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Lyapunov exponents and the dimension of periodic incompressible Navier—Stokes flows: numerical measurements

Published online by Cambridge University Press:  26 April 2006

Roland Grappin  and
Jacques Léorat
Roland Grappin
Affiliation:
Observatoire de Meudon, F-92190 Meudon, France
Jacques Léorat
Affiliation:
Observatoire de Meudon, F-92190 Meudon, France
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Abstract

In this paper we study the quasi-stationary turbulent state developed by an incompressible flow submitted to a constant periodic force. The turbulent state is described by its Lyapunov exponents and Lyapunov dimension. Our aim is to investigate in particular if the dimension is, as several lines of argument seem to indicate, given in one way or another by the number of ‘turbulent’ modes present in the flow.

The exponents may be viewed as the asymptotic limits of local exponents, which are the divergence rates between the actual state of the fluid and nearby states. These local exponents fluctuate as the fluid suffers chaotic changes: they are systematically larger during bursts of excitation at large scale. In one case, we find that the flow oscillates repeatedly, on a very long timescale, between two distinct turbulent states, which have distinct local Lyapunov spectra and also distinct energy spectra. At the moderate resolutions studied here (642, 1282 and 163), we find that the dimension of the attractor in flows with standard dissipation is bounded above by the number of degrees of freedom contained in the large scales, i.e. at wavenumbers smaller than that of the injection wavenumber kf. We also consider two-dimensional flows with artificial terms (hyperviscosity), which concentrate small-scale dissipation in a narrow band of wavenumbers, and allow the formation of an inertial range. The dimension in these flows is no longer bounded by the number of large scales; it grows with, but remains significantly smaller than, the number of modes contained in the inertial range. It is conjectured that this is due to the presence of coherent structures in the flow, and that at higher resolutions also, the presence of coherent structures in turbulent flows will lead to an effective attractor dimension significantly lower than the estimations based on self-similar Kolmogorov theories.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 222 , January 1991 , pp. 61 - 94
Copyright
© 1991 Cambridge University Press

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References

Atten, P., Caputo, J. G., Malraison, B. & Gagne, Y., 1984 Détermination de dimension d'attracteurs pour différents écoulements. J. Méc. Theor. Appl., Numéro Spécial, p. 133.Google Scholar
Babiano, A., Basdevant, C., Legras, B. & Sadourny, R., 1987 Vorticity and passive-scalar dynamics in two-dimensional turbulence. J. Fluid Mech. 183, 379.Google Scholar
Basdevant, C., Legras, B., Sadourny, R. & Beland, M., 1981 A study of barotropic model flows: intermittency waves and predictability. J. Atmos. Sci. 38, 2305.Google Scholar
Benettin, G., Galgani, L., Giorgilli, A. & Strelcyn, J.-M. 1980 Meccanica 15, 10.
Brachet, M. E., Meneguzzi, M., Politano, H. & Sulem, P. L., 1988 The dynamics of freely decaying two-dimensional turbulence. J. Fluid Mech. 194, 333.Google Scholar
Constantin, P., Foias, C., Manley, O. P., Temam, R.: 1985 Determining modes and fractal dimension of turbulent flows. J. Fluid Mech. 150, 427.Google Scholar
Galloway, D. & Frisch, U., 1987 A note on the stability of space-periodic Beltrami flows. J. Fluid Mech. 180, 557.Google Scholar
Grappin, R. & Léorat, J. 1987 Computation of the dimension of two-dimensional turbulence. Phys. Rev. Lett. 59, 1100 (referred to herein as GL87).Google Scholar
Grappin, R. & Leorat, J., 1989 Lyapunov dimension of homogeneous shear flows. In Conf. On Nonlinear Dynamics, Bologna (ed. G. Turchetti). Singapore: World Scientific.
Grappin, R., Léohat, J. & Londrillo, P. 1988 Onset and development of turbulence in two-dimensional periodic shear flows. J. Fluid Mech. 195, 239.Google Scholar
Grappin, R., Léohat, J. & Pouquet, A. 1986 Computation of the dimension of a model of fully developed turbulence. J. Phys. Paris. 47, 1127.Google Scholar
Green, J. S. A.: 1974 Two-dimensional turbulence near the viscous limit. J. Fluid Mech. 62, 273.Google Scholar
Kaplan, J. L. & Yorke, J. A., 1979 In Functional Differential Equations and Approximations of Fixed Points, p. 228. Springer.
Keefe, L.: 1989 Properties of Ginzburg—Landau attractors associated with their Lyapunov vectors and spectra. Phys Lett. A 140, 317.Google Scholar
Keefe, L., Moin, P. & Kim, J., 1989 Applications of chaos theory to shear turbulence. NASA Ames Res. Center, preprint MS202A-1.Google Scholar
Kolmogorov, A. N.: 1960 In Seminar Notes edited by Arnold, V. I. & Meshalkin, L. D., Usp. Mat. Nauk 15, 247.Google Scholar
Kraichnan, R. H.: 1967 Inertial ranges in two- and three-dimensional turbulence. Phys. Fluids 10, 1417.Google Scholar
Kraichnan, R. H.: 1987 Reduced descriptions of hydrodynamic turbulence. J. Statist. Phys. 51, 949.Google Scholar
Lafon, A.: 1985 Etude des attracteurs pour des éeoulements bidimensionnels de fluides visqueux incompressibles. Thèse d'Etat Université Paris 6.
Landau, L. D. & Lifshitz, E. M., 1971 Mécanique des Fluides, chap. 3, p. 154. Moscow: Mir.
Legras, B., Santangelo, P. & Benzi, R., 1988 High resolution numerical experiments for forced two-dimensional turbulence. Europhys. Lett. 5, 37.Google Scholar
Manneville, P.: 1985 Lyapunov exponents for the Kuramoto-Sivashinsky model. In Macroscopic Modeling of Turbulent Flows (ed. U. Frisch, J. B. Keller, G. Papanicolaou, O. Pironneau). Springer.
Meshalkin, L. D. & Sinai, Y. A. & Ya, G. 1961 Z. Angew. Math. Mech. 25, 1700.
Obukhov, A. M.: 1983 Russ. Math. Survey 38, 113.
Ohkitani, K. & Yamada, M., 1989 Prog. Theor. Phys. 81, 329.Google Scholar
Oseledec, V. I.: 1968 A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Maths Soc. 19, 197.Google Scholar
She, Z. S.: 1987 Metastability and vortex pairing in the Kolmogorov flow. Phys. Lett. A 124, 161.Google Scholar
Yamada, M. & Ohkitani, K., 1988 Lyapunov spectrum of a model of two-dimensional turbulence. Phys. Rev. Lett. 60, 983.Google Scholar