Primary and secondary nonlinear global instability

Physica D: Nonlinear Phenomena

Volumn 132, Issue 4, 1999, Pages 428-456

  Couairon, A ^a   Chomaz, J M ^a  

^a ECOLE POLYTECHNIQUE   (France)

Author keywords

47.20. k; 47.20.Ft; 47.20.Ky; 47.20.Lz; 47.54.+r; Hydrodynamic stability; Instability of shear flows; Nonlinearity; Pattern selection; Secondary instability

Indexed keywords

EID: 0001644243     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(99)00062-7     Document Type: Article

References (64)

1. Cross M.C., Hohenberg P.C. Pattern formation outside of equilibrium. Rev. Mod. Phys. 65:1993;851-1112.
2. Huerre P., Monkewitz P.A. Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22:1990;473-537.
3. Couairon A., Chomaz J.-M. Global instability in nonlinear systems. Phys. Rev. Lett. 77:1996;4015-4018.
4. Couairon A., Chomaz J.-M. Absolute and convective instabilities, front velocities and global modes in nonlinear systems. Physica D. 108:1997;236-276.
5. A. Couairon, J.-M. Chomaz, Pattern selection in the presence of a cross flow, Phys. Rev. Lett. 79 (14) (1997) 2666-2669.
6. C. Mathis, M. Provansal, L. Boyer, The Bénard von Kármán instability: an experimental study near the threshold, J. Phys. (Paris) Lett. 45 (1984) 483.
7. Goujon-Durand S., Jenffer P., Wesfreid J.E. Downstream evolution of the Bénard-von Kármán instability. Phys. Rev. E. 50:1994;308.
8. Zielinska B.J.A., Wesfreid J.E. On the spatial structure of global modes in wake flow. Phys. Fluids. 7(6):1995;1418-1424.
9. Wesfreid J.E., Goujon-Durand S., Zielinska B.J.A. Global mode behavior of the streamwise velocity in wakes. J. Phys. II France. 6:1996;1-14.
10. A. Couairon, J.-M. Chomaz, Fully nonlinear global modes in slowly varying flows, Phys. Fluids, 1998, submitted for publication.
11. Büchel P., Lücke M., Roth D., Schmitz R. Pattern selection in the absolutely unstable regime as a nonlinear eigenvalue problem: Taylor vortices in axial flow. Phys. Rev. E. 53:1996;4764-4777.
12. Müller H.W., Lücke M., Kamps M. Convective patterns in horizontal flow. Europhys. Lett. 10:1989;451-456.
13. Müller H.W., Lücke M., Kamps M. Transversal convection patterns in horizontal shear flow. Phys. Rev. A. 45:1992;3714-3726.
14. Tobias S., Proctor M.R.E., Knobloch E. Convective and absolute instabilities of fluid flows in finite geometries. Physica D. 113:1998;43.
15. van Saarloos W., Hohenberg P.C. Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations. Physica D. 56:1992;303-367.
16. Kolodner P., Glazier J.A. Interaction of localized pulses of traveling-wave convection with propagating disturbances. Phys. Rev. A. 42:1990;7504.
17. Ahlers G., Cannel D.S. Vortex-front propagation in rotating Couette-Taylor flow. Phys. Rev. Lett. 50:1983;1583-1586.
18. Hanna A., Saul A., Showalter K. Detailed studies of propagating fronts in the iodate oxidation of arsenous acid. J. Am. Chem. Soc. 104:1982;3838.
19. Moloney J.V., Newell A.C. Nonlinear optics. Physica D. 44:1990;1-37.
20. Bélanger P.A., Gagnon L., Paré C. Solitary pulses in an amplified nonlinear dispersive medium. Opt. Lett. 14:1989;943.
21. Chaté H. Spatio-temporal intermittency regimes of the one-dimensional complex Ginzburg-Landau equation. Nonlinearity. 7:1994;185.
22. A. Pumir, B.I. Shraiman, W. van Saarloos, P.C. Hohenberg, H. Chaté, M. Holen, Phase vs. defect turbulence in the 1-d complex Ginzburg-Landau equation, in: C.D. Andereck, F. Hayot (Eds.), Ordered and Turbulent Patterns in Taylor-Couette flow, Plenum Press, New York, 1992.
23. Gil L. Space and time intermittency behavior of a 1-d complex Ginzburg-Landau equation. Nonlinearity. 4:1991;1213.
24. van Hecke M. Building blocks of spatiotemporal intermittency. Phys. Rev. Lett. 80:1997;1896-1899.
25. Fineberg J., Steinberg V. Vortex-front propagation in Rayleigh-Bénard convection. Phys. Rev. Lett. 58:1987;1332-1335.
26. Newell A.C., Whitehead J.A. Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38:1969;279.
27. Segel L.A. Distance side-walls cause slow amplitude modulation of cellular convection. J. Fluid Mech. 38:1969;203.
28. Stewartson K., Stuart J.T. A nonlinear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech. 48:1971;529.
29. Sakaguchi H. Instability of the hole solution in the complex Ginzburg-Landau equation. Prog. Theor. Phys. 85:1991;417-421.
30. Chaté H., Manneville P. Stability of the Bekki-Nozaki hole solutions to the one-dimensional complex Ginzburg-Landau equation. Phys. Lett. A. 171:1992;183-188.
31. Popp S., Stiller O., Aranson I., Kramer L. Hole solutions in the 1D complex Ginzburg-Landau equation. Physica D. 84:1995;398-423.
