Boundary layer thickness effects of the hydrodynamic instability along an impedance wall

S.W. Rienstra, M. Darau

Research output: Book/ReportReportAcademic

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Abstract

Abstract: The Ingard-Myers condition, modelling the effect of an impedance wall under a mean fl ow by assuming a vanishingly thin boundary layer, is known to lead to an ill-posed problem in time-domain. By analysing the stability of a linear-then-constant mean flow over a mass-spring-damper liner in a 2D incompressible limit, we show that the fl ow is absolutely unstable for h smaller than a critical hc and convectively unstable or stable otherwise. This critical hc is by nature independent of wave length or frequency and is a property of liner and mean flow only. An analytical approximation of hc is given, which is complemented by a contourplot covering all parameter values. For an aeronautically relevant example, hc is shown to be extremely small, which explains why this instability has never been observed in industrial practice. A systematically regularised boundary condition, to replace the Ingard-Myers condition, is proposed that retains the effects of a finite h, such that the stability of the approximate problem correctly follows the stability of the real problem. Keywords: Aeroacoustics, Boundary layer stability, Impedance wall.
Original languageEnglish
Place of PublicationEindhoven
PublisherTechnische Universiteit Eindhoven
Number of pages15
Publication statusPublished - 2010

Publication series

NameCASA-report
Volume1069
ISSN (Print)0926-4507

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boundary layer thickness
hydrodynamics
impedance
linings
boundary layer stability
aeroacoustics
dampers
boundary layers
coverings
boundary conditions
approximation

Cite this

Rienstra, S. W., & Darau, M. (2010). Boundary layer thickness effects of the hydrodynamic instability along an impedance wall. (CASA-report; Vol. 1069). Eindhoven: Technische Universiteit Eindhoven.
Rienstra, S.W. ; Darau, M. / Boundary layer thickness effects of the hydrodynamic instability along an impedance wall. Eindhoven : Technische Universiteit Eindhoven, 2010. 15 p. (CASA-report).
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Rienstra, SW & Darau, M 2010, Boundary layer thickness effects of the hydrodynamic instability along an impedance wall. CASA-report, vol. 1069, Technische Universiteit Eindhoven, Eindhoven.

Boundary layer thickness effects of the hydrodynamic instability along an impedance wall. / Rienstra, S.W.; Darau, M.

Eindhoven : Technische Universiteit Eindhoven, 2010. 15 p. (CASA-report; Vol. 1069).

Research output: Book/ReportReportAcademic

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T1 - Boundary layer thickness effects of the hydrodynamic instability along an impedance wall

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AU - Darau, M.

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N2 - Abstract: The Ingard-Myers condition, modelling the effect of an impedance wall under a mean fl ow by assuming a vanishingly thin boundary layer, is known to lead to an ill-posed problem in time-domain. By analysing the stability of a linear-then-constant mean flow over a mass-spring-damper liner in a 2D incompressible limit, we show that the fl ow is absolutely unstable for h smaller than a critical hc and convectively unstable or stable otherwise. This critical hc is by nature independent of wave length or frequency and is a property of liner and mean flow only. An analytical approximation of hc is given, which is complemented by a contourplot covering all parameter values. For an aeronautically relevant example, hc is shown to be extremely small, which explains why this instability has never been observed in industrial practice. A systematically regularised boundary condition, to replace the Ingard-Myers condition, is proposed that retains the effects of a finite h, such that the stability of the approximate problem correctly follows the stability of the real problem. Keywords: Aeroacoustics, Boundary layer stability, Impedance wall.

AB - Abstract: The Ingard-Myers condition, modelling the effect of an impedance wall under a mean fl ow by assuming a vanishingly thin boundary layer, is known to lead to an ill-posed problem in time-domain. By analysing the stability of a linear-then-constant mean flow over a mass-spring-damper liner in a 2D incompressible limit, we show that the fl ow is absolutely unstable for h smaller than a critical hc and convectively unstable or stable otherwise. This critical hc is by nature independent of wave length or frequency and is a property of liner and mean flow only. An analytical approximation of hc is given, which is complemented by a contourplot covering all parameter values. For an aeronautically relevant example, hc is shown to be extremely small, which explains why this instability has never been observed in industrial practice. A systematically regularised boundary condition, to replace the Ingard-Myers condition, is proposed that retains the effects of a finite h, such that the stability of the approximate problem correctly follows the stability of the real problem. Keywords: Aeroacoustics, Boundary layer stability, Impedance wall.

M3 - Report

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Rienstra SW, Darau M. Boundary layer thickness effects of the hydrodynamic instability along an impedance wall. Eindhoven: Technische Universiteit Eindhoven, 2010. 15 p. (CASA-report).