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Resonant Interactions in Shear Flows

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Laminar-Turbulent Transition

Abstract

Raetz [1,2] and Stuart [3] first showed theoretically that resonant triads of Tollmein-Schlichting waves can occur in boundary-layer flows. At about the same time, interest arose in resonance (triads and quartets) among surface waves in water (Phillips [4], McGoldrick [5]) and among surface and internal waves in stratified fluid (Ball [6], Thorpe [7], Davis & Acrivos [8]). Meanwhile, similar advances were taking place in other fields, notably nonlinear optics and plasma physics. Subsequently, Kelly [9,10] examined subharmonic and three-wave resonance among neutrally-stable inviscid modes in jet and shear-layer profiles; while Craik [11] demonstrated that resonant triads of surface gravity waves exist if a sufficiently strong mean shear flow is present.

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References

  1. Raetz, G.S.: 1959 Norair Rep. NOR-59–383. Hawthorne, Calif. A new theory of the cause of transition in fluid flows.

    Google Scholar 

  2. Raetz, G.S.: 1964 Norair Rep. NOR-64–111. Hawthorne, Calif. Current status of resonance theory of transition.

    Google Scholar 

  3. Stuart, J.T.: 1962 Adv. Aero. Sci. 3–4, 121–142. On three-dimensional nonlinear effects in the stability of parallel flows.

    Google Scholar 

  4. Phillips, O.M.: 1960 J. Fluid Mech. 9, 193–217. On the dynamics of unsteady gravity waves of finite amplitude, Part 1. The elementary interactions.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. McGoldrick, L.F.: 1965 J. Fluid Mech. 21, 305–331. Resonant interactions among capillary-gravity waves.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. Ball, F.K.: 1964 J. Fluid Mech. 20, 457–479. Energy transfer between external and internal gravity waves.

    Article  ADS  MathSciNet  Google Scholar 

  7. Thorpe, S.A.: 1966 J. Fluid Mech. 24, 737–751. On wave interactions in a stratified fluid.

    Article  MATH  ADS  Google Scholar 

  8. Davis, R.E. & Acrivos, A.: 1967 J. Fluid Mech. 30, 723–736. The stability of oscillatory internal waves.

    Article  MATH  ADS  Google Scholar 

  9. Kelly, R.E.: 1967 J. Fluid Mech. 27, 657–689. On the stability of an inviscid shear layer which is periodic in space and time.

    Article  MATH  ADS  Google Scholar 

  10. Kelly, R.E.: 1968 J. Fluid Mech. 31, 789–799. On the resonant interaction of neutral disturbances in two inviscid shear flows.

    Article  MATH  ADS  Google Scholar 

  11. Craik, A.D.D.: 1968 J. Fluid Mech. 34, 531–549. Resonant gravity-wave interactions in a shear flow.

    Article  MATH  ADS  Google Scholar 

  12. Craik, A.D.D.: 1971 J. Fluid Mech. 50, 393–413. Nonlinear resonant instability in boundary layers.

    Article  MATH  ADS  Google Scholar 

  13. Usher, J.R. & Craik, A.D.D.: 1974 J. Fluid Mech. 66, 209–221. Nonlinear wave interactions in shear flows. Part 1. A variational formulation.

    Article  MATH  ADS  Google Scholar 

  14. Usher, J.R. & Craik, A.D.D.: 1975 J. Fluid Mech. 70, 437–461. Nonlinear wave interactions in shear flows. Part 2. Third-order theory.

    Article  MATH  ADS  Google Scholar 

  15. Kelbanoff, P.S., Tidstrom, K.D. & Sargent, L.M.: 1962 J. Fluid Mech. 12, 1–34. The three-dimensional nature of boundary-layer instability.

    Article  ADS  Google Scholar 

  16. Orszag, S.A. & Kells, L.C.: 1980 J. Fluid Mech. 96, 159–205. Transition to turbulence in plane Poiseuille and plane Couette flow.

    Article  MATH  ADS  Google Scholar 

  17. Orszag, S.A. & Patera, A.T.: 1983 J. Fluid Mech. 128, 347–385. Second- ary instability of wall-bounded shear flows.

    Article  MATH  ADS  Google Scholar 

  18. Miksad, R.W.: 1972 J. Fluid Mech. 56, 695–719. Experiments on the nonlinear stages of free-shear-layer transition.

    Article  ADS  Google Scholar 

  19. Miksad, R.W.: 1973 J. Fluid Mech. 59, 1–21. Experiments on nonlinear interactions in the transition of a free-shear-layer.

    Article  ADS  Google Scholar 

  20. Kachanov, Yu.S., Kozlov, V.V. & Levchenko, V.Ya.: 1977 Fluid Dyn. 3, 383–390. [Mekh. Zhid. i Gaza 3, 49–53.] Nonlinear development of a wave in a boundary layer (Engl. trs.)

