Determining modes and fractal dimension of turbulent flows

Constantin, Peter; Foiaş, Ciprian; Manley, O. P.; Témam, Roger

doi:10.1017/s0022112085000209

Cited by 199 publications

(117 citation statements)

References 7 publications

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“…Global regularity of these equations on a periodic box and subsequence convergence to Leray solutions of the NSE as α 1 → 0 (and α 0 → 1) is established in [15], as well as estimates on the Hausdorff and fractal dimension of the attractor. In particular it is shown in [15] 3 for a generic constant c. The power on l 0 /l matches the Landau-Lifschitz prediction and also matches the estimates on invariant sets bounded in V = P H 1 (Ω) for weak solutions of the 3D NSE as shown in [11][12][13]. There is also no potential to "absorb" the growth term (1/α 1 ) 3/2 , so the estimate simply grows without bound as α 1 → 0.…”

Section: Resultssupporting

confidence: 74%

“…[11,14,31,32]), first developed for the 2D NSE, that relies both on trace formulas and the Lieb-Thirring inequality (LTI), the latter being first used in this context in [31,32]. Related results for weak solutions of the NSE in 3D can be found in [4,12,13,29]. In [15] for strong solutions of the model known variously as the NS-α, 3D LANS-α, or 3D Camassa-Holm equations, the CFT/LTI methodology is applied toward estimating attractor dimension and in the conclusion of this paper we will compare the attractor estimates developed in [15] with (1.10a-1.10d).…”

Section: Introductionmentioning

confidence: 99%

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The Asymptotic Finite-dimensional Character of a Spectrally-hyperviscous Model of 3D Turbulent Flow

Avrin

2008

J Dyn Diff Equat

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We obtain attractor and inertial-manifold results for a class of 3D turbulent flow models on a periodic spatial domain in which hyperviscous terms are added spectrally to the standard incompressible Navier-Stokes equations (NSE). Let P m be the projection onto the first m eigenspaces of A = − , let µ and α be positive constants with α 3/2, and let Q m = I − P m , then we add to the NSE operators µA ϕ in a general family such that A ϕ Q m A α in the sense of quadratic forms. The models are motivated by characteristics of spectral eddy-viscosity (SEV) and spectral vanishing viscosity (SVV) models. A distinguished class of our models adds extra hyperviscosity terms only to high wavenumbers past a cutoff λ m 0 where m 0 m, so that for large enough m 0 the inertial-range wavenumbers see only standard NSE viscosity. We first obtain estimates on the Hausdorff and fractal dimensions of the attractor A (respectively dim H A and dim F A). For a constant K α on the order of unity we show ifrepresents characteristic macroscopic length, and l is the Kolmogorov length scale, i.e. l = (ν 3 / ) where is Kolmogorov's mean rate of dissipation of energy in turbulent flow. All bracketed constants and K α are dimensionless and scale-invariant. The estimate grows in m due to the term λ m /λ 1 but at a rate lower than m 3/5 , and the estimate grows in µ as the relative size of ν to µ. The exponent on l 0 /l is significantly less than the Landau-Lifschitz predicted value of 3. If we impose the condition λ m (1/l ) 2 , the estimates become K α [l 0 /l ] 3 for µ ν and As a corollary, for most of the cases of the operators A ϕ in the distinguished-class case that we expect will be typically used in practice we also obtain an M, now of dimension m 0 for m 0 large enough, though under conditions requiring generally larger m 0 than the m in the special class. In both cases, for large enough m (respectively m 0 ), we have an inertial manifold for a system in which the inertial range essentially behaves according to standard NSE physics, and in particular trajectories on M are controlled by essentially NSE dynamics.KEY WORDS: 3D turbulent flow models; attractor dimension; inertial manifolds; degrees of freedom.

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Section: Resultssupporting

confidence: 74%

Section: Introductionmentioning

confidence: 99%

The Asymptotic Finite-dimensional Character of a Spectrally-hyperviscous Model of 3D Turbulent Flow

Avrin

2008

J Dyn Diff Equat

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“…In various works [6,8,9,19,24], several estimates on the dimension of global attractor of the three-dimensional Navier-Stokes equations were derived, although it is still an outstanding open problem to show that it exists. In particular, it has been shown that if A ⊂ H is invariant and bounded in V, then dim H A ≤ cG 3/2 .…”

Section: Remark 54mentioning

confidence: 99%

Singular Limits of Voigt Models in Fluid Dynamics

Zelati

Gal

2015

J. Math. Fluid Mech.

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ABSTRACT. We investigate the long-term behavior, as a certain regularization parameter vanishes, of the three-dimensional Navier-Stokes-Voigt model of a viscoelastic incompressible fluid. We prove the existence of global and exponential attractors of optimal regularity. We then derive explicit upper bounds for the dimension of these attractors in terms of the three-dimensional Grashof number and the regularization parameter. Finally, we also prove convergence of the (strong) global attractor of the 3D Navier-Stokes-Voigt model to the (weak) global attractor of the 3D Navier-Stokes equation. Our analysis improves and extends recent results obtained by Kalantarov and Titi in [33].

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“…This is relevant to the widely held view that chaotic, dissipative, dynamical systems are eventually drawn into a strange attractor which is of relatively low dimension [14,15]. Indeed a number of fluid experiments support this view [16,17,18], Theoretical estimates also support this view but greatly overestimate the dimension and alas provide no clue to the parametrization of the attractor [19]. While we do not confront this issue directly it will be seen that a practical, and in a sense optimal, description of the attractor is furnished.…”

mentioning

confidence: 95%