THE DISCRETE TEMPORAL EIGENVALUE SPECTRUM OF THE GENERALIZED HIEMENZ FLOW AS SOLUTION OF THE ORR-SOMMERFELD EQUATION

A spectral collocation method is used to obtain the solution to the Or r-Sommerfeld stability equation. The accuracy of the method is establi shed by comparing against well documented flows, such as the plane Poi seuille and the Blasius Boundary layers. The focus is then placed on t he generalised Hiemenz flow, an exact solution to the Navier-Stokes eq uations constituting the base flow at the leading edge of swept cylind ers and aerofoils. The spanwise profile of this flow is very similar t o that of Blasius but, unlike the latter case, there is no rational ap proximation leading to the Orr-Sommerfeld equation. We will show that if, based on experimentally obtained intuition, a nonrational reductio n of the full system of linear stability equations is attempted and th e resulting Orr-Sommerfeld equation is solved, the linear stability cr itical Reynolds number is overestimated, as has indeed been done in th e past. However, as shown by recent Direct Numerical Simulation result s, the frequency eigenspectrum of instability waves may still be obtai ned through solution of the Orr-Sommerfeld equation. This fact lends s ome credibility to the assumption under which the Orr-Sommerfeld equat ion is obtained insofar as the identification of the frequency regime responsible for linear growth is concerned. Finally, an argument is pr esented pointing towards potential directions in the ongoing research for explanation of subcriticality in the leading edge boundary layer.