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Linear stability of Poiseuille flow in a circular pipe

Published online by Cambridge University Press:  19 April 2006

Harold Salwen ,
Fredrick W. Cotton  and
Chester E. Grosch
Harold Salwen
Affiliation:
Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030
Fredrick W. Cotton
Affiliation:
Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030
Chester E. Grosch
Affiliation:
Department of Oceanography and Department of Mathematics, Old Dominion University, Norfolk, Virginia 23508
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Abstract

Correction of an error in the matrix elements used by Salwen & Grosch (1972) has brought the results of the matrix-eigenvalue calculation of the linear stability of Hagen–Poiseuille flow into complete agreement with the numerical integration results of Lessen, Sadler & Liu (1968) for azimuthal index n = 1. The n = 0 results were unaffected by the error and the effect of the error for n > 1 is smaller than for n = 1. The new calculations confirm the conclusion that the flow is stable to infinitesimal disturbances.

Further calculations have led to the discovery of a degeneracy at Reynolds number R = 61·452 ± 0·003 and wavenumber α = 0·9874 ± 0·0001, where the second and third eigenmodes have equal complex wave speeds. The variation of wave speed for these two modes has been studied in the vicinity of the degeneracy and shows similarities to the behaviour near the degeneracies found by Cotton and Salwen (see Cotton 1977) for rotating Hagen-Poiseuille flow. Finally, new results are given for n = 10 and 30; the n = 1 results are extended to R = 106; and new results are presented for the variation of the wave speed with αR at high Reynolds number. The high-R results confirm both Burridge & Drazin's (1969) slow-mode approximation and more recent fast-mode results of Burridge.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 98 , Issue 2 , 29 May 1980 , pp. 273 - 284
Copyright
© 1980 Cambridge University Press

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References

Antosiewicz, H. A. 1970 Bessel functions of fractional order. In Handbook of Mathematical Functions (ed. M. Abramowitz & I. A. Stegun). Washington: National Bureau of Standards.
Burridge, D. M. & Drazin, P. G. 1969 Phys. Fluids 12, 264–265.
Cotton, F. W. 1977 A Study of the Linear Stability Problem for Viscous Incompressible Fluids in Circular Geometries by Means of a Matrix-Eigenvalue Approach. Ph.D. dissertation, Stevens Institute of Technology. University Microfilms International, Ann Arbor, Mi. 48106 (Order no. 77-26, 931).
Cotton, F. W. & Salwen, H. 1976 beslib (decus 10-272) and index (decus 10-273). DEC Users Library, Maynard, Mass. 01754.
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Salwen, H. & Grosch, C. E. 1972 J. Fluid Mech. 54, 93–112.