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The European Physical Journal Special Topics

, Volume 204, Issue 1, pp 119–131 | Cite as

Modal decomposition and normal form for hydrodynamic flows: Examples from cellular flame patterns

  • J. D. Kelley
  • G. H. GunaratneEmail author
  • A. Palacios
  • J. Shulman
Review

Abstract

Spatio-temporal complexity of hydrodynamic flows may be reduced through modal decomposition, especially in systems with symmetries. The symmetries of the most significant modes can then be used to deduce normal form equations associated with the observed state. In turn, the normal form equations can be used to deduce bifurcations to and from the given state.

We illustrate this process using two spatio-temporal cellular states on a circular flame front. The first example contains a pair of uniformly rotating cells. Principle component analysis shows that two coherent structures capture most of the dynamics and suggests that the state is a broken-parity traveling mode. Other experimentally observed states, such as modulated rotating states and a heteroclinic cycle between two spatially orthonormal states result from secondary bifurcations from the rotating state. The second example, referred to as the hopping mode, visually appears to have significantly more complicated dynamics. However, modal decomposition shows that it consists of two parity broken states moving at different angular velocities. The corresponding normal form contains a codimension-three steady-state bifurcation leading to a homoclinic cycle whose spatio-temporal characteristics are similar to those of hopping states.

We use these examples to propose a methodology to combine coherent structures that form a single, possibly time-dependent entity which we refer to as a generalized coherent structure. The process can reduce the number of entities needed to expand complex spatio-temporal states.

The paper is dedicated to the memory of Michael Gorman, whose experiments on cellular flame fronts and relentless demands for better theoretical understanding of the patterns motivated the study.

