The Görtler vortex instability mechanism in three-dimensional boundary layers

Abstract
It is well known that the two-dimensional boundary layer on a concave wall is centrifugally unstable with respect to vortices aligned with the basic flow for sufficiently high values of the Gortler number. However, in most situations of practical interest the basic flow is three-dimensional and previous theoretical investigations do not apply. In this paper the linear stability of the flow over an infinitely long swept wall of variable curvature is considered. If there is no pressure gradient in the boundary layer it is shown that the instability problem can always be related to an equivalent two-dimensional calculation. However, in general, this is not the case and even for small values of the crossflow velocity field dramatic differences between the two- and three-dimensional problems emerge. In particular, it is shown that when the relative size of the crossflow and chordwise flow is O(Re$^{-\frac{1}{2}}$), where Re is the Reynolds number of the flow, the most unstable mode is time-dependent. When the size of the crossflow is further increased, the vortices in the neutral location have their axes locally perpendicular to the vortex lines of the basic flow. In this regime the eigenfunctions associated with the instability become essentially `centre modes' of the Orr-Sommerfeld equation destabilized by centrifugal effects. The critical Gortler number for such modes can be predicted by a large wavenumber asymptotic analysis; the results suggest that for order unity values of the ratio of the crossflow and chordwise velocity fields, the Gortler instability mechanism is almost certainly not operational.

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