Distribution of complex phase velocities for small disturbances to pipe Poiseuille flow

Abstract

It is numerically known that normal modes for small disturbances to pipe Poiseuille flow have complex phase velocities c which form a Y-shaped set of discrete points in the fourth quadrant, when the Reynolds number R is large. In this paper, the eigenvalue problem of determining c for axisymmetric torsional disturbances is treated, and the Y-shaped distribution of these c is studied analytically (with a little aid from numerics) by using some asymptotic forms of a Whittaker function. As a result, a Y-shaped contour on which eigenvalues c are approximately located is obtained, independent of the wavenumber α and R, from simple equations which contain elementary functions only. Naturally, the location of each individual c depends on α and R. How it changes on the contour when large R is becoming still larger is explained. Several approximate values of c on the contour are compared with c computed by Schmid & Henningson (1994), and their agreement is seen to be good. Furthermore, the limit Re c → 2/3 as Im c → −∞ is rigorously proved when R is fixed at an arbitrary number, which is not required to be large. This limit is shown to be true also with eigenvalues c for axisymmetric meridional disturbances. The alternate distribution of c for torsional and meridional disturbances on a branch of the Y-shaped contour is explained.