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Applications of Symmetry Analysis in Stability Theory

Nold, Andreas and Oberlack, Martin and Cheviakov, Alexei
Dillmann, Andreas and Heller, Gerd and Klaas, Michael and Kreplin, Hans-Peter and Nitsche, Wolfgang and Schröder, Wolfgang (eds.) (2010):
Applications of Symmetry Analysis in Stability Theory.
In: New Results in Numerical and Experimental Fluid Mechanics VII, Berlin Heidelberg, Springer, pp. 257-264, [Online-Edition: http://dx.doi.org/10.1007/978-3-642-14243-7_32],
[Book Section]

Abstract

For decades the stability of nearly parallel shear flows was primarily analyzed employing the Orr-Sommerfeld-Equation (OSE).We show that the OSE is solely based on three symmetries of the linearized Navier-Stokes-Equation for two-dimensional perturbations. In fact, the OSE is a similarity reduction using the latter three symmetries. Though rather successful in boundary layer flows the OSE does not give proper results for the plane channel flow with the classical parabolic flow profile. For this special case we found a new symmetry. It leads to a new ansatz considerably distinct from the OSE with two new similarity variables. We analyzed the scope in which the new ansatz could be used. Finally, we derived a technique to solve the equation via the new ansatz function.

Item Type: Book Section
Erschienen: 2010
Editors: Dillmann, Andreas and Heller, Gerd and Klaas, Michael and Kreplin, Hans-Peter and Nitsche, Wolfgang and Schröder, Wolfgang
Creators: Nold, Andreas and Oberlack, Martin and Cheviakov, Alexei
Title: Applications of Symmetry Analysis in Stability Theory
Language: English
Abstract:

For decades the stability of nearly parallel shear flows was primarily analyzed employing the Orr-Sommerfeld-Equation (OSE).We show that the OSE is solely based on three symmetries of the linearized Navier-Stokes-Equation for two-dimensional perturbations. In fact, the OSE is a similarity reduction using the latter three symmetries. Though rather successful in boundary layer flows the OSE does not give proper results for the plane channel flow with the classical parabolic flow profile. For this special case we found a new symmetry. It leads to a new ansatz considerably distinct from the OSE with two new similarity variables. We analyzed the scope in which the new ansatz could be used. Finally, we derived a technique to solve the equation via the new ansatz function.

Title of Book: New Results in Numerical and Experimental Fluid Mechanics VII
Series Name: Notes on Numerical Fluid Mechanics and Multidisciplinary Design
Volume: 112
Place of Publication: Berlin Heidelberg
Publisher: Springer
Divisions: 16 Department of Mechanical Engineering > Fluid Dynamics (fdy)
16 Department of Mechanical Engineering
Date Deposited: 24 Aug 2011 17:56
Official URL: http://dx.doi.org/10.1007/978-3-642-14243-7_32
Additional Information:

dio: 10.1007/978-3-642-14243-7_32

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