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Instability of flow through pipes of general cross-section, Part 1

Part of: Incompressible viscous fluids Hydrodynamic stability

Published online by Cambridge University Press:  26 February 2010

F. T. Smith
F. T. Smith
Affiliation:
University of Western Ontario, London, Ontario, Canada.
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Summary

The temporal and spatial linear instability of Poiseuille flow through pipes of arbitrary cross-section is discussed for large Reynolds numbers (R). For a pipe whose aspect ratio is finite, neutral stability (lower branch) is found to be governed by disturbance modes of large axial wavelength (of order hR, where h is a characteristic cross-sectional dimension). By contrast, spatial instability for finite aspect ratios is governed by length scales between O(h) and O(hR). When the aspect ratio is increased to O(R1/7), however, these two characteristic length scales both become O(R1/7h) and a match with plane channel flow instability is achieved. Thus the general cross-section produces temporal and spatial instability if the aspect ratio is O(R1/7). Further, in the flow in a rectangular pipe neutral stability (lower branch) exists for some finite aspect ratios, while for the flow in any non-circular elliptical pipe spatial instability is possible. It is suggested that both temporal and spatial instability occur for a wide range of pipe cross-sections of finite aspect ratio. Part 2 (Smith 1979a), which studies the upper branch neutral stability, confirms the importance of the O(hR) scale modes in neutral stability for finite aspect ratios.

MSC classification

Secondary: 76D99: None of the above, but in this section 76E05: Parallel shear flows
Type
Research Article
Information
Mathematika , Volume 26 , Issue 2 , December 1979 , pp. 187 - 210
Copyright
Copyright © University College London 1979

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