neglect of critical layers is sometimes, but certainly not always, justified. The model is that of a linear-then-constant velocity profile with uniform density in a cylindrical duct, allowing exact Green’s function solutions in terms of Bessel functions and Frobenius expansions. For sources outside the sheared flow, the contribution of the critical layer is found to decay algebraically along the duct as O(1/x^4), where x is the distance downstream of the source. For sources within the sheared ¿ow, the contribution from the critical layer is found to consist of a nonmodal disturbance of constant amplitude representing the hydrodynamic trailing vorticity of the source, and a disturbance decaying algebraically as O(1/x^5). For thin boundary layers, these disturbances trigger the inherent convective instability of the flow. Extra care is required for high frequencies as the critical layer can be neglected only in combination with a particular downstream pole. The advantages of Frobenius expansions over direct numerical calculation are also demonstrated, especially with regard to spurious modes around the critical layer.

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