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Parametrization of global attractors, experimental observations, and turbulence

Published online by Cambridge University Press:  26 April 2007

JAMES C. ROBINSON
JAMES C. ROBINSON*
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
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Abstract

This paper is concerned with rigorous results in the theory of turbulence and fluid flow. While derived from the abstract theory of attractors in infinite-dimensional dynamical systems, they shed some light on the conventional heuristic theories of turbulence, and can be used to justify a well-known experimental method.

Two results are discussed here in detail, both based on parametrization of the attractor. The first shows that any two fluid flows can be distinguished by a sufficient number of point observations of the velocity. This allows one to connect rigorously the dimension of the attractor with the Landau–Lifschitz ‘number of degrees of freedom’, and hence to obtain estimates on the ‘minimum length scale of the flow’ using bounds on this dimension. While for two-dimensional flows the rigorous estimate agrees with the heuristic approach, there is still a gap between rigorous results in the three-dimensional case and the Kolmogorov theory.

Secondly, the problem of using experiments to reconstruct the dynamics of a flow is considered. The standard way of doing this is to take a number of repeated observations, and appeal to the Takens time-delay embedding theorem to guarantee that one can indeed follow the dynamics ‘faithfully’. However, this result relies on restrictive conditions that do not hold for spatially extended systems: an extension is given here that validates this important experimental technique for use in the study of turbulence.

Although the abstract results underlying this paper have been presented elsewhere, making them specific to the Navier–Stokes equations provides answers to problems particular to fluid dynamics, and motivates further questions that would not arise from within the abstract theory itself.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 578 , 10 May 2007 , pp. 495 - 507
Copyright
Copyright © Cambridge University Press 2007

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References

REFRENCES

Constantin, P., Foias, C. & Temam, R. 1985 Attractors representing turbulent flows. Memoirs of AMS, Vol. 53, no. 314.Google Scholar
Constantin, P., Foias, C. & Temam, R. 1988 On the dimension of attractors in two-dimensional turbulence. Physica D 30, 284–296.CrossRefGoogle Scholar
Doerin, C. R. & Gibbon, J. D. 1995 Applied Analysis of the Navier–Stokes Equations. Cambridge University Press.CrossRefGoogle Scholar
Falconer, K. 1990 Fractal Geometry. Wiley.Google Scholar
Foias, C. & Temam, R. 1984 Determination of the solutions of the Navier–Stokes equations by a set of nodal values. Math. Comput. 43, 117–133.CrossRefGoogle Scholar
Foias, C. & Temam, R. 1989 Gevrey class regularity for the solutions of the Navier–Stokes equations. J. Func. Anal. 87, 359–369.Google Scholar
Friz, P. K. & Robinson, J. C. 1999 Smooth attractors have zero ‘thickness’. J. Math. Anal. Appl. 240, 37–46Google Scholar
Friz, P. K. & Robinson, J. C. 2001 Parametrising the attractor of the two-dimensional Navier–Stokes equations with a finite number of nodal values. Physica D 148, 201–220.Google Scholar
Gibbon, J. D. & Titi, E. S. 1997 Attractor dimension and small length scale estimates for the three-dimensional Navier–Stokes equations. Nonlinearity 10, 109–119.CrossRefGoogle Scholar
Grappin, R. & Léorat, J. 1991 Lyapunov exponents and the dimension of periodic incompresible Navier–Stokes flows: numerical measurements. J. Fluid Mech. 222, 66–94.CrossRefGoogle Scholar
Henry, D. 1981 Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer.CrossRefGoogle Scholar
Hunt, B. R. & Kaloshin, V. Y. 1999 Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces. Nonlinearity 12, 1263–1275.Google Scholar
Hunt, B. R., Sauer, T. & Yorke, J. A. 1992 Prevalence: a translation-invariant almost every for infinite dimensional spaces. Bull. Am. Math. Soc. 27, 217–238.CrossRefGoogle Scholar
Keefe, L., Moin, P. & Kim, J. 1992 The dimension of attractors underlying periodic turbulent Poiseuille flow. J. Fluid Mech. 242, 1–29.CrossRefGoogle Scholar
Kraichnan, R. H. 1967 Inertial rangesin two-dimensional turbulence. Phys. Fluids 10, 1417–1423.CrossRefGoogle Scholar
Kukavica, I. 1994 An absence of a certain class of periodic solutions in the Navier–Stokes equations. J. Dyn. Diffl Equat. 6, 175–183.CrossRefGoogle Scholar
Kukavica, I. & Robinson, J. C. 2004 Distinguishing smooth functions by a finite number of point values, and a version of the Takens embedding theorem. Physica D 196, 45–66.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Course of Theoretical Physics, vol. 6. Pergamon.Google Scholar
Miranville, A. & Wang, X. 1997 Attractors for nonautonomous nonhomogeneous Navier–Stokes equations. Nonlinearity 10, 1047–1061.CrossRefGoogle Scholar
Robinson, J. C. 2001 Infinite-dimensional Dynamical Systems. Cambridge University Press.CrossRefGoogle Scholar
Robinson, J. C. 2003 Low dimensional attractors arise from forcing at small scales. Phys. D 181, 39–44.Google Scholar
Robinson, J. C. 2005 A topological delay embedding theorem for infinite-dimensional dynamical systems. Nonlinearity 18, 2135–2143.CrossRefGoogle Scholar
Robinson, J. C. & Vidal-López, A. 2004 Minimal periods of semilinear evolution equations with Lipschitz nonlinearity. J. Diffl Equat. 220, 396–406.CrossRefGoogle Scholar
Sauer, T., Yorke, J. A. & Casdagli, M. 1993 Embedology J. Statist. Phys. 71, 529–547.Google Scholar
Takens, F. 1981 Detecting Strange Attractors in Turbulence. Springer Lecture Notes in Mathematics, vol. 898, pp. 366–381.CrossRefGoogle Scholar
Temam, R. 1988 Infinite-dimensional Dynamical Systems in Mechanics and Physics. Springer.Google Scholar
Tran, C. V., Shepherd, T. G. & Cho, H.-R. 2004 Extensivity of two-dimensional turbulence. Physica D 192, 187–195.Google Scholar
Yorke, J. A. 1969 Periods of periodic solutions and the Lipschitz constant. Proc. AMS bf22, 509–512Google Scholar