The discrete temporal eigenvalue spectrum of the generalised Hiemenz flow as solution of the Orr-Sommerfeld equation

Abstract

A spectral collocation method is used to obtain the solution to the Orr-Sommerfeld stability equation. The accuracy of the method is established by comparing against well documented flows, such as the plane Poiseuille and the Blasius Boundary layers. The focus is then placed on the generalised Hiemenz flow, an exact solution to the Navier-Stokes equations constituting the base flow at the leading edge of swept cylinders and aerofoils. The spanwise profile of this flow is very similar to that of Blasius but, unlike the latter case, there is no rational approximation leading to the Orr-Sommerfeld equation. We will show that if, based on experimentally obtained intuition, a nonrational reduction of the full system of linear stability equations is attempted and the resulting Orr-Sommerfeld equation is solved, the linear stability critical Reynolds number is overestimated, as has indeed been done in the past. However, as shown by recent Direct Numerical Simulation results, the frequency eigenspectrum of instability waves may still be obtained through solution of the Orr-Sommerfeld equation. This fact lends some credibility to the assumption under which the Orr-Sommerfeld equation is obtained insofar as the identification of the frequency regime responsible for linear growth is concerned. Finally, an argument is presented pointing towards potential directions in the ongoing research for explanation of subcriticality in the leading edge boundary layer.