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On the stability of attachment-line boundary layers. Part 1. The incompressible swept Hiemenz flow

Published online by Cambridge University Press:  26 April 2006

Ray-Sing Lin  and
Mujeeb R. Malik
Ray-Sing Lin
Affiliation:
High Technology Corporation, PO Box 7262, Hampton VA 23666, USA
Mujeeb R. Malik
Affiliation:
High Technology Corporation, PO Box 7262, Hampton VA 23666, USA
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Abstract

The stability of the incompressible attachment-line boundary layer is studied by solving a partial-differential eigenvalue problem. The basic flow near the leading edge is taken to be the swept Hiemenz flow which represents an exact solution of the Navier-Stokes (N-S) equations. Previous theoretical investigations considered a special class of two-dimensional disturbances in which the chordwise variation of disturbance velocities mimics the basic flow and renders a system of ordinary-differential equations of the Orr-Sommerfeld type. The solution of this sixth-order system by Hall, Malik & Poll (1984) showed that the two-dimensional disturbance is stable provided that the Reynolds number $\overline{R} < 583.1 $. In the present study, the restrictive assumptions on the disturbance field are relaxed to allow for more general solutions. Results of the present analysis indicate that unstable perturbations other than the special symmetric two-dimensional mode referred to above do exist in the attachment-line boundary layer provided $\overline{R} > 646 $. Both symmetric and antisymmetric two- and three-dimensional eigenmodes can be amplified. These unstable modes with the same spanwise wavenumber travel with almost identical phase speeds, but the eigenfunctions show very distinct features. Nevertheless, the symmetric two-dimensional mode always has the highest growth rate and dictates the instability. As far as the special two-dimensional mode is concerned, the present results are in complete agreement with previous investigations. One of the major advantages of the present approach is that it can be extended to study the stability of compressible attachment-line flows where no satisfactory simplified approaches are known to exist.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 311 , 25 March 1996 , pp. 239 - 255
Copyright
© 1996 Cambridge University Press

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