Hostname: page-component-84b7d79bbc-4hvwz Total loading time: 0 Render date: 2024-07-27T18:22:24.591Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear temporal-spatial modulation of near-planar Rayleigh waves in shear flows: formation of streamwise vortices

Published online by Cambridge University Press:  26 April 2006

Xuesong Wu
Xuesong Wu
Affiliation:
Department of Mathematics, Imperial College, 180 Queens Gate, London SW7 2BZ, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The nonlinear temporal-spatial modulation of a near-planar Rayleigh instability wave is studied. The amplitude of the wave is allowed to be a slowly varying function of spanwise position as well as of time (or streamwise variable in the spatial evolution case). It is shown that the development of the disturbance is controlled by critical-layer nonlinear effects when the linear growth rate decreases to O(ε⅖), where ε is the magnitude of the disturbance. Nonlinear interactions influence the evolution by producing spanwise dependent mean-flow distortions. The evolution is governed by an integro-partial-differential equation containing history-dependent nonlinear terms of Hickernell (1984) type. A notable feature of the amplitude equation is that the highest derivative with respect to spanwise position appears in the nonlinear terms. These terms are associated with three-dimensionality. The possible properties of the amplitude equation are discussed. Numerical solutions show that a disturbance initially centred at a spanwise position can propagate laterally to form concentrated, quasi-periodic streamwise vortices. This qualitatively captures the phenomena observed in experiments. The focusing of vorticity may be associated with a localized singularity which can occur at a finite distance downstream or within a finite time. It is noted that the amplitude equation is rather generic and applies to a broad class of shear flows which is inviscidly unstable.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 256 , November 1993 , pp. 685 - 719
Copyright
© 1993 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ashurst, W. T. & Meiburg, E. 1988 Three-dimensional shear layers via vortex dynamics. J. Fluid Mech. 189, 87.Google Scholar
Bell, J. H. & Mehta, R. D. 1992 Measurements of the streamwise vortical structures in a plane mixing layer. J. Fluid Mech. 239, 213.Google Scholar
Bernal, L. P. 1981 The coherent structure of turbulent mixing layers. PhD thesis, California Institute of Technology.
Bernal, L. P. & Roshko, A. 1986 Streamwise vortex structure in plane mixing layers. J. Fluid Mech. 170, 499.Google Scholar
Bodonyi, R.J. & Smith, F.T. 1981 The upper branch stability of the Blasius boundary layer, including non-parallel effects. Proc. R. Soc. Lond. A 375, 65.Google Scholar
Breidenthal, R. E. 1981 Structure in turbulent mixing layers and wakes using a chemical reaction. J. Fluid Mech. 109, 1.Google Scholar
Breuer, K. S. & J. H. Haritonidis 1990 The evolution of a localized disturbance in a laminar boundary layer. Part 1. Weak disturbances. J. Fluid Mech. 220, 569Google Scholar
Breuer, K. S. & Landahl, M. T. 1990 The evolution of a localized disturbance in a laminar boundary layer. Part 2. Strong disturbances. J. Fluid Mech. 220, 595.Google Scholar
Brown, P., Brown, S. N. & Smith, F.T. 1993 On the starting process of strongly nonlinear vortex/Rayleigh wave interactions. Mathematika (to appear.)Google Scholar
Brown, S. N. & Stewartson, K. 1978 The evolution of the critical layer of a Rossby wave. Part II. Geophys. Astrophys. Fluid Dyn. 10, 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
Corcos, G. M. & Lin, S. J. 1984 The mixing layer: deterministic models of a turbulent flow. Part 2. The origin of the three-dimensional motion. J. Fluid Mech. 139, 67.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
Cowley, S.J., Van Dommelen, L. L.& Lam, S.T. 1990 On the use of Lagrangian variables in descriptions of unsteady boundary-layer separations. ICASE Rep. 90–47.
Davey, A., Hocking, L. M. & Stewartson, K. 1974 On the nonlinear evolution of three-dimensional disturbances in plane Poiseuille flow. J. Fluid. Mech. 63, 529.Google Scholar
Gaster, M. & Grant, I. 1975 An experimental investigation of the formation and development of a wave packet in a laminar boundary layer. Proc. R. Soc. Lond. A 347, 253.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.Google Scholar
Goldstein, M. E. & Hultgren, L. S. 1989 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
Haberman, R. 1972 Critical layers in parallel shear flows. Stud. Appl. Maths 50, 139.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. & Smith, F.T. 1989 Nonlinear Tollmien-Schlichting/vortex interaction in boundary layers. Eur. J. Mech. B 8, 179.Google Scholar
Hall, P. & Smith, F.T. 1990 Near-planar TS waves and longitudinal vortices in channel flows: nonlinear interaction and focusing. In Instability and Transition II (ed. M.Y. Hussaini & R.G. Voigt). Springer.
