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Weakly nonlinear stability of viscous vortices in three-dimensional boundary layers

Published online by Cambridge University Press:  26 April 2006

Andrew P. Bassom  and
S. R. Otto
Andrew P. Bassom
Affiliation:
Department of Mathematics, University of Exeter, North Park Road, Exeter, Devon EX4 4QE, UK
S. R. Otto
Affiliation:
ICASE, Mailstop 132c, NASA Langley Research Center, Hampton, VA 23665-5225, USA
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Abstract

Recently it has been demonstrated that three-dimensionality can play an important role in dictating the stability of any Görtler vortices which a particular boundary layer may support. According to a linearized theory, vortices within a high Görtler number flow can take one of two possible forms within a two-dimensional flow supplemented by a small crossflow of size O(Re-½G⅗), where Re is the Reynolds number of the flow and G the Görtler number. Bassom & Hall (1991) showed that these forms are characterized by O(1)-wavenumber inviscid disturbances and larger $O(G^{\frac {1}{5}})$-wave-number modes which are trapped within a thin layer adjacent to the bounding surface. Here we concentrate on the latter, essentially viscous, vortices. These modes are unstable in the absence of crossflow but the imposition of small crossflow has a stabilizing effect. Bassom & Hall (1991) demonstrated the existence of neutrally stable vortices for certain crossflow/wavenumber combinations and here we describe the weakly nonlinear stability properties of these disturbances. It is shown conclusively that the effect of crossflow is to stabilize the nonlinear modes and the calculations herein allow stable finite-amplitude vortices to be found. Predictions are made concerning the likelihood of observing some of these viscous modes within a practical setting and asymptotic work permits discussion of the stability properties of modes with wavenumbers that are small relative to the implied $O(G^{\frac {1}{5}})$ scaling.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 249 , April 1993 , pp. 597 - 618
Copyright
© 1993 Cambridge University Press

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