ON THE STABILITY OF ATTACHMENT-LINE BOUNDARY-LAYERS .1. THE INCOMPRESSIBLE SWEPT HIEMENZ FLOW

The stability of the incompressible attachment-line boundary layer is studied by solving a partial-differential eigenvalue problem. The basi c flow near the leading edge is taken to be the swept Hiemenz flow whi ch 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 disturban ce velocities mimics the basic flow and renders a system of ordinary-d ifferential equations of the Orr-Sommerfeld type. The solution of this sixth-order system by Hall, Malik & Poll (1984) showed that the two-d imensional disturbance is stable provided that the Reynolds number (R) over bar < 583.1. In the present study, the restrictive assumptions o n the disturbance field are relaxed to allow for more general solution s. Results of the present analysis indicate that unstable perturbation s other than the special symmetric two-dimensional mode referred to ab ove do exist in the attachment-line boundary layer provided (R) over b ar > 646. Both symmetric and antisymmetric two- and three-dimensional eigenmodes can be amplified. These unstable modes with the same spanwi se wavenumber travel with almost identical phase speeds, but the eigen functions 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 investigat ions. One of the major advantages of the present approach is that it c an be extended to study the stability of compressible attachment-line flows where no satisfactory simplified approaches are known to exist.