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

Published online by Cambridge University Press:  10 February 2009

TAKAHIRO NISHIYAMA
TAKAHIRO NISHIYAMA*
Affiliation:
Department of Applied Science, Yamaguchi University, Ube 755-8611, Japan
*
†Email address for correspondence: t-nishi@yamaguchi-u.ac.jp
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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.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 620 , 10 February 2009 , pp. 299 - 312
Copyright
Copyright © Cambridge University Press 2009

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References

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