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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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References

Baker, N. H., Moore, D. W. and Spiegel, E. A.. 1971, Quart. J. Mech. Appl. Maths., 24, 391.CrossRefGoogle Scholar
Daniels, P. G.. 1977, Proc. Roy. Soc., A358, 173.Google Scholar
Garg, V. K. and Rouleau, W. T.. 1972, J. Fluid Mech., 54, 113.Google Scholar
Gaster, M.. 1962, J. Fluid Mech., 14, 222.Google Scholar
Gaster, M.. 1965, J. Fluid Mech, 22, 433.CrossRefGoogle Scholar
Gaster, M.. 1974, J. Fluid Mech, 66, 465.CrossRefGoogle Scholar
Gill, A. E.. 1965, J. Fluid Mech, 21, 145.CrossRefGoogle Scholar
Hall, P. and Walton, I. C.. 1977, Proc. Roy. Soc., A358, 199.Google Scholar
Hocking, L. M.. 1977, Quart. J. Meek Appl. Maths., 30, 343.CrossRefGoogle Scholar
Lin, C. C.. 1955, The theory of hydrodynamic stability (Cambridge University Press).Google Scholar
McLachlan, N. W.. 1947, Theory and application of Mathieu functions (Cambridge University Press).Google Scholar
Reid, W. H.. 1965, in “Basic developments in fluid dynamics”, Vol. 1 (ed. Holt, M.) (Academic Press).Google Scholar
Salwen, H. and Grosch, C. E.. 1972, J. Fluid Mech, 54, 93.Google Scholar
Segel, L. A.. 1969, J. Fluid Mech, 38, 203.Google Scholar
Smith, F. T.. 1976a, Mathematika, 23, 62.CrossRefGoogle Scholar
Smith, F. T.. 1976b, Quart. J. Meek Appl. Maths., 29, 365.Google Scholar
Smith, F. T.. 1977, J. Fluid Mech, 79, 631.Google Scholar
Smith, F. T.. 1979a (Part 2), Mathematika, 26, 211–223.CrossRefGoogle Scholar
Smith, F. T.. 1979b, Proc. Roy. Soc., A, 366, 91.Google Scholar
Stewartson, K. and Stuart, J. T.. 1971, J. Fluid Mech, 48, 529.Google Scholar
Stuart, J. T.. 1963, Ch. IX of “Laminar boundary layers” {ed. Rosenhead, L.).Google Scholar