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
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
Hocking, L. M. & Stewartson, K. 1972 On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance. Proc. R. Soc. Lond. A 326, 289.Google Scholar
Hocking, L. M., Stewartson, K. & Stuart, J. T. 1972 A nonlinear instability burst in plane parallel flow. J. Fluid Mech. 51, 707.Google Scholar
Huerre, P. 1987 On the Landau Coefficient in the mixing layer. Proc. R. Soc. Lond. A 409, 308.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
Jimenez, J. 1983 A spanwise structure in the plane shear layer. J. Fluid Mech. 132, 319.Google Scholar
Jimenez, J., Cogollos, M. & Bernal, L. P. 1985 A perspective view of the plane mixing layer. J. Fluid Mech. 152, 125.Google Scholar
Kim, H.T., Kline, S. J. & Reynolds, W.C. 1971 The production of turbulence near a smooth wall in a turbulent boundary layer. J. Fluid Mech. 50, 133.Google Scholar
Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 The three-dimensional nature of boundary layer instability. J. Fluid Mech. 12, 1Google Scholar
Kleiser, L. & Zang, T. A. 1991 Numerical simulation of transition in wall-bounded shear flows. Ann. Rev. Fluid Mech. 23, 495.Google Scholar
Kline, S.J., Reynolds, W.C., Schraub, F.A. & Runstadler, P.W. The structure of turbulent boundary layers. J. Fluid Mech. 30, 741.
Konrad, J. H. 1976 An experimental investigation of mixing in two-dimensional flows with application to diffusion-limited chemical reactions. PhD thesis, California Institute of Technology.
Landahl, M. T. 1975 Wave breakdown and turbulence. SIAM J. Appl. Maths 28, 735.Google Scholar
Lasheras, J.S. & Choi, H. 1988 Three-dimensional instability of a plane free shear layer: an experimental study of the formation and the evolution of streamwise vortices. J. Fluid Mech. 189, 53.CrossRefGoogle Scholar
Lasheras, J.S., Cho, J.S. & Maxworthy, T. 1986 On the origin and evolution of streamwise vortical structure in a plane free shear layer. J. Fluid Mech. 172, 231 (referred to herein as LCM.)Google Scholar
Lin, S.J. & Corcos, G.M. 1984 The mixing layer: deterministic models of a turbulent flow. Part 3. The effect of plane strain on the dynamics of streamwise vortices. J. Fluid Mech. 141, 139–178.CrossRefGoogle Scholar
Maslowe, S. A. 1986 Critical layers in shear flows. Ann. Rev. Fluid Mech. 18, 406.Google Scholar
Nygaard, K.J. & Glezer, A. 1991 Evolution of streamwise vortices and generation of small-scale motion in plane mixing layer. J. Fluid Mech. 231, 257.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.Google Scholar
Pullin, D. I. & Jacobs, P. A. 1986 Inviscid evolution of stretched vortex arrays. J. Fluid Mech. 171, 377.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. & Blennerhassett, P. 1992 Nonlinear interaction of oblique three-dimensional Tollmien-Schlichting waves and longitudinal vortices, in channel flows and boundary layers. Proc. R. Soc. Lond. A 436, 585.Google Scholar
Smith, F. T. & Bowles, R. I. 1992 Transition theory and experimental comparisons on (I) amplification into streets and (II) a strongly nonlinear break-up criterion. Proc. R. Soc. Lond. A 439, 163.Google Scholar
Smith, F.T., Brown, S. N. & Brown, P. G. 1993 Initiation of three-dimensional nonlinear transition paths from an inflexional profile. Eur. J. Mech. (to appear.)Google Scholar
Smith, F.T. & Walton, A. G. 1989 Nonlinear interaction of near-planar TS waves and longitudinal vortices in boundary-layer transition. Mathematika 36, 262.Google Scholar
Stewart, P.A. & Smith, F.T. 1992 Development of three-dimensional nonlinear blow-up from a nearly planar initial disturbance, in boundary layer transition. J. Fluid Mech. 244, 79.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. 1971 A nonlinear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech. 48, 529.Google Scholar
Stuart, J. T. 1984 Instability of laminar flows, non-linear growth of fluctuations and transition to turbulence. In Turbulence and Chaotic Phenomena in Fluids, IUTAM Symp. Kyoto (ed. T. Tatsumi), pp. 17–26. North-Holland.
Stuart, J.T. 1987 Nonlinear Euler partial differential equations: singularity in their solution. In Proc. Symp. to Honor C.C. Lin (ed. D.J. Benney, F.H. Shu & Yuan Chi). World Scientific.
Stuart, J.T. 1990 The Lagrangian picture of fluid motion and its implication for flow structures. IMA J. Appl. Maths 46, 147.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. 1993 On critical-layer and diffusion layer nonlinearity in the three-dimensional stage of boundary-layer transition. Proc. R. Soc. Lond. A 443, 95.Google Scholar
Wu, X. & Cowley, S.J. 1993 On the nonlinear evolution of instability modes in unsteady shear flows: the Stokes layer as a paradigm. Q. J. Mech. Appl. Maths (submitted).Google Scholar
Wu, X., Lee, S.S. & Cowley, S.J. 1993 On the weakly nonlinear three-dimensional instability of shear layers to pairs of oblique waves: the Stokes layer as a paradigm. J. Fluid Mech. 253, 681.CrossRefGoogle Scholar