Keywords

European Physical Journal Special Topic Bifurcation Diagram Coherent Structure Principle Component Analysis Pure Mode 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Dover Publications, New York, 1981)Google Scholar
  2. 2.
    P.G. Drazin, W.H. Reid, Hydrodynamic Stability (Cambridge University Press, Cambridge, 1981)Google Scholar
  3. 3.
    M. Golubitsky, I. Stewart, D.G. Schaeffer, Singularities and Groups in Bifurcation Theory, vol. 2 (New York: Springer-Verlag, New York, 1988)Google Scholar
  4. 4.
    M. Gorman, M. elHamdi, K. Robbins, Combust. Sci. Technol. 98, 37 (1994)CrossRefGoogle Scholar
  5. 5.
    M. Gorman, M. elHamdi, K. Robbins, Combust. Sci. Technol. 98, 47 (1994)CrossRefGoogle Scholar
  6. 6.
    M. Gorman, M. elHamdi, K. Robbins, Combust. Sci. Technol. 98, 71 (1994)CrossRefGoogle Scholar
  7. 7.
    M. Gorman, M. elHamdi, K. Robbins, Combust. Sci. Technol. 98, 79 (1994)CrossRefGoogle Scholar
  8. 8.
    M. Gorman, C. Hamill, M. elHamdi, K. Robbins, Combust. Sci. Technol. 98, 25 (1994)CrossRefGoogle Scholar
  9. 9.
    Y.B. Zeldovich, Theory of Combustion and Detonation of Glass (Moscow: Academy of Sciences (USSR), Moscow, 1944)Google Scholar
  10. 10.
    F.A. Williams, Combustion Theory (Menlo Park, CA: Benjamin Cummins, Menlo Park, 1985)Google Scholar
  11. 11.
    G. Joulin, P. Clavin, Acta Astronautica 3, 223 (1976)CrossRefGoogle Scholar
  12. 12.
    G. Joulin, P. Clavin, Combust. Flame 35, 139 (1979)CrossRefGoogle Scholar
  13. 13.
    G. Joulin, G. Sivashinsky, Combust. Sci. Technol. 31, 75 (1983)CrossRefGoogle Scholar
  14. 14.
    P. Clavin, G. Joulin, J. Phys. Lett. 44, L1 (1983)CrossRefGoogle Scholar
  15. 15.
    A. Bayliss, B. Matkowsky, SIAM J. Appl. Math. 52, 396 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    A. Bayliss, B. Matkowsky, H. Riecke, Physica D 74, 1 (1994)ADSzbMATHCrossRefGoogle Scholar
  17. 17.
    L. Sirovich, Quart. Appl. Math. 45, 561 (1987)MathSciNetADSzbMATHGoogle Scholar
  18. 18.
    L. Sirovich, Quart. Appl. Math. 45, 573 (1987)MathSciNetGoogle Scholar
  19. 19.
    L. Sirovich, Quart. Appl. Math. 45, 583 (1987)MathSciNetGoogle Scholar
  20. 20.
    W. Mullins, R. Sekerka, J. Appl. Phys. 34, 323 (1963)ADSCrossRefGoogle Scholar
  21. 21.
    W. Mullins, R. Sekerka, J. Appl. Phys. 35, 444 (1964)ADSCrossRefGoogle Scholar
  22. 22.
    A. Simon, J. Bechhoefer, A. Libchaber, Phys. Rev. Lett. 61, 2574 (1988)ADSCrossRefGoogle Scholar
  23. 23.
    G. Faivre, S. DeCheveigne, C. Guthmann, P. Kurowski, Europhys. Lett. 9, 779 (1989)ADSCrossRefGoogle Scholar
  24. 24.
    F. Daviaud, M. Bonetti, M. Dubois, Phys. Rev. A 42, 3388 (1990)ADSCrossRefGoogle Scholar
  25. 25.
    D. Bensimon, P. Kolodner, C. Surko, H. Williams, V. Croquette, J. Fluid Mech. 217, 441 (1990)ADSCrossRefGoogle Scholar
  26. 26.
    M. Rabaud, S. Michalland, Y. Couder, Phys. Rev. Lett. 64, 184 (1990)ADSCrossRefGoogle Scholar
  27. 27.
    P. Coullet, R. Goldstein, G. Gunaratne, Phys. Rev. Lett. 63, 1954 (1989)ADSCrossRefGoogle Scholar
  28. 28.
    R. Goldstein, G. Gunaratne, L. Gil, Phys. Rev. A 41, 5731 (1990)ADSCrossRefGoogle Scholar
  29. 29.
    R. Goldstein, G. Gunaratne, L. Gil, P. Coullet, Phys. Rev. A 43, 6700 (1991)ADSCrossRefGoogle Scholar
  30. 30.
    P. Coullet, G. Iooss, Phys. Rev. Lett. 64, 866 (1990)MathSciNetADSzbMATHCrossRefGoogle Scholar
  31. 31.
    G. Dangelmayr, D. Armbruster, Proc. London Math. Soc. 46, 517 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    G. Dangelmayr, E. Knobloch, Nonlinearity 4, 399 (1991)MathSciNetADSzbMATHCrossRefGoogle Scholar
  33. 33.
    A. Palacios, G. Gunaratne, M. Gorman, K. Robbins, Chaos 7, 463 (1997)ADSzbMATHCrossRefGoogle Scholar
  34. 34.
    A. Palacios, G. Gunaratne, M. Gorman, K. Robbins, Phys. Rev. E 57, 5958 (1998)ADSCrossRefGoogle Scholar
  35. 35.
    I. Melbourne, P. Chossat, M. Golubitsky, Proc. Roy Soc. Edinburgh 113A, 315 (1989)MathSciNetCrossRefGoogle Scholar
  36. 36.
    M. Krupa, I. Melbourne, Ergod. Th. & Dynam. Sys. 15, 121 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    M. Field, Trans. Amer. Math. Soc. 259, 185 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    M. Krupa, J. Nonlin. Sci. 7, 129 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    A. Palacios, M. Gorman, G. Gunaratne, Chaos 9, 755 (1999)ADSzbMATHCrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer 2012

Authors and Affiliations

  • J. D. Kelley
    • 1
  • G. H. Gunaratne
    • 1
    Email author
  • A. Palacios
    • 2
  • J. Shulman
    • 1
  1. 1.Department of PhysicsUniversity of HoustonHoustonUSA
  2. 2.Department of MathematicsSan Diego State UniversitySan DiegoUSA

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