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Finite amplitude instability of plane Couette flow

Published online by Cambridge University Press:  12 April 2006

Terence Coffee
Terence Coffee
Affiliation:
Department of Mathematical Sciences, New Mexico State University, Las Cruces
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Abstract

The Orr–Sommerfeld equation is solved numerically using expansions in Chebyshev polynomials and the generalized Rayleigh quotient iteration. Accurate results for large values of the parameters are obtained, and these further verify the belief that plane Couette flow is stable to infinitesimal disturbances. For finite disturbances, a formal expansion based on the method of Stuart and Watson as modified by Reynolds & Potter is used. This method shows a transition to instability for a large enough amplitude.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 83 , Issue 3 , 5 December 1977 , pp. 401 - 413
Copyright
© 1977 Cambridge University Press

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References

Davey, A. 1973 On the stability of plane Couette flow to infinitesimal disturbances. J. Fluid Mech. 57, 369.Google Scholar
Davey, A. & Nguyen, H. P. F. 1971 Finite-amplitude stability of pipe flow. J. Fluid Mech. 45, 701.Google Scholar
Deardorff, J. W. 1963 On the stability of viscous plane Couette flow. J. Fluid Mech. 15, 623.Google Scholar
Ellingsen, T., Gjevik, G. & Palm, E. 1970 On the non-linear stability of plane Couette flow. J. Fluid Mech. 40, 97.Google Scholar
Fox, L. & Parker, I. 1972 Chebyshev Polynomials in Numerical Analysis. Oxford University Press.
Gallagher, A. & Mercer, A. 1962 On the behaviour of small disturbances in plane Couette flow. J. Fluid Mech. 13, 91.Google Scholar
Gary, J. & Helgason, R. 1970 A matrix method for ordinary differential eigenvalue problems. J. Comp. Phys. 5, 169.Google Scholar
Orszag, S. 1971 Accurate solution of the Orr — Sommerfeld stability equation. J. Fluid Mech. 50, 689.Google Scholar
Ostrowski, A. 1958–9 On the convergence of the Rayleigh quotient iteration for the computation of characteristic roots and vectors. Arch. Rat. Mech. Anal. 1, 233; 2, 423; 3, 325, 341, 472; 4, 153.Google Scholar
Reichardt, H. 1956 Über die Geschwindigkeitsverteilung in einer geradlinigen turbulenten Couetteströmung. Z. angew. Math. Mech. Suppl. 36, 26.Google Scholar
Reynolds, W. C. & Potter, M. C. 1967 Finite-amplitude instability of parallel shear flows. J. Fluid Mech. 27, 465.Google Scholar
Wilkinson, J. 1965 The Algebraic Eigenvalue Problem. Oxford: Clarendon Press.