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Nonlinear Development of Viscous Gertler Vortices in a Three‐Dimensional Boundary Layer
Author(s) -
Otto S. R.,
Bassom Andrew P.
Publication year - 1994
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm199492117
Subject(s) - wavenumber , vortex , reynolds number , boundary layer , physics , nonlinear system , mechanics , classical mechanics , flow (mathematics) , turbulence , optics , quantum mechanics
In many practical situations where Görder vortices are known to arise, the underlying basic velocity profile is three‐dimensional. Only in recent years have studies been made of the stability of vortices in three‐dimensional flows, and it has been shown that only a small crossflow velocity component is required in order to stabilize the Görder mechanism completely. For large Görder number (G ≫ 1) flows, the most unstable linear vortex within a two‐dimensional boundary layer has a wavenumber of O (G ⅕ ) and a corresponding growth rate of O (G ⅗ ). Imposition of a crossflow component of size O ( R e −½ G ⅗ ) (where R e is the Reynolds number of the flow) is sufficient to cause these higher wavenumber Gertler modes to decay. Indeed, for certain crossflow/vortex wavenumber combinations, the vortices can be made neutrally stable. A weakly nonlinear analysis of near neutral modes reveals that this slight nonlinearity is stabilizing and so can lead to finite amplitude equilibrium states. In the present work, we give a nonlinear account of the fate of the O (G ⅕ ) wavenumber vortices as they evolve downstream. A study of the large wavenumber modes within a two‐dimensional boundary layer [5], has shown that the effect of this strong nonlinearity is destabilizing and leads to a finite distance breakdown in the flow structure. Here we include the influence of the crossflow component and demonstrate how the stabilizing effects of crossflow and the destabilizing nature of nonlinearity compete. Our calculations can also describe unsteadiness in the vortex structure and they allow us to speculate upon the relative likelihoods of observing various members of the nonlinear Görder modes in practice.