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On Itoh's finite amplitude stability theory for pipe flow

Published online by Cambridge University Press:  12 April 2006

A. Davey
A. Davey
Affiliation:
School of Mathematics, University of Newcastle upon Tyne, England
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Abstract

In two recent papers Itoh has developed a finite amplitude stability theory which indicates that nonlinearity increases the damping rate of a small but finite amplitude disturbance to flow in a circular pipe when the disturbance is concentrated near the axis of the pipe. For such a centre mode, which is the only mode considered by Itoh, Davey & Nguyen found, in an earlier paper, the opposite result that nonlinearity decreases the damping rate. We examine the reasons for this discrepancy and we explain the subtle difference between Itoh's method and the method of Reynolds & Potter, which was used by Davey & Nguyen.

We suggest that for the centre mode of pipe flow neither Itoh's result nor Davey & Nguyen's result is a reliable guide to the true situation. However, for the wall mode of pipe flow, and especially for plane Couette flow, both methods give very similar results and we suggest that this similarity indicates that in these cases the damping rate is decreased by nonlinearity. For a particular problem we believe that it is only when the results of the two methods are very similar that either method is likely to be useful.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 86 , Issue 4 , 28 June 1978 , pp. 695 - 703
Copyright
© 1978 Cambridge University Press

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References

Davey, A. & Nguyen, H. P. F. 1971 J. Fluid Mech. 45, 701.
Gill, A. E. 1965 J. Fluid Mech. 21, 145.
Herbert, Th. 1977 Laminar–Turbulent Transition. AGARD Conf. Proc. no. 224.
Itoh, N. 1977a J. Fluid Mech. 82, 455.
Itoh, N. 1977b J. Fluid Mech. 82, 469.
Reynolds, W. C. & Potter, M. C. 1967 J. Fluid Mech. 27, 465.