Distribution patterns of eigenvalues of laminar pipe flow. 2nd Report. Classification of modes based on dynamics of the system.

Abstract

This study aims to clarify the structure and dynamic behavior of the linear system which describes the small perturbation of a laminar pipe flow. In the preceding paper the eigenvalue problem was formulated for a Hilbert space, based on the spectrum theory. And a numerical method for calculating the eigenvalues was proposed, together with a measure of accuracy. Applying the proposed method, this paper discusses the distribution of eigenvalues and the mode of perturbations for the Poiseuille pipe flow. The wave perturbations for various azimuthal and axial wave numbers are investigated with a fixed Reynolds number. It is shown that the distribution of eigenvalues in a complex phase velocity plane has a tree like shape. The mode of perturbations is divided into three classes : slow, fast and mean modes by the axial phase velocity, or wall, center and neutral modes by the radial distribution of the magnitude of the eigenfunction. For each type of mode, the location of the corresponding eigenvalue in the complex phase velocity plane is clarified, and the dependence of the eigenvalue on the original linear dynamic system is also clarified by computer calculations.