A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer

Abstract

A numerical study is made of the temporal eigenvalue spectrum of the Orr-Sommerfeld equation for the Blasius boundary layer. Unlike channel flows, there is no mathematical proof that this flow has an infinite spectrum of discrete eigenvalues. The Orr-Sommerfeld equation is integrated numerically, and the eigenvalues located by tracing out the contour lines in the complex wave velocity (c = c_r + ic_i) plane on which the real and imaginary parts of the secular determinant are zero. This method gives only a finite and small number of discrete eigenvalues for a wide range of Reynolds numbers and wavenumbers. The spectrum of plane Poiseuille flow is used as a guide to study the spectrum of an artificial two wall flow which consists of two Blasius boundary layers. As the upper boundary of this flow moves to infinity, it is found that the portion of the spectrum with an infinite number of eigenvalues moves towards c_r = 1 and the spacing between eigenvalues goes to zero. It is concluded, on the basis of this result and the contour method, that the original few eigenvalues found are the only discrete eigenvalues that exist for Blasius flow over a wide portion of the c plane for c_r < 1 and c_r > 1. It is suggested that the discrete spectrum is supplemented by a continuous spectrum which lies along the c_r = 1 axis for c_i < −α/R.