The evolution of freely-decaying, isotropic, two-dimensional turbulence

A. J. Lowe, P. A. Davidson

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This paper discusses the decay of freely-evolving, two-dimensional turbulence. We present numerical simulations in which particular care is taken to avoid unphysical pollution from periodic boundary conditions. Batchelor's classical theory, which assumes that the kinetic energy is the only invariant of the flow, predicts that the integral scale grows as l∼t. In line with the results of Herring et al. [Evolution of Decaying Two-dimensional Turbulence and Self-similarity, Trends in Mathematics, Birkhäuser Verlag Basel, Switzerland, 1999], and many others, our DNS results show that l grows approximately as t1/2. Bartello and Warn [J. Fluids Mech. 326 (1996) 357-372], and McWilliams [J. Fluids Mech. 146 (1984) 21-43], suggested that, in the limit Re→∞, two-dimensional turbulence possesses a second invariant: the peak in vorticity. It is now widely accepted that this explains the failure of Batchelor's theory. However, as yet, there is no satisfactory explanation for the t1/2 growth in l. Periodic boundary conditions impose mirror image long-range correlations of velocity and vorticity. Lilly [J. Fluid Mech. 45 (2) (1971) 395-415], and Davidson [Turbulence: An Introduction for Scientists and Engineers, Oxford University Press, 2004], note that these correlations have the potential to influence the dynamical behavior of the turbulence. Varying the energy-containing length-scale of the turbulence relative to the size of the periodic domain allows this effect to be investigated. To this end, we introduce the box-ratio, ldomain/lturbulence, as a measure of the number of energy containing eddies in our simulations. Over a wide range of box-ratios we show that l grows approximately as t1/2, the enstrophy decays as ∼t-1 (at least for large Re), and the l-normalised vorticity correlations more or less collapse onto a single, self-similar curve. We provide one possible explanation for the observed t1/2 growth of l. © 2005 Elsevier SAS. All rights reserved.
Original languageEnglish
