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Parametrization of global attractors experimental observations and turbulence
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Robinson, James C.. (2007) Parametrization of global attractors experimental observations and turbulence. Journal of Fluid Mechanics, Vol.57 . pp. 495507. ISSN 00221120

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Official URL: http://dx.doi.org/10.1017/S0022112007005137
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 infinitedimensional dynamical systems, they shed some light on the conventional heuristic theories of turbulence, and can be used to justify a wellknown 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 twodimensional flows the rigorous estimate agrees with the heuristic approach, there is still a gap between rigorous results in the threedimensional 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 timedelay 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.
Item Type:  Journal Article  

Subjects:  Q Science > QA Mathematics T Technology > TA Engineering (General). Civil engineering (General) 

Divisions:  Faculty of Science > Mathematics  
Library of Congress Subject Headings (LCSH):  Turbulence, Fluid dynamics, Attractors (Mathematics)  
Journal or Publication Title:  Journal of Fluid Mechanics  
Publisher:  Cambridge University Press  
ISSN:  00221120  
Official Date:  26 April 2007  
Dates: 


Volume:  Vol.57  
Page Range:  pp. 495507  
Identifier:  10.1017/S0022112007005137  
Status:  Peer Reviewed  
Access rights to Published version:  Open Access  
Funder:  Royal Society (Great Britain)  
References:  Constantin, P., Foias, C. & Temam, R. 1985 Attractors representing turbulent flows. Memoirs of AMS, Vol. 53, no. 314. 
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