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Fully coupled resonant-triad interactions in a free shear layer

Published online by Cambridge University Press:  26 April 2006

R. Mallier  and
S. A. Maslowe
R. Mallier
Affiliation:
Department of Mathematics, McGill University, Montreal, P. Q., H3A 2K6, Canada
S. A. Maslowe
Affiliation:
Department of Mathematics, McGill University, Montreal, P. Q., H3A 2K6, Canada
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Abstract

We report the results of an investigation of the weakly nonlinear evolution of a triad of waves, each slightly amplified on a linear basis, that are superimposed on a tanh y mixing layer. The triad consists of a plane wave and a pair of oblique modes that act as a subharmonic of order 1/2. The oblique modes are inclined at approximately ±60°. to the mean flow direction and because the resonance conditions are satisfied exactly the analysis is entirely self-consistent as an asymptotic theory. The nonlinearity first occurs within the critical layer and the initial interaction is of the parametric resonance type. This produces faster than exponential growth of the oblique waves, behaviour observed recently in the experiments of Corke & Kusek (1993). The critical-layer dynamics lead subsequently to coupled integro-differential equations governing the amplitude evolution and, as first shown in related work by Goldstein & Lee (1992) on boundary layers in an adverse pressure gradient, these equations develop singularities in a finite time.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 278 , 10 November 1994 , pp. 101 - 121
Copyright
© 1994 Cambridge University Press

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References

Benney, D. J. 1961 A non-linear theory for oscillations in a parallel flow. J. Fluid Mech. 10, 209–236.Google Scholar
Benney, D. J. & Maslowe, S. A. 1975 The evolution in space and time of nonlinear waves in parallel shear flows. Stud. Appl. Maths 54, 181–205.Google Scholar
Churilov, S. M. & Shukhman, I. G. 1987 Note on weakly nonlinear stability theory of a free mixing layer. Proc. R. Soc. Lond. A 409, 351–367.Google Scholar
Corke, T. C. & Kusek, S. M. 1993 Resonance in axisymmetric jets with controlled helical-mode input. J. Fluid Mech. 249, 307–336.Google Scholar
Craik, A. D. D. 1986 Exact solutions of non-conservative equations for three-wave and second-harmonic resonance. Proc. R. Soc. Lond. A 406, 1–12.Google Scholar
Curle, N. 1956 Hydrodynamic stability of the laminar mixing region between parallel streams. Aero. Res. Coun. Rep. 18426 (unpublished).
Dallard, T. & Browand, F. K. 1993 The growth of large scales at defect sites in the plane mixing layer. J. Fluid Mech. 247, 339–368.Google Scholar
Freymuth, P. 1966 On transition in a separated laminar boundary layer. J. Fluid Mech. 25, 632–704.Google Scholar
Goldstein, M. E. 1994 Nonlinear interactions between oblique instability waves on nearly parallel shear flows. Phys. Fluids 6, 724–735.Google Scholar
Goldstein, M. E. & Choi, S.-W. 1989 Nonlinear evolution of interacting oblique waves on two-dimensional shear layers. J. Fluid Mech. 207, 97–120.Google Scholar
Goldstein, M. E. & Lee, S. S. 1992 Fully coupled resonant-triad interaction in an adverse-pressure-gradient boundary layer. J. Fluid Mech. 245, 523–551.Google Scholar
Hickernell, F. J. 1984 Time-dependent critical layers in shear flows on the beta-plane. J. Fluid Mech. 142, 431–449.Google Scholar
Ho, C. M. & Huang, L. S. 1982 Subharmonics and vortex merging in mixing layers. J. Fluid Mech. 119, 443–473.Google Scholar
Ho, C. M. & Huerre, P. 1984 Perturbed free shear layers. Ann. Rev. Fluid Mech. 16, 365–424.Google Scholar
Huerre, P. 1987 On the Landau constant in mixing layers. Proc. R. Soc. Lond. A 409, 369–381.Google Scholar
Hultgren, L. S. 1992 Nonlinear spatial equilibrium of an externally excited instability wave in a free shear layer. J. Fluid Mech. 236, 635–664.Google Scholar
Kelly, R. E. 1967 On the stability of an inviscid shear layer which is periodic in space and time. J. Fluid Mech. 27, 657–689.Google Scholar
Lasheras, J. C., Cho, J. S. & Maxworthy, T. 1986 On the origin and scale of streamwise vortical structures in a plane, free shear layer. J. Fluid Mech. 172, 231–258.Google Scholar
Mallier, R. & Maslowe, S. A. 1994 Parametric resonant triad interactions in a free shear layer. Can. Appl. Math. Quart. 2, no. 1 (Winter) 91–113.Google Scholar
Maslowe, S. A. 1986 Critical layers in shear flows. Ann. Rev. Fluid Mech. 18, 405–432.Google Scholar
Monkewitz, P. A. 1988 Subharmonic resonance, pairing and shredding in the mixing layer. J. Fluid Mech. 188, 223–252.Google Scholar
Nygaard, K. J. & Glezer, A. 1991 Evolution of streamwise vortices and generation of small-scale motion in a plane mixing layer. J. Fluid Mech. 231, 257–301.Google Scholar
Nygaard, K. J. & Glezer, A. 1994 The effect of phase variations and cross shear on vortical structures in a plane mixing layer. J. Fluid Mech. 276, 21–59.Google Scholar
Patnaik, P. C., Sherman, F. S. & Corcos, G. M. 1976 A numerical simulation of Kelvin-Helmholtz waves of finite amplitude. J. Fluid Mech. 73, 215–240.Google Scholar
Pierrehumbert, R. T. & Widnall, S. E. 1982 The two- and three-dimensional instabilities of a spatially periodic shear layer. J. Fluid Mech. 114, 59–82.Google Scholar
Sato, H. 1959 Further investigation on the transition of two-dimensional separated layer at subsonic speeds. J. Phys. Soc. Japan 14, 1797–1810.Google Scholar
Schoppe, W., Hussain, F. & Metcalfe, R. E. 1994 A new mechanism of small-scale transition in a plane mixing layer; core dynamics instability. J. Fluid Mech. (submitted).Google Scholar
Wu, X 1992 The nonlinear evolution of high-frequency resonant-triad waves in an oscillatory Stokes layer at high Reynolds number. J. Fluid Mech. 245, 553–597.Google Scholar
Wu, X., Lee, S. S. & Cowley, S. J. 1993 On the weakly non-linear three-dimensional instability of shear layers to pairs of oblique waves: The Stokes layer as paradigm. J. Fluid Mech. 253, 681–721.Google Scholar
Wundrow, D. W., Hultgren, L. S. & Goldstein, M. E. 1994 Interaction of oblique instability waves with a nonlinear plane wave. J. Fluid Mech. 264, 343–372.Google Scholar