On the eigenvalues of the Orr-Sommerfeld equation

Abstract

The characteristic function defining the eigenvalues of the Orr-Sommerfeld equation is discussed and it is shown how the expected analytic properties of this function can be exploited to generate series expansions defining eigenvalues within the circle of convergence. This technique is applied to the modes arising in the Blasius flat-plate boundary layer (treated as a parallel flow), for which the complex wavenumber α can be expanded as a convergent power series in the complex frequency parameter β in various regions of the β plane. Such power series are effectively equivalent to Fourier expansions and the properties of the latter are used to find the coefficients.

A square-root singularity in the relationship between α and β is found and it is shown how α can, nevertheless, be expressed in terms of β as the sum of one regular series and the square root of a second regular series. The loci of the real and imaginary parts of α have been computed from these series and show the behaviour in the neighbourhood of the branch point.

The series description provides a particularly simple and rapid method of evalauting eigenvalues and their derivatives within any given region.