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On the weakly nonlinear three-dimensional instability of shear layers to pairs of oblique waves: the Stokes layer as a paradigm

Published online by Cambridge University Press:  26 April 2006

Xuesong Wu ,
Sangsoo Lee  and
Stephen J. Cowley
Xuesong Wu
Affiliation:
Department of Mathematics, Imperial College, 180 Queens Gate, London SW7 2BZ, UK
Sangsoo Lee
Affiliation:
Sverdrup Technology, Inc., Lewis Research Center Group, Cleveland, OH 44135, USA
Stephen J. Cowley
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
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Abstract

The nonlinear evolution of a pair of initially linear oblique waves in a high-Reynolds-number shear layer is studied. Attention is focused on times when disturbances of amplitude ε have O($\epsilon^{\frac{1}{3}} R $) growth rates, where R is the Reynolds number. The development of a pair of oblique waves is then controlled by nonlinear critical-layer effects (Goldstein & Choi 1989). Viscous effects are included by studying the distinguished scaling ε = O(R-1). When viscosity is not too large, solutions to the amplitude equation develop a finite-time singularity, indicating that an explosive growth can be induced by nonlinear effects; we suggest that such explosive growth is the precursor to certain of the bursts observed in experiments on Stokes layers and other shear layers. Increasing the importance of viscosity generally delays the occurrence of the finite-time singularity, and sufficiently large viscosity may lead to the disturbance decaying exponentially. For the special case when the streamwise and spanwise wavenumbers are equal, the solution can evolve into a periodic oscillation. A link between the unsteady critical-layer approach to high-Reynolds-number flow instability, and the wave/vortex approach of Hall & Smith (1991), is identified.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 253 , August 1993 , pp. 681 - 721
Copyright
© 1993 Cambridge University Press

