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A computational study of Rayleigh–Bénard convection. Part 2. Dimension considerations

Published online by Cambridge University Press:  26 April 2006

Lawrence Sirovich  and
Anil E. Deane
Lawrence Sirovich
Affiliation:
Center for Fluid Mechanics and The Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
Anil E. Deane
Affiliation:
Center for Fluid Mechanics and The Division of Applied Mathematics, Brown University, Providence, RI 02912, USA Present address: Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA
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Abstract

A study is made of the number of dimensions needed to specify chaotic Rayleigh–Bénard convection, over a range of Rayleigh numbers (γ = Ra/Rac < 102). This is based on the calculation of Lyapunov dimension over the range, as well as the notion of Karhunen–Loéve dimension. An argument suggesting a universal relation between these estimates and supporting numerical evidence is presented. Numerical evidence is also presented that the reciprocal of the largest Lyapunov exponent and the correlation time are of the same order of magnitude. Several other universal features are suggested. In particular it is suggested that the intrinsic attractor dimension is $O(Ra^{\frac{2}{3}})$, which is sharper than previous results.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 222 , January 1991 , pp. 251 - 265
Copyright
© 1991 Cambridge University Press

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References

Badii, R., Heinzelmann, K., Meier, P. F. & Politi, A., 1988 Correlation functions and generalized Lyapunov exponents. Phys. Rev. A 37, 1323.Google Scholar
Balachandar, S., Maxey, M. & Sirovich, L., 1989 Numerical simulation of high Rayleigh number convection. J. Sci. Comput. 4, 219.Google Scholar
Batchelor, G. K. & Davis, R. H., 1956 Surveys in Mechanics, p. 352. Cambridge University Press.
Benettin, G., Galgani, L., Giorgilli, A. & Strelcyn, J.-M. 1980 Lyapunov characteristic exponents for smooth dynamical systems and Hamiltonian systems: A method for computing all of them. Mecanica 15, 9.Google Scholar
Bergé, P., Pomeau, Y. & Vidal, C., 1986 Order Within Chaos. Wiley-Interscience.
Brandstater, A. & Swinney, H. L., 1987 Strange attractors in weakly turbulent Couette—Taylor flow. Phys. Rev. A 35, 2207.Google Scholar
Castaing, B., Gunaratne, G., Heslot, F., Kadanoff, L., Libchaber, A., Thomae, S., Wu, X-Z., Zeleski, S. & Zanetti, G., 1989 Scaling of hard thermal turbulence in Rayleigh-Bénard Convection. J. Fluid Mech. 204, 1.Google Scholar
Constantin, P., Foiaş, C., Manley, O. P. & Temam, R. 1985 Determining modes and fractal dimension of turbulent flows. J. Fluid Mech. 150, 427.Google Scholar
Deane, A. & Sirovich, L., 1991 A computational study of Rayleigh-Bénard convection. Part 1. Rayleigh-number scaling. J. Fluid Mech. 222, 231 (referred to herein as I).Google Scholar
Farmer, D., Crutchfield, J., Froehling, M., Packard, N. & Shaw, R., 1980 Power spectra and mixing properties of strange attractors. Ann. NY Acad. Sci. 357, 453.Google Scholar
Foiaş, C., Manley, O. P. & Temam, R., 1986 Physical estimates of the number of degrees of freedom in free convection. Phys. Fluids 29, 3101.Google Scholar
Foiaş, C., Manley, O. P. & Temam, R., 1987 Nonlin. Anal. Theory, Meth. & Applies. 11, 939–967.
Foiaş, C., Manley, O. P., Temam, R. & Treve, M. T., 1983 Physica 9D, 157.
Goldhirsh, I., Sulem, P. L. & Orszag, S. A., 1987 Stability and Lyapunov stability of dynamical systems: A differential approach and a numerical method. Physica 27D, 311.Google Scholar
Herring, J. R. & Wyngaard, J. C., 1987 Convection with a simple chemically reactive passive scalar. In Turbulent Shear Flows 5, p. 328. Springer.
Kaplan, J. L., mallet—Paret, J. & Yorke, J. A. 1984 The Lyapunov dimension of a nowhere differentiable attracting torus. Ergod. Theor Dynam. Syst. 4, 261.Google Scholar
Kaplan, J. & Yorke, J., 1979 Chaotic behavior in multi-dimensional difference equations. In Functional Differential Equations and the Approximation of Fixed Points (ed. H. O. Peitgen & H. O. Walther). Lecture Notes in Mathematics, vol. 730, p. 228. Springer.
Keefe, L. & Moin, P., 1987 Bull. Am. Phys. Soc. II, 32, 2026.
Landau, L. D.: 1944 Turbulence. Dokl. Akad. Nauk. SSSR 44, 339.Google Scholar
Manneville, P.: 1985 Lyapunov exponents for the Kuramoto-Sivashinsky model. In Macroscopic Modelling of Turbulent Flows. Lecture Notes in Physics, vol. 230, p. 319. Springer.
Nicolaenko, B.: 1986 Some mathematical aspects of flame chaos and flame multiplicity. Physica 20D, 109.Google Scholar
Ruelle, D.: 1982 On the nature of turbulence. Commun. Math. Phys. 87, 287.Google Scholar
Russel, D. A., Hanson, J. D. & Ott, E., 1980 Dimension of strange attractors. Phys. Rev. Lett. 45, 1175.Google Scholar
Schuster, G. S.: 1984 Deterministic Chaos: An Introduction. Weinheim, FRG: Physik.
Shaw, R.: 1981 Strange attractors, chaotic behavior and information flows. Z. Naturforsh. 36a, 80.Google Scholar
Shimada, I. & Nagashima, T., 1979 A numerical approach to the ergodic problem of dessipative dynamical systems. Prog. Theor. Phys. 61, 1605.Google Scholar
Sirovich, L.: 1987a Turbulence and the dynamics of coherent tructures, Part I. Coherent Structures. Q. Appl. Maths XLV, 561.Google Scholar
Sirovich, L.: 1987b Turbulence and the dynamics of coherent structures, Part II. Symmetries and transformations. Q. Appl. Maths XLV, 573.Google Scholar
Sirovich, L.: 1987c Turbulence and the dynamics of coherent structures, Part III. Dynamics and scaling. Q. Appl. Maths XLV, 583.Google Scholar
Sirovich, L.: 1989 Chaotic dynamics of coherent structures. Physica D 37, 126.Google Scholar
Sirovich, L., Balachandar, S. & Maxey, M. R., 1989 Simulations of turbulent thermal convection, Phys. Fluids A 1, 1911.Google Scholar
Sirovich, L. & Rodriguez, J. D., 1987 Coherent structures and chaos: A model problem. Phys. Lett. A 120, 211.Google Scholar
Sirovich, L. & Sirovich, C., 1989 Low dimensional description of complicated phenomena. In Contemporary Mathematics (ed. B. Nicolaenko), p. 277. American Mathematical Society.
Tbnnekes, H. & Lumley, J. L., 1972 A First Course in Turbulence. MIT Press.
Wolf, A., Swift, J. B., Swinney, H. L. & Vastano, J. A., 1985 Determining Lyapunov exponents from a time series. Physica 16D. 285.Google Scholar