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Finite-amplitude stability of pipe flow

Published online by Cambridge University Press:  29 March 2006

A. Davey  and
H. P. F. Nguyen
A. Davey
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
H. P. F. Nguyen
Affiliation:
Computer Science Department, Johns Hopkins University. With an appendix by A. E. GILL, Department of Applied Mathematics and Theoretical Physics, University of Cambridge
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Abstract

In this paper we present some results concerning the stability of flow in a circular pipe to small but finite axisymmetric disturbances. The flow is unstable if the amplitude of a disturbance exceeds a critical value, the equilibrium amplitude, which we have calculated for a wide range of wave-numbers and Reynolds numbers. For large values of the Reynolds number, R, and for a real value of the wave-number, α, we indicate that the energy density of a critical disturbance is of order c2i, where −ααci is the damping rate of the associated infinitesimal disturbance. The energy, per unit length of the pipe, of a critical disturbance which is concentrated near the axis of the pipe is of order R−2, and the wave-number α is of order R1/3 For a critical disturbance which is concentrated near the wall of the pipe the energy is of order $R^{-\frac{3}{2}}$ and α is of order R½. This suggests that non-linear instability is most likely to be caused by a ‘centre’ mode rather than by a ‘wall’ mode. The wall mode solution is also essentially the solution for the problem of plane Couette flow when αR is large. We compare it with the true solution.

In an appendix Dr A. E. Gill indicates how some of the results of this paper may be inferred from a simple scale analysis.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 45 , Issue 4 , 26 February 1971 , pp. 701 - 720
Copyright
© 1971 Cambridge University Press

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References

Benney, D. J. & Bergeron, R. F. 1969 Stud. Appl. Math. 48, 181.
Betchov, R. & Criminale, W. O. 1967 Stability of Parallel Flows. New York: Academic.
Conte, S. D. 1966 S.I.A.M. Review, 8, 309.
Davey, A. & Drazin, P. G. 1969 J. Fluid Mech. 36, 209.
Ellingsen, T., Gjevik, B. & Palm, E. 1970 J. Fluid Mech. 40, 97.
Fox, J. A., Lessen, M. & Bhat, W. V. 1968 Phys. Fluids, 11, 1.
Gaster, M. 1963 J. Fluid Mech. 14, 222.
Gill, A. E. 1963 Ph.D. Thesis, University of Cambridge.
Gill, A. E. 1965 J. Fluid Mech. 21, 145.
Hasselmann, K. 1967 J. Fluid Mech. 30, 737.
Kaplan, R. E. 1964 M.I.T. Aero-Elastic and Structures Research Laboratory, ASRL-TR 116.
Landau, L. 1944 C.R. Acad. Sci. U.R.S.S. 44, 311.
Leite, R. J. 1959 J. Fluid Mech. 5, 81.
Malkus, W. V. R. & Vebonis, G. 1958 J. Fluid Mech. 4, 225.
Pekeris, C. L. 1948 Proc. U.S. Nat. Acad. Sci. 34, 285.
Pekeris, C. L. & Shkoller, B. 1969 J. Fluid Mech. 39, 611.
Reynolds, W. C. & Potter, M. C. 1967 J. Fluid Mech. 27, 465.
Sharma, R. 1968 Ph.D. Thesis, University of Southampton.
Stuart, J. T. 1960 J. Fluid Mech. 9, 353.
Watson, J. 1960 J. Fluid Mech. 9, 371.
Wilkinson, J. H. 1965 The Algebraic Eigenvalue Problem. Oxford: Clarendon.