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On the stability of plane Poiseuille flow to finite-amplitude disturbances, considering the higher-order Landau coefficients

Published online by Cambridge University Press:  20 April 2006

P. K. Sen  and
D. Venkateswarlu
P. K. Sen
Affiliation:
Department of Applied Mechanics, Indian Institute of Technology, New Delhi – 110016
D. Venkateswarlu
Affiliation:
Department of Mechanical Engineering, Delhi College of Engineering, Delhi – 110006
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Abstract

In this work a study has been made of the Stuart (1960)–Watson (1960) formalism as applied to plane Poiseuille flow. In particular, the higher-order Landau coefficients have been calculated for the Reynolds & Potter (1967) method and for the Watson (1960) method. The results have been used to study the convergence of the Stuart–Landau series. A convergence curve in the (α, R)-plane has been obtained by using suitable Domb–Sykes plots. In the region of poor convergence of the series, and also in a part of the divergent region of the series, it has been found that the Shanks (1955) method, using the em1 transformation, serves as a very effective way of finding the proper sum of the series, or of finding the proper antilimit of the series. The results for the velocity calculations at R = 5000 are in very good agreement with Herbert's (1977) Fourier-truncation method using N = 4. The Watson method and the Reynolds & Potter method have also been compared inthe subcritical and supercritical regions. It is found in the supercritical region that there is not much difference in the results by the ‘true problem’ of Watson and the ‘false problem’ of Reynolds & Potter when the respective series in both methods are summed by the Shanks method. This fact could possibly be capitalized upon in the subcritical region, where the Watson method is difficult to apply.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 133 , August 1983 , pp. 179 - 206
Copyright
© 1983 Cambridge University Press

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References

Davey, A. & Nguyen, H. P. F. 1971 Finite-amplitude stability of pipe flow J. Fluid Mech. 45, 701–720.Google Scholar
Eckhaus, W. 1965 Studies in Non-Linear Stability Theory. Springer.
Herbert, T. 1977 Finite amplitude stability of plane parallel flows. In Proc. AGARD Symp. on Laminar-Turbulent Transition, Paper no. 3, AGARD-CP-224.Google Scholar
Herbert, T. 1980 Non-linear stability of parallel flows by higher order amplitude expansions AIAA J. 18, 243–248.Google Scholar
Itoh, N. 1974a Spatial growth of finite wave disturbances in parallel and nearly parallel flows, part 1. The theoretical analysis and the numerical results for plane-Poiseuille flow Trans. Japan Soc. Aero. Space Sci. 17, 160–174.Google Scholar
Itoh, N. 1974b Spatial growth of finite wave disturbances in parallel and nearly parallel flows, part 2. The numerical results for the flat-plate boundary layer Trans. Japan Soc. Aero. Space Sci. 17, 175–186.Google Scholar
Itoh, N. 1977a Nonlinear stability of parallel flows with subcritical Reynolds numbers. Part 1. An asymptotic theory valid for small amplitude disturbances J. Fluid Mech. 82, 455–467.Google Scholar
Itoh, N. 1977b Nonlinear stability of parallel flows with subcritical Reynolds numbers. Part 2. Stability of pipe Poiseuille flow to finite axisymmetric disturbances J. Fluid Mech. 82, 469–479.Google Scholar
Landau, L. D. 1944 On the problem of turbulence C.R. Acad. Sci. URSS 44, 311–314.Google Scholar
Nishioka, M., Iida, S. & Ichikawa, Y. 1975 An experimental investigation of the stability of plane Poiseuille flow J. Fluid Mech. 72, 731–751.Google Scholar
Pekeris, C. L. & Shkoller, B. 1967 Stability of plane Poiseuille flow to periodic disturbances of finite amplitude in the vicinity of the neutral curve J. Fluid Mech. 29, 31–38.Google Scholar
Reynolds, W. C. & Potter, M. C. 1967 Finite-amplitude instability of parallel shear flows J. Fluid Mech. 27, 465–492.Google Scholar
Shanks, D. 1955 Non-linear transformations of divergent and slowly convergent sequences J. Maths & Phys. 34, 1–42.Google Scholar
Stewartson, K. & Stuart, J. T. 1971 A non-linear instability theory for a wave system in plane Poiseuille flow J. Fluid Mech. 48, 529–545.Google Scholar
Stuart, J. T. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 1. The basic behaviour in plane Poiseuille flow J. Fluid Mech. 9, 353–370.Google Scholar
Thomas, L. H. 1953 The stability of plane Poiseuille flow Phys. Rev. 91, 780–784.Google Scholar
Watson, J. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 2. The development of a solution for plane Poiseuille flow and for plane Couette flow J. Fluid Mech. 9, 371–389.Google Scholar
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics. Parabolic.
Zhou, H. 1982 On the nonlinear theory of stability of plane Poiseuille flow in the subcritical range Proc. R. Soc. Lond. A381, 407–418.Google Scholar