Robust heteroclinic cycles in two-dimensional Rayleigh-Bénard convection without Boussinesq symmetry

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 12, Issue 11, 2002, Pages 2501-2522

  Mercader, Isabel ^a   Prat, Joana ^a   Knobloch, Edgar ^b  

^a UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

^b UNIVERSITY OF LEEDS   (United Kingdom)

Author keywords

Convection; Heteroclinic cycles; Mode interactions

Indexed keywords

COMPUTER SIMULATION; COOLING; HEAT CONVECTION; PARAMETER ESTIMATION; PROBLEM SOLVING;

HETEROCLINIC CYCLES;

BIFURCATION (MATHEMATICS);

EID: 0036876571     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127402006047     Document Type: Article

References (43)

1. Armbruster, D. [1987] "O(2)-symmetric bifurcation theory for convection rolls," Physica D27, 433-439.
2. Armbruster, D., Guckenheimer, J. & Holmes, P. [1988] "Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry," Physica D29, 257-282.
3. Armbruster, D., Guckenheimer, J. & Holmes, P. [1989] "Kuramoto-Sivashinsky dynamics on the center-unstable manifold," SIAM J. Appl. Math. 49, 676-691.
4. Aubry, N., Holmes, P., Lumley, J. L. & Stone, E. [1988] "The dynamics of coherent structures in the wall region of a turbulent boundary layer," J. Fluid Mech. 192, 115-173.
5. Berkooz, G., Holmes, P., Lumley, J. L., Aubry, N. & Stone, E. [1994] "Observation regarding 'Coherence and chaos in a model of turbulent boundary layer' by X. Zhou and L. Sirovich [Phys. Fluids A4, 2855 (1992)]," Phys. Fluids 6, 1574-1578.
6. Busse, F. H. [1981] "Transition to turbulence in Rayleigh-Bénard convection," in Hydrodynamic Instabilities and the Transition to Turbulence, eds. Swinney, H. L. & Gollub, J. P. (Springer-Verlag), pp. 97-137.
7. Campbell, S. A. & Holmes, P. [1991] "Bifurcation from O(2) symmetric heteroclinic cycles with three interacting modes," Nonlinearity 4, 697-726.
8. Chossat, P. [2001] "The bifurcation of heteroclinic cycles in systems of hydrodynamic type," Dyn. Contin. Discr. Impul. Syst., Ser. A. Math. Anal. 8, 575-590.
9. Clune, T. L. [1993] "Pattern selection in convecting systems," Ph.D. Thesis, University of California at Berkeley.
10. Cox, S. M. [1996] "Mode interactions in Rayleigh-Bénard convection," Physica D95, 50-61.
11. Dangelmayr, G. [1986] "Steady-state mode interactions in the presence of O(2)-symmetry," Dyn. Stab. Syst. 1, 159-185.
12. Deane, A. E., Knobloch, E. & Toomre, J. [1987] "Traveling waves and chaos in thermosolutal convection," Phys. Rev. A36, 2862-2869.
13. Drissi, A., Net, M. & Mercader, I. [1999] "Subharmonic instabilities of Tollmien-Schlichting waves in two-dimensional Poiseuille flow," Phys. Rev. E60, 1781-1791.
14. Echebarría, B., Krmpotić, D. & Pérez-García, C. [1997] "Resonant interactions in Bénard-Marangoni convection in cylindrical containers," Physica D99, 487-502.
15. Holmes, P., Lumley, J. L. & Berkooz, G. [1996] Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University Press, Cambridge).
16. Jones, C. A. [1982] "On the flow between counter-rotating cylinders," J. Fluid Mech. 120, 433-450.
17. Karniadakis, G. E., Israeli, M. & Orszag, S. A. [1991] "High-order splitting methods for the incompressible Navier-Stokes equations," J. Comput. Phys. 97, 414-443.
18. Kevrekidis, I. G., Nicolaenko, B. & Scovel, J. C. [1990] "Back in the saddle again: A computer assisted study of the Kuramoto-Sivashinsky equation," SIAM J. Appl. Math. 50, 760-790.
19. Knobloch, E. [1990] "Pattern selection in long-wavelength convection," Physica D41, 450-479.
