Applications of Symmetry Analysis in Stability Theory

  • Andreas Nold
  • Martin Oberlack
  • Alexei F. Cheviakov
Part of the Notes on Numerical Fluid Mechanics and Multidisciplinary Design book series (NNFM, volume 112)


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.


Reynolds Number Stability Theory Invariant Solution Symmetry Analysis Additional Symmetry 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Andreas Nold
    • 1
  • Martin Oberlack
    • 1
  • Alexei F. Cheviakov
    • 2
  1. 1.Fachgebiet StrömungsmechanikTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Department of Mathematics and StatisticsUniversity of SaskatchewanSaskatoonCanada

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