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References

Akhavan, R., Kamm, R. D. & Shapiro, A. H. 1991a An investigation of transition to turbulence in bounded oscillatory Stokes flow. Part 1. Experiments. J. Fluid Mech. 225, 395.Google Scholar
Akhavan, R., Kamm, R. D. & Shapiro, A. H. 1991b An investigation of transition to turbulence in bounded oscillatory Stokes flow. Part 2. Numerical simulations. J. Fluid Mech. 225, 423.Google Scholar
Benney, D. J. 1961 A nonlinear theory for oscillations in a parallel flow. J. Fluid Mech. 10, 209.Google Scholar
Benney, D. J. & Bergeron, R. F. 1969 A new class of non-linear waves in parallel flows. Stud. Appl. Maths 48, 181.Google Scholar
Betchov, R. 1960 On the mechanism of turbulent transition. Phys. Fluids 3, 1026.Google Scholar
Bodonyi, Q. J., Smith, F. T. & Gajjar, J. 1983 Amplitude-dependent stability of a boundary layer with a strong non-linear critical layer. IMA J. Appl. Maths 30, 1–19.Google Scholar
Brown, P., Brown, S. N. & Smith, F. T. 1993 On the starting process of strongly nonlinear vortex/Rayleigh wave interactions. Mathematica (to appear).Google Scholar
Brown, S. N. & Stewartson, K. 1978 The evolution of the critical layer of Rossby wave. Part II. Geophys. Astrophys. Fluid Dyn. 10, 1.Google Scholar
Churilov, S. M. & Shukhman, I.G. 1987a The nonlinear development of disturbances in a zonal shear flow. Geophys. Astrophys. Fluid Dyn. 38, 145–175.Google Scholar
Churilov, S. M. & Shukhman, I.G. 1987b Nonlinear stability of a stratified shear flow: a viscous critical layer. J. Fluid Mech. 180, 1.Google Scholar
Churilov, S. M. & Shukhman, I.G. 1988 Nonlinear stability of a stratified shear flow in the regime with an unsteady critical layer. J. Fluid Mech. 194, 187.Google Scholar
Cowley, S.J. 1987 High frequency Rayleigh instability of Stokes layers. In Stability of Time Dependent and Spatially Varying Flows (ed. D.L. Dwoyer & M.Y. Hussaini), p. 261. Springer.
Gajjar, J. S. B. 1990 Amplitude-dependent neutral modes in compressible boundary layer flows. NASA Tech. Mem. 102524.
Gajjar, J. S. B. & Cole, J.W. 1989 Upper branch stability of compressible boundary layer flows. Theor. Comp. Fluid Dyn. 1, 105.Google Scholar
Gajjar, J. S. B. & Smith, F.T. 1985 On the global instability of free disturbances with a time-dependent nonlinear viscous critical layer. J. Fluid Mech. 157, 53.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. Corrigendum, J. Fluid Mech. 216, 1990, 659–(referred to herein as GC.)Google Scholar
Goldstein, M. E., Durbin, P.A. & Leib, S.J. 1987 Roll-up of vorticity in adverse-pressure-gradient boundary layers. J. Fluid Mech. 183, 325.Google Scholar
Goldstein, M. E. & Hultgren, L.S. 1988 Nonlinear spatial evolution of an externally excited instability wave in a free shear layer. J. Fluid. Mech. 197, 295.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.Google Scholar
Goldstein, M. E. & Leib, S.J. 1988 Nonlinear roll-up of externally excited free shear layers. J. Fluid Mech. 191, 481.Google Scholar
Goldstein, M. E. & Leib, S.J. 1989 Nonlinear evolution of oblique waves on compressible shear layers. J. Fluid Mech. 207, 73.Google Scholar
Goldstein, M. E. & Wundrow, D.W. 1990 Spatial evolution of nonlinear acoustic mode instabilities on hypersonic boundary layers. J. Fluid Mech. 219, 585.Google Scholar
Greenspan, H. P. & Benney, D.J. 1963 On shear layer instability, breakdown and transition. J. Fluid Mech. 15, 133.Google Scholar
Hall, P. 1978 The linear instability of flat Stokes layers. Proc. R. Soc. Lond. A 359, 151.Google Scholar
Hall, P. 1983 On the nonlinear stability of slowly varying time-dependent viscous flows. J. Fluid Mech. 126, 357.Google Scholar
Hall, P. 1991 Görtler vortices in growing boundary layers: the leading edge receptivity problem, linear growth and the nonlinear breakdown stage. Mathematika 37, 151.Google Scholar
Hall, P. & Horseman, N.J. 1991 The linear inviscid secondary instability of longitudinal vortex structures in boundary-layers. J. Fluid Mech. 232, 357.Google Scholar
Hall, P. & Smith, F.T. 1991 On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641. See also ICASE Rep. 89–22.Google Scholar
Haynes, P. H. 1985 Nonlinear instability of a Rossby-wave critical layer. J. Fluid Mech. 161, 493.Google Scholar
Haynes, P. H. & Cowley, S.J. 1986 The evolution of an unsteady translating nonlinear Rossby-wave critical layer. Geophys. Astrophys. Fluid Dyn. 35, 1.Google Scholar
Hickernell, F. J. 1984 Time-dependent critical layers in shear flows on the beta-plane. J. Fluid Mech. 142, 431.Google Scholar
Hino, M., Kashiwayanagi, M., Nakayama, A. & Hara, T. 1983 Experiments on the turbulence statistics and structure of a reciprocating oscillatory flow. J. Fluid Mech. 131. 363.Google Scholar
