Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-25T22:19:46.028Z Has data issue: false hasContentIssue false
  • English
  • Français

On an active resonant triad of mixed modes in symmetric shear flows: a plane wake as a paradigm

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

In this paper, we identify a new type of resonant triad which operates in a parallel or nearly parallel shear flow with a symmetric profile. The triad consists of a planar sinuous mode, an oblique sinuous mode and an oblique varicose mode, but is not of the usual subharmonic-resonance form. The development of the triad is studied in the non-equilibrium critical-layer régime. The equations governing the evolution of the modes are derived. We show that the quadratic resonance can significantly enhance the growth of both the oblique sinuous and varicose modes, and may cause them to grow super-exponentially. This can lead to a subsequent stage in which the oblique sinuous mode produces a back reaction on the oblique varicose mode through a phase-locked interaction, causing both oblique modes to evolve even more rapidly. We suggest that the resonant triad is a viable mechanism for the development of three-dimensional structures and varicose components observed in the later stage of plane wake transition.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 317 , 25 June 1996 , pp. 337 - 368
Copyright
© 1996 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

Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.
Chen, J. H., Cantwell, B. J. & Mansour, N. N. 1990 The effect of Mach number on the stability of a plane supersonic wake. Phys. Fluids A 2, 984–1004.Google Scholar
Cimbala, J. M., Nagib, H. M. & Roshko, A. 1988 Large structure in the far wake of two-dimensional bluff bodies. J. Fluid Mech. 190, 265–298.Google Scholar
Corke, T. C., Krull, J. D. & Ghassemi, M. 1992 Three-dimensional-mode resonance in far wakes. J. Fluid Mech. 239, 99–132.Google Scholar
Cowley, S. J. & Wu, X. 1994 Asymptotic approaches to transition modelling. In Progress in Transition Modelling, AGARD Rep. 793.
Craik, A. D. D. 1971 Non-linear resonant instability in boundary layers. J. Fluid Mech. 50, 393–413.Google Scholar
Drazin, P. G. & Howard, L. N. 1966 Hydrodynamic stability of parallel flow of inviscid flow. Adv. Appl. Mech. 9, 1–89.Google Scholar
Goldstein, M. E. 1994 Nonlinear interaction between oblique waves on nearly planar shear flows. Phys. Fluids A 6, 42–65.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. Corrigendum, J. Fluid Mech. 216, 1990, 659–663.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, 295–330.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
Goldstein, M. E. & Lee, S. S. 1993 Oblique instability waves in nearly parallel shear flows. In Nonlinear Waves and Weak Turbulence with Applications in Oceanography and Condensed Matter Physics (ed. N. Fitzmaurice, D. Gurarie, F. McCaughan & W. Woyczynski), pp. 159–177. Birkhauser.
Goldstein, M. E. & Leib, S. J. 1989 Nonlinear evolution of oblique waves on compressible shear layers. J. Fluid Mech. 207, 73–96.Google Scholar
Haberman, R. 1972 Critical layers in parallel shear flows. Stud. Appl. Maths 50, 139.Google Scholar
Herbert, T. 1988 Secondary instability of boundary layers. Ann. Rev. Fluid Mech. 20, 487–526.Google Scholar
Kelly, R. E. 1968 On the resonant interaction of neutral disturbances in two inviscid shear flows. J. Fluid Mech. 31, 789–799.Google Scholar
Ko, D., Kubota, T. & Lees, L. 1970 Finite disturbance effect on the stability of a laminar incompressible wake behind a flat plate. J. Fluid Mech. 40, 315–341.Google Scholar
