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Revisiting Batchelor's theory of two-dimensional turbulence

Published online by Cambridge University Press:  30 October 2007

DAVID G. DRITSCHEL ,
CHUONG V. TRAN  and
RICHARD K. SCOTT
DAVID G. DRITSCHEL
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK
CHUONG V. TRAN
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK
RICHARD K. SCOTT
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK
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Abstract

Recent mathematical results have shown that a central assumption in the theory of two-dimensional turbulence proposed by Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) is false. That theory, which predicts a χ2/3k−1 enstrophy spectrum in the inertial range of freely-decaying turbulence, and which has evidently been successful in describing certain aspects of numerical simulations at high Reynolds numbers Re, assumes that there is a finite, non-zero enstrophy dissipation χ in the limit of infinite Re. This, however, is not true for flows having finite vorticity. The enstrophy dissipation in fact vanishes.

We revisit Batchelor's theory and propose a simple modification of it to ensure vanishing χ in the limit Re → ∞. Our proposal is supported by high Reynolds number simulations which confirm that χ decays like 1/ln Re, and which, following the time of peak enstrophy dissipation, exhibit enstrophy spectra containing an increasing proportion of the total enstrophy 〈ω2〉/2 in the inertial range as Re increases. Together with the mathematical analysis of vanishing χ, these observations motivate a straightforward and, indeed, alarmingly simple modification of Batchelor's theory: just replace Batchelor's enstrophy spectrum χ2/3k−1 with 〈ω2〉 k−1 (ln Re)−1).

Type
Papers
Information
Journal of Fluid Mechanics , Volume 591 , 25 November 2007 , pp. 379 - 391
Copyright
Copyright © Cambridge University Press 2007

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