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Finite-amplitude steady waves in plane viscous shear flows

Published online by Cambridge University Press:  20 April 2006

F. A. Milinazzo  and
P. G. Saffman
F. A. Milinazzo
Affiliation:
Applied Mathematics, California Institute of Technology, Pasadena, CA 91125 Present address: Royal Roads Military College, Victoria, B.C., Canada.
P. G. Saffman
Affiliation:
Applied Mathematics, California Institute of Technology, Pasadena, CA 91125
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Abstract

Computations of two-dimensional solutions of the Navier–Stokes equations are carried out for finite-amplitude waves on steady unidirectional flow. Several cases are considered. The numerical method employs pseudospectral techniques in the streamwise direction and finite differences on a stretched grid in the transverse direction, with matching to asymptotic solutions when unbounded. Earlier results for Poiseuille flow in a channel are re-obtained, except that attention is drawn to the dependence of the minimum Reynolds number on the physical constraint of constant flux or constant pressure gradient. Attempts to calculate waves in Couette flow by continuation in the velocity of a channel wall fail. The asymptotic suction boundary layer is shown to possess finite-amplitude waves at Reynolds numbers orders of magnitude less than the critical Reynolds number for linear instability. Waves in the Blasius boundary layer and unsteady Rayleigh profile are calculated by employing the artifice of adding a body force to cancel the spatial or temporal growth. The results are verified by comparison with perturbation analysis in the vicinity of the linear-instability critical Reynolds numbers.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 160 , November 1985 , pp. 281 - 295
Copyright
© 1985 Cambridge University Press

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References

Couder, V., Basdevant, C. & Thome, H. 1984 Solitary vortex couples in two-dimensional wakes. C. r. Acad. Sci., Paris 299, 89–93.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Goldshtik, M. A., Lifshits, A. M. & Shtern, V. N. 1983 The transition Reynolds number for a plane channel. Dokl. Akad. Nauk SSSR 273, 75–79.Google Scholar
Herbert, T. 1978 Die neutrale Flache der ebenen Poiseuille-Strömung. Habilitationsschrift, Universität Stuttgart.
Herbert, T. 1981 Stability of plane Poiseuille flow – theory and experiment. Fluid Dyn. Trans. 11, 77–126.Google Scholar
Hocking, L. M. 1975 Non-linear instability of the asymptotic suction velocity profile. Q. J. Mech. Appl. Maths 28, 341–353.Google Scholar
Keller, H. B. 1977 Numerical solution of bifurcation and nonlinear eigenvalue problems. In Applications of Bifurcation Theory (ed. P. H. Rabinowitz). p. 359. Academic.
Noether, F. 1921 Das Turbulenzproblem. Z. angew. Math. Mech. 1, 125–138, 218–219.Google Scholar
Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347–385.Google Scholar
Rozhdestvensky, B. L. & Simakin, I. N. 1984 Secondary flow in a plane channel: their relationship and comparison with turbulent flows. J. Fluid Mech. 147, 261–289.Google Scholar
Saffman, P. G. 1983 Vortices, stability and turbulence. In Proc. 4th Intl Conf. Physico-Chemical Hydrodynamics (ed. R. Pfeffer); Ann. NY Acad. Sci. 404, 12–24.
Smith, F. T. & Bodonyi, R. J. 1982 Amplitude dependent neutral modes in the Hagen–Poiseuille flow through a circular pipe. Proc. R. Soc. Lond. A 384, 463–489.Google Scholar