32. I.S. Aranson, L. Aranson, L. Kramer, A. Weber, Stability limits of spirals and traveling waves in nonequilibrium media, Phys. Rev. A 46 (1992) R2992-R2995.
33. R.J. Briggs, Electron-Stream Interaction with Plasmas, MIT Press, Cambridge, MA, 1964.
34. A. Bers, Space time evolution of plasma instabilities - absolute and convective, in: M.N. Rosenbluth, R.Z. Sagdeev (Eds.), Handbook of Plasma Physics, North-Holland, Amsterdam, 1983, pp. 451-517.
35. E.M. Lifshitz, L.P. Pitaevskii, Physical Kinetics, Pergamon, London, 1981.
36. Matkowsky B.J., Volpert V. Stability of plane wave solutions of complex Ginzburg-Landau equations. Quart. Appl. Math. 51:1993;265-281.
37. J.-M. Chomaz, P. Huerre, L.G. Redekopp, The effect of nonlinearity and forcing on global modes, In: P. Coullet, P. Huerre (Eds.), New trends in Nonlinear Dynamics and Pattern-forming Phenomena, Plenum Press, New York, 1990, pp. 227-240.
38. J.-M. Chomaz, P. Huerre, L.G. Redekopp, Models of hydrodynamic resonances in separated shear flows, in: Proc. 6th Symp. on Turbulent Shear Flows, Toulouse, France, 1987, pp. 321-326.
39. Landman M.J. Solutions of the Ginzburg-Landau equation of interest in shear flow transition. Stud. Appl. Math. 76:1987;187.
40. J.A. Powell, C.K.R.T. Jones, T.M. Kapitula, Nearly real fronts in a Ginzburg-Landau equation, Proc. R. Soc. Edinburgh 116A (1990) 193.
41. Koppel N., Howard L.N. Plane wave solutions to reaction-diffusion equations. Stud. Appl. Math. 56:1973;291.
42. V.I. Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, Heidelberg, 1983.
43. Kolmogorov A., Petrovsky I., Piskunov N. Investigation of a diffusion equation connected to the growth of materials, and application to a problem in biology. Bull. Univ. Moscow, Ser. Int. Sec. A. 1:1937;1.
44. Newell A.C., Passot T., Lega J. Order parameter equations for patterns. Annu. Rev. Fluid Mech. 25:1993;399-453.
45. van Saarloos W. Front propagation into unstable states: marginal stability as a dynamical mechanism for velocity selection. Phys. Rev A. 37:1988;211.
46. van Saarloos W. Front propagation into unstable states: II linear versus nonlinear marginal stability and rate of convergence. Phys. Rev A. 39:1989;6367.
47. C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978.
48. Brunet E., Derrida B. Shift in the velocity of a front due to a cutoff. Phys. Rev. E. 56:1997;2597-2604.
49. Ebert U., van Saarloos W. Universal algebraic relaxation of fronts propagating into an unstable state and implications for moving boundary approximations. Phys. Rev. Lett. 80:1998;1650-1653.
50. Chaté H., Manneville P. Phase diagram of the two-dimensional complex Ginzburg-Landau equation. Physica A. 224:1996;348-368.
51. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical recipes in fortran, Cambridge University Press, Cambridge, 1994.
52. Cossu C., Loiseleux T. On the convective and absolute nature of instabilities in finite difference numerical simulations of open flows. J. Comput. Phys. 144:1998;98-108.
53. Recktenwald A., Lücke M., Müller H.W. Taylor vortex formation in axial through-flow: linear and weakly nonlinear analysis. Phys. Rev. E. 48:1993;4444-4454.
54. Tuckerman L.S., Barkley D. Bifurcation analysis of the Eckhaus instability. Physica D. 46:1990;57-86.
55. Tuckerman L.S., Barkley D. Comment on bifurcation structure and the Eckhaus instability. Phys. Rev. Lett. 67:1991;1051-1054.
56. J.-M. Chomaz, A. Couairon, Against the wind, Phys. Fluids, 1998, submitted for publication.
57. Nozaki K., Bekki N. Exact solutions of the generalized Ginzburg-Landau equation. J. Phys. Soc. Jpn. 53:1984;1581-1582.
58. Lega J., Fauve S. Traveling hole solutions to the complex Ginzburg-Landau equation as perturbations of nonlinear Schrödinger dark solitons. Physica D. 102:1997;234-252.
59. Müller H.W., Tveitereid M. Absolute and convective nature of the Eckhaus and zigzag instability. Phys. Rev. Lett. 74(9):1995;1582-1585.
60. J.-M. Chomaz, A. Couairon, S. Julien, Absolute and convective nature of the Eckhaus and zigzag instability for an arbitrarily oriented throughflow, Phys. Fluids, 1999, submitted for publication.
61. Brancher P., Chomaz J.-M. Absolute and convective secondary instabilities in spatially periodic shear flows. Phys. Rev. Lett. 78:1997;658-661.
62. P. Huerre, M. Rossi, in: P. Manneville (Ed.), Hydrodynamics and Nonlinear Instabilities, Cambridge University Press, Cambridge, 1998.
63. Brevdo L., Bridges T.J. Absolute and convective instabilities of spatially periodic flows. Phil. Trans. R. Soc. London, Ser. A. 354:1996;1027-1064.
64. Weber A., Kramer L., Aranson I.S., Aranson L. Stability limits of traveling waves and the transition to spatiotemporal chaos in the complex Ginzburg-Landau equation. Physica D. 61:1992;279-283.