Linear Stability Theory of Boundary Layer along a Cone Rotating in Axial Flow

A linear stability theory is treated for an incompressible laminar boundary layer along a cone rotating about its axis of symmetry with a constant angular speed in an external forced flow field. Small perturbations are assumed to be of spiral vortices. This leads to one set of perturbation equations including an order of magnitude (δ1/R0), where δ1 denotes a displacement thickness of the boundary layer, calculated from a velocity component in the meridional section, and R0 is a local radius of the cone at instability. A numerical procedure for solving the eigenvalue problem is shown. As an example, calculations are carried out for a cone having a total included angle of 30° at a rotating speed ratio 3, where the rotating speed ratio is defined as the ratio of the circumferential velocity at the cone surface to the external flow velocity at the outer edge of the boundary layer. From stability diagrams obtained herein, critical Reynolds number, spiral angle and the associated eigenvalues are determined. The critical Reynolds number is compared with an experiment.