Hino, M., Sawamoto, M. & Takasu, S. 1976 Experiments on transition to turbulence in an oscillatory pipe flow. J. Fluid Mech. 75, 193.Google Scholar
Huerre, P. & Scott, J.F. 1980 Effects of critical layer structure on the nonlinear evolution of waves in free shear layers. Proc. R. Soc. Lond. A 371, 509.Google Scholar
Hultgren, L. S. 1992 Nonlinear spatial equilibration of an externally excited instability wave in a free shear layer. J. Fluid Mech. 236, 635.Google Scholar
Kachanov, Yu. S. 1987 On the resonant nature of the breakdown of a laminar boundary layer. J. Fluid Mech. 184, 43.Google Scholar
Kachanov, YuS. & Levchenko, V.Ya. 1984 The resonant interaction of disturbances at laminar turbulent transition in a boundary layer. J. Fluid Mech. 138, 209.Google Scholar
Kelly, R. E. & Maslowe, S.A. 1970 The nonlinear critical layer in a slightly stratified shear flow. Stud. Appl. Maths 49, 301.Google Scholar
Kerczek, C. Von & Davis, S.H. 1974 Linear stability theory of oscillatory Stokes layers. J. Fluid Mech. 62, 753.Google Scholar
Killworth, P. D. & McIntyre, M.E. 1985 Do Rossby-wave critical layers absorb, reflect or overreflect? J. Fluid Mech. 161, 449.Google Scholar
Klebanoff, P. S., Tidstrom, K.D. & Sargent, L.M. 1962 The three-dimensional nature of boundary layer instability. J. Fluid Mech. 12, 1.Google Scholar
Leib, S. J. 1991 Nonlinear evolution of subsonic and supersonic disturbances on a compressible free shear layer. J. Fluid Mech. 224, 551.Google Scholar
Leib, S. J. & Goldstein, M. E. 1989 Nonlinear interaction between the sinuous and varicose instability modes in a plane wake. Phys. Fluids A 1, 513.Google Scholar
Mankbadi, R. R., Wu, X. & Lee, S. S. 1993 A critical-layer analysis of the resonant triad in boundary-layer transition: nonlinear interactions. J. Fluid Mech. (to appear).Google Scholar
Maslowe, S. A. 1986 Critical layers in shear flows. Ann. Rev. Fluid Mech. 18, 406.Google Scholar
Merkli, P. & Thomann, H. 1975 Transition to turbulence in oscillating pipe flow. J. Fluid Mech. 68, 567.Google Scholar
Monkewitz, P. A. & Bunster, A. 1987 The stability of the Stokes layer: visual observations and some theoretical considerations. In Stability of Time dependent and spatially Varying Flows (ed. D. L. Dwoyer & M. Y. Hussaini), p. 244.
Nishioka, M., Iida, S. & Ichikawa, J. 1975 An experimental investigation of the stability of plane Poiseuille flow. J. Fluid Mech. 72, 731.Google Scholar
Saric, W. S. & Thomas, A. S. W. 1984 Experiments on subharmonic route to turbulence in boundary layers. In Turbulence and Chaotic Phenomena in Fluids (ed. T. Tatsumi), p. 117. North-Holland.
Schubauer, G. B. & Skramstad, H. K. 1947 Laminar boundary-layer oscillations and transition on a flat plate. NACA Rep. No. 909.
Shukhman, I. G. 1989 Nonlinear stability of a weakly supercritical mixing layer in a rotating fluid. J. Fluid Mech. 200, 425.Google Scholar
Shukhman, I. G. 1991 Nonlinear evolution of spiral density waves generated by the instability of the shear layer in rotating compressible fluid. J. Fluid Mech. 233, 587.Google Scholar
Smith, F. T. 1979 On the non-parallel flow stability of the Blasius boundary layer. Proc. R. Soc. Land. A 366, 91.Google Scholar
Smith, F. T. & Blennerhassett, P. 1992 Nonlinear interaction of oblique three-dimensional Tollmien-waves and longitudinal vortices, in channel flows and boundary layers. Proc. R. Soc. Lond. A 436, 585.Google Scholar
Smith, F. T. & Bodonyi, R. J. 1982a Nonlinear critical layers and their development in streaming-flow stability. J. Fluid Mech. 118, 165.Google Scholar
Smith, F.T. & Bodonyi, R. J. 1982b Amplitude-dependent neutral modes in the Hagen–Poiseuille flow through a circular pipe. Proc. R. Soc. Lond. A 384, 463.Google Scholar
Stewartson, K. 1981 Marginally stable inviscid flows with critical layers. IMA J. Appl. Maths 27, 133.Google Scholar
Stewartson, K. & Stuart, J.T. 1972 A nonlinear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech. 48, 529.Google Scholar
Tromans, P. 1979 Stability and transition of periodic pipe flows. PhD thesis, University of Cambridge.
Wu, X. 1991 Nonlinear instability of Stokes layers. PhD thesis, University of London.
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.Google Scholar
Wu, X. 1993a Nonlinear temporal-spatial modulation of near-planar disturbances in shear flows: formation of streamwise vortices J. Fluid Mech., (to appear).Google Scholar
Wu, X. 1993b On critical-layer and diffusion-layer nonlinearity in the three-dimensional stage of boundary layer transition. Proc. R. Soc. Lond. A, (to appear).Google Scholar
Wu, X. & Cowley, S. J. 1993 On the nonlinear evolution of instability modes in unsteady shear layers: the Stokes layer as a paradigm. In preparation.