    Google Scholar 

  21. Kachanov, Yu.S. & Levchenko, V.Ya.: 1983 J. Fluid Mech. 138, 209–247. The resonant interaction of disturbances at laminar-turbulent transition in a boundary layer.

    Article  ADS  Google Scholar 

  22. Saric, W.S. Thomas, A.S.W.: 1984 Proc. IUTAM Symp. ‘Turbulence and Chaotic Phenomena in Fluids’, Kyoto 1983. Experiments on the sub-harmonic route to turbulence in boundary layers.

    Google Scholar 

  23. Zakharov, V.E.: 1976 Dokl. Akad. Nauk. S.S.S.R. 228, 1314–1316 [Soviet Phys. Dokl. 21, 322–323]. Exact solutions to the problem of the parametric interaction of three-dimensional wave packets.

    ADS  Google Scholar 

  24. Kaup, D.J., Reiman, A. & Bers, A.: 1979 Rev. Mod. Phys. 51, 275–310. Errata, in Rev. Mod. Phys. 51, 915, are corrected in reprints. Space-time evolution of nonlinear three-wave interactions. I Interactions in a homogeneous medium.

    Article  ADS  MathSciNet  Google Scholar 

  25. Kaup, D.J.: 1981 J. Math. Phys. 22, 1176–1181. The lump solutions and the Backlund transformation for the three-dimensional three-wave resonant interaction.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. Craik, A.D.D.: 1978 Proc. Roy. Soc. Lond. A363, 257–269. Evolution in space and time of resonant wave triads. II A class of exact solutions.

    ADS  MathSciNet  Google Scholar 

  27. Craik, A.D.D., Adam, J.A.: 1979 J. Fluid Mech. 92, 15–33. ‘Explosive’ resonant wave interactions in a three-layer fluid flow.

    Article  ADS  Google Scholar 

  28. Craik, A.D.D.: 1983 In Nonlinear Waves (ed. L. Debnath) pp 116–132. Two-and three-wave resonance.

    Google Scholar 

  29. Wersinger, J-M., Finn, J.M., Cott, E.: 1980 Phys. Fluids 23, 1142–1154. Bifurcation and ‘strange’ behaviour in instability saturation by nonlinear three-wave mode coupling.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  30. Volodin, A.G. & Zelman, M.B.: 1979 Fluid Dyn. (Trs. of Russian Mekh. Zhid. i Gaza 5, 81.) Three-wave resonant interaction of disturbances in a boundary layer.

    Google Scholar 

  31. Nayfeh, A.H. & Bozatli, A.N.: 1979a Phys. Fluids 22, 805–813. Secondary instability in boundary-layer flows.

    Article  MATH  ADS  Google Scholar 

  32. Nayfeh, A.H. & Bozatli, A.N.: 1980 Phys, Fluids 23, 448–458. Nonlinear interaction of two waves in boundary-layer flows.

    Article  MATH  ADS  Google Scholar 

  33. Nayfeh, A.H. i, & Bozatli, A.N.: 1979h Proc. AIAA 12th Fluid 4 Plasma Dyn. Conf., Williamsburg, Va.: AIAA Paper 79–1456. Nonlinear wave interactions in boundary layers.

    Google Scholar 

  34. Herbert, T.: 1983a Phys. Fluids 26, 871–874. Secondary instability of plane channel flow to subharmonic three-dimensional disturbances.

    Article  ADS  Google Scholar 

  35. Herbert, T.: 1983b AIAA 16th Fluid and Plasma Dyn. Conf., Danvers, Mass. AIAA-83–1759. Subharmonie three-dimensional disturbances in unstable plane shear flows.

    Google Scholar 

  36. Herbert, T.: 1984 Proc. IUTAM Symp. ‘Turbulence and Chaotic Phenomena in Fluids’, Kyoto, 1983. Modes of secondary instability in plane Poiseuille flow.

    Google Scholar 

  37. Gustaysson, L.H. & Ilultgren, L.S.: 1980 J. Fluid Mech. 98, 149–159. A resonance mechanism in plane Couette flow.

    Article  ADS  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Applied Mathematics, St Andrews University, St Andrews, Fife, KY16 9SS, Scotland, UK

    A. D. D. Craik

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  1. A. D. D. Craik
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Editors and Affiliations

  1. Institute of Theoretical and Applied Mechanics, Institutskaya, 4/1, Novosibirsk-90, USSR

    V. V. Kozlov

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© 1985 Springer-Verlag, Berlin, Heidelberg

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Craik, A.D.D. (1985). Resonant Interactions in Shear Flows. In: Kozlov, V.V. (eds) Laminar-Turbulent Transition. International Union of Theoretical and Applied Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-82462-3_1

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  • DOI: https://doi.org/10.1007/978-3-642-82462-3_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-82464-7

  • Online ISBN: 978-3-642-82462-3

  • eBook Packages: Springer Book Archive

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