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On perturbation methods in nonlinear stability theory

Published online by Cambridge University Press:  20 April 2006

Thorwald Herbert
Thorwald Herbert
Affiliation:
Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061
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Abstract

This paper reconsiders formal expansion methods for the analysis of the nonlinear properties of a modal disturbance. A survey is given of the various types of expansions based on different assumptions, and their range and shortcomings are discussed. By introducing a well-defined amplitude, Watson's expansion in a time-dependent amplitude is developed into a rational method for uniquely determining Landau constants of arbitrary order. Complementary to the common orthogonality condition for points at the neutral curve, an alternative definition of the Landau constants is given for points in the unstable domain. The method is not restricted to small amplification rates but is invalid in the stable domain. The method of Reynolds and Potter for a direct attack on equilibrium states is extended into a class of rational methods. The methods in this class agree to within a rearrangement of the infinite expansion series but differ in their respective range of validity.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 126 , January 1983 , pp. 167 - 186
Copyright
© 1983 Cambridge University Press

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References

Busse, F. H. 1978 Rep. Prog. Phys. 41, 1929.
Chen, T. S. & Joseph, D. D. 1973 J. Fluid Mech. 58, 337.
Coffee, T. 1977 J. Fluid Mech. 83, 401.
Davey, A. 1962 J. Fluid Mech. 14, 336.
Davey, A. 1978 J. Fluid Mech. 86, 695.
Davey, A. & Nguyen, H. P. F. 1971 J. Fluid Mech. 45, 701.
Eagles, P. M. 1971 J. Fluid Mech. 49, 529.
Eckhaus, W. 1965 Studies in Nonlinear Stability Theory. Springer.
Ellingsen, T., Gjevik, B. & Palm, E. 1970 J. Fluid Mech. 40, 97.
Fasel, H. 1974 Untersuchungen zum Problem des Grenzschichtumschlages durch numerische Integration der Navier–Stokes–Gleichungen. Dissertation, Universität Stuttgart.
Gaunt, D. S. & Guttmann, A. J. 1974 In Phase Transitions and Critical Phenomena (ed. C. Domb & M. S. Green), vol. 3, p. 181. Academic.
Gorkov, L. P. 1957 Sov. Phys. JETP 6, 311.
Herbert, TH. 1977 Laminar–Turbulent Transition. AGARD Conf. Proc. no. 224, paper no. 3.
Herbert, TH. 1978a Die neutrale Fläche der ebenen Poiseuille–-Strömung. Habilitationsschrift, Universität Stuttgart.
Herbert, TH. 1978b A.I.A.A. Paper no. 78-1125.
Herbert, TH. 1980 A.I.A.A. J. 18, 243.
Herbert, TH. 1981 In Seventh International Conference on Numerical Methods in Fluid Dynamics (ed. W. C. Reynolds & R. W. MacCormack). Lecture Notes in Physics, vol. 141, p. 200. Springer.
Itoh, N. 1974 Trans. Japan Soc. Aero. Space Sci. 17, 160.
Itoh, N. 1977b J. Fluid Mech. 82, 455.
Itoh, N. 1977b J. Fluid Mech. 82, 469.
Joseph, D. D. & Sattinger, D. H. 1972 Arch. Rat. Mech. Anal. 45, 79.
Kirchgässner, K. & Sorger, P. 1969 Q. J. Mech. Appl. Math. 22, 183.
Kleiser, L. 1982 Numerische Simulationen zum laminar-turbulenten Umschlagsprozeß der ebenen Poiseuille–Strömung. Dissertation, Universität Karlsruhe.
Kuo, H. L. 1961 J. Fluid Mech. 10, 611.
Kuo, H. L. & Platzman, G. W. 1961 Beitr. Phys. Atmos. 33, 137.
Malkus, W. V. R. & Veronis, G. 1958 J. Fluid Mech. 4, 225.
Orszag, S. A. & Kells, L. C. 1980 J. Fluid Mech. 96, 159.
Orszag, S. A. & Patera, A. T. 1981 In Transition and Turbulence (ed. R. E. Meyer), p. 127. Academic.
Palm, E. 1960 J. Fluid Mech. 8, 183.
Pekeris, C. L. & Shkoller, B. 1967 J. Fluid Mech. 29, 31.
Reynolds, W. C. & Potter, M. C. 1967a A finite-amplitude state-selection theory for Taylor-vortex flow. Unpublished Stanford Univ. Paper.
Reynolds, W. C. & Potter, M. C. 1967b J. Fluid Mech. 27, 465.
Rosenblat, S. & Davis, S. H. 1979 SIAM J. Appl. Math. 37, 1.
Shanks, D. 1955 J. Math. & Phys. 34, 1.
Stewartson, K. 1975 Fluid Dyn. Trans. 7, 101.
Stuart, J. T. 1960 J. Fluid Mech. 9, 353.
Stuart, J. T. 1962 Adv. Aero. Sci. 3, 121.
Stuart, J. T. 1971 Ann. Rev. Fluid Mech. 3, 347.
VAN DYKE, M. 1974 Q. J. Mech. Appl. Math. 27, 423.
VAN DYKE, M. 1975 SIAM J. Appl. Math. 28, 720.
VAN DYKE, M. 1978 J. Fluid Mech. 86, 129.
Watson, J. 1960 J. Fluid Mech. 9, 371.
Zahn, J. P., Toomre, J., Spiegel, E. A. & Gough, D. O. 1974 J. Fluid Mech. 64, 319.