The critical layer in sheared flow

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

Research output: Book/ReportReportAcademic

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Abstract

Critical layers arise as a singularity of the linearized Euler equations when the phase speed of the disturbance is equal to the mean flow velocity. They are usually ignored when estimating the sound field, with their contribution assumed to be negligible. It is the aim of this paper to study fully both numerically and analytically a simple yet typical sheared ducted flow in order to distinguish between situations when the critical layer may or may not be ignored. The model is that of a linear-then-constant velocity profile with uniform density in a cylindrical duct, allowing for exact Green’s function solutions in terms of Bessel functions and Frobenius expansions. It is found that the critical layer contribution decays algebraically in the constant flow part, with an additional contribution of constant amplitude when the source is in the boundary layer, an additional contribution of constant amplitude is excited, representing the hydrodynamic trailing vorticity of the source. This immediately triggers, for thin boundary layers, the inherent convective instability in the flow. Extra care is required for high frequencies as the critical layer can be neglected only together with the pole beneath it. For low frequencies this pole is trapped in the critical layer branch cut.
Original languageEnglish
Place of PublicationEindhoven
PublisherTechnische Universiteit Eindhoven
Number of pages19
Publication statusPublished - 2011

Publication series

NameCASA-report
Volume1135
ISSN (Print)0926-4507

Fingerprint

ducted flow
boundary layers
poles
Bessel functions
sound fields
ducts
vorticity
estimating
disturbances
Green's functions
flow velocity
velocity distribution
actuators
hydrodynamics
low frequencies
expansion
decay

Cite this

Brambley, E. J., Darau, M., & Rienstra, S. W. (2011). The critical layer in sheared flow. (CASA-report; Vol. 1135). Eindhoven: Technische Universiteit Eindhoven.
Brambley, E.J. ; Darau, M. ; Rienstra, S.W. / The critical layer in sheared flow. Eindhoven : Technische Universiteit Eindhoven, 2011. 19 p. (CASA-report).
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author = "E.J. Brambley and M. Darau and S.W. Rienstra",
year = "2011",
language = "English",
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Brambley, EJ, Darau, M & Rienstra, SW 2011, The critical layer in sheared flow. CASA-report, vol. 1135, Technische Universiteit Eindhoven, Eindhoven.

The critical layer in sheared flow. / Brambley, E.J.; Darau, M.; Rienstra, S.W.

Eindhoven : Technische Universiteit Eindhoven, 2011. 19 p. (CASA-report; Vol. 1135).

Research output: Book/ReportReportAcademic

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AB - Critical layers arise as a singularity of the linearized Euler equations when the phase speed of the disturbance is equal to the mean flow velocity. They are usually ignored when estimating the sound field, with their contribution assumed to be negligible. It is the aim of this paper to study fully both numerically and analytically a simple yet typical sheared ducted flow in order to distinguish between situations when the critical layer may or may not be ignored. The model is that of a linear-then-constant velocity profile with uniform density in a cylindrical duct, allowing for exact Green’s function solutions in terms of Bessel functions and Frobenius expansions. It is found that the critical layer contribution decays algebraically in the constant flow part, with an additional contribution of constant amplitude when the source is in the boundary layer, an additional contribution of constant amplitude is excited, representing the hydrodynamic trailing vorticity of the source. This immediately triggers, for thin boundary layers, the inherent convective instability in the flow. Extra care is required for high frequencies as the critical layer can be neglected only together with the pole beneath it. For low frequencies this pole is trapped in the critical layer branch cut.

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Brambley EJ, Darau M, Rienstra SW. The critical layer in sheared flow. Eindhoven: Technische Universiteit Eindhoven, 2011. 19 p. (CASA-report).