Lasheras, J. C. & Meiburg, E. 1990 Three-dimensional vorticity modes in the wake of a flat plate. Phys. Fluids A 2, 371–380.Google Scholar
Lee, S. S. 1994 Critical-layer analysis of fully coupled resonant-triad interaction in a boundary layer. Submitted to J. Fluid Mech.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, 513–521.Google Scholar
Mallier, R. 1995 Fully coupled resonant triad interactions in a Bickley jet. Submitted to Eur. J. Mech.Google Scholar
Mallier, R. & Maslowe, S. A. 1994 Fully coupled resonant triad interactions in a free shear layer. J. Fluid Mech. 278, 101–121.Google Scholar
Mankbadi, R. R., Wu, X. & Lee, S. S. 1993 A critical–layer analysis of the resonant triad in Blasius boundary-layer transition: nonlinear interactions. J. Fluid Mech. 256, 85–106.Google Scholar
Marasli, B., Champagne, F. H. & Wygnanski, I. J. 1989 Modal decomposition of velocity signals in a plane, turbulent wake. J. Fluid Mech. 198, 255–273.Google Scholar
Mattingly, G. E. & Criminale, W. O. 1972 The stability of an incompressible two-dimensional wake. J. Fluid Mech. 51, 233–272.Google Scholar
Meiburg, E. 1987 On the role of subharmonic perturbations in the far wake. J. Fluid Mech. 177, 83–107.Google Scholar
Miksad, R. W., Jones, F. L., Powers, E. J., Kim, Y. C. & Khadra, L. 1982 Experiments on the role of amplitude and phase modulation during transition to turbulence. J. Fluid Mech. 123, 1–29.Google Scholar
Raetz, G. S. 1959 A new theory of the cause of transition in fluid flows. Northrop Corp. NOR-59-383 BLC-121.Google Scholar
Sato, H. 1970 An experimental study of nonlinear interaction of velocity fluctuations in the transition region of a two-dimensional wake. J. Fluid Mech. 44, 741–765.Google Scholar
Sato, H. & Kuriki, K. 1961 The mechanism of transition in the wake of a thin flat plate placed parallel to a uniform flow. J. Fluid Mech. 11, 321–353.Google Scholar
Sato, H. & Saito, H. 1975 Fine-structure of energy spectra of velocity fluctuations in the transition region of a two-dimensional wake. J. Fluid Mech. 67, 539–559.Google Scholar
Sato, H. & Saito, H. 1978 Artificial control of the laminar-turbulent transition of a two-dimensional wake by external sound. J. Fluid Mech. 84, 657–572.Google Scholar
Williamson, C. H. K. & Prasad, A. 1993a A new mechanism for oblique wave resonance in the ‘natural’ far wake. J. Fluid Mech. 256, 269–313.Google Scholar
Williamson, C. H. K. & Prasad, A. 1993b Acoustic forcing of oblique wave resonance in the far wake. J. Fluid Mech. 256, 315–341.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. 1993 On critical-layer and diffusion-layer nonlinearity in the three-dimensional stage of boundary-layer transition. Proc. R. Soc. Lond. A 433, 95–106.Google Scholar
Wu, X. 1995 Viscous effects on fully-coupled resonant triad interactions: an analytical approach. J. Fluid Mech. 292, 377–407.Google Scholar
Wu, X., Lee, S. S. & Cowley, S. J. 1993 On the weakly nonlinear three-dimensional instability of shear flows to pairs of oblique waves: the Stokes layer as a paradigm. J. Fluid Mech. 253, 681–721.Google Scholar
Wu, X. & Stewart, P. A. 1996a Interaction of phase-locked modes: a new mechanism for the rapid growth of three-dimensional disturbances. J. Fluid Mech. 316, 335–372.Google Scholar
Wu, X. & Stewart, P. A. 1996b Side-band instability of growing waves. To be submitted to Phys. Fluids.Google Scholar
Wundrow, D. W., Hultgren, L. S. & Goldstein, M. E. 1994 Interaction of oblique stability waves with a nonlinear planar wave. J. Fluid Mech. 264, 343–372.Google Scholar
Wygnanski, I., Champagne, F. & Marasli, B. 1986 On the large-scale structures in two-dimensional small-deficit, turbulent wakes. J. Fluid Mech. 168, 31–71.Google Scholar