20. Krupa, M. & Melbourne, I. [1995] "Asymptotic stability of heteroclinic cycles in systems with symmetry," Ergod. Theor. Dynam. Syst. 15, 121-147.
21. Manogg, G. & Metzener, P. [1994] "Strong resonance in two-dimensional non-Boussinesq convection," Phys. Fluids 6, 2944-2955.
22. Mercader, I., Prat, J. & Knobloch, E. [2001] "The 1: 2 mode interaction in Rayleigh-Bénard convection with weakly broken midplane symmetry," Int. J. Bifurcation and Chaos 11, 27-41.
23. Nore, C., Tuckerman, L., Daube, O. & Xin, S. [2001] "The 1: 2 mode interaction in exactly counter-rotating von Kármán swirling flow," J. Fluid Mech., in press.
24. Petrescu, C.-D. [1998] "Sélection des grandes structures horizontal de la convection de Rayleigh-Bénard," Ph.D. Thesis, Ecole Polytechnique Fédérale de Lausanne.
25. Pismen, L. M. [1986] "Inertial effects in long-scale thermal convection," Phys. Lett. A116, 241-244.
26. Platten, J. K. & Legros, J. C. [1984] Convection in Liquids (Springer-Verlag, Berlin).
27. Pontes, J., Christov, C. I. & Velarde, M. G. [1996] "Numerical study of patterns and their evolution in finite geometries," Int. J. Bifurcation and Chaos 6, 1883-1890.
28. Pontes, J., Christov, C. I. & Velarde, M. G. [1999] "Numerical approach to pattern selection in a model problem for Bénard convection in a finite fluid layer," Annuaire de l'Université de Sofia "St. Kliment Ohridski" 93, 157-175.
29. Porter, J. & Knobloch, E. [2000] "Complex dynamics in the 1: 3 spatial resonance," Physica D143, 138-168.
30. Porter, J. & Knobloch, E. [2001] "New type of complex dynamics in the 1: 2 spatial resonance," Physica D159, 125-154.
31. Prat, J., Massaguer, J. M. & Mercader, I. [1995] "Large-scale flows and resonances in 2-D thermal convection," Phys. Fluids 7, 121-134.
32. Prat, J., Mercader, I. & Knobloch, E. [1998] "Resonant mode interaction in Rayleigh-Bénard convection," Phys. Rev. E58, 3145-3156.
33. Prat, J., Mercader, I. & Knobloch, E. [2001] "Rayleigh-Bénard convection with experimental boundary conditions," Proc. Conf. Bifurcations, Symmetry and Patterns (Birkhäuser-Verlag), in press.
34. Prat, J., Mercader, I. & Knobloch, E. [2002] "The 1: 2 mode interaction in Rayleigh-Bénard convection with and without Boussinesq symmetry," Int. J. Bifurcation and Chaos 12, 281-308.
35. Proctor, M. R. E. [1981] "Planform selection by finite-amplitude thermal convection between poorly conducting slabs," J. Fluid Mech. 113, 469-485.
36. Proctor, M. R. E. & Jones, C. A. [1988] "The interaction of two spatially resonant patterns in thermal convection. Part 1. Exact 1: 2 resonance," J. Fluid Mech. 188, 301-335.
37. Recktenwald, A. & Lücke, M. [1998] "Thermoconvection in magnetized ferrofluids: The influence of boundaries with finite heat conductivity," J. Magn. Magn. Mater. 188, 326-332.
38. Sanghi, S. & Aubry, N. [1993] "Mode interaction models for near-wall turbulence," J. Fluid Mech. 247, 455-488.
39. Steinberg, V., Fineberg, J., Moses, E. & Rehberg, I. [1989] "Pattern selection and transition to turbulence in propagating waves," Physica D37, 359-383.
40. 4He mixture," Phys. Rev. Lett. 61, 78-81.
41. Surko, C. M., Kolodner, P., Passner, A. & Walden, R. W. [1986] "Finite-amplitude traveling-wave convection in binary fluid mixtures," Physica D23, 220-229.
42. Villers, D. & Platten, J. K. [1984] "Heating curves in the two-component Bénard problem," J. Non-Equilib. Thermodyn. 9, 131-146.
43. Zhou, X. & Sirovich, L. [1992] "Coherence and chaos in a model of turbulent boundary layer," Phys. Fluids A4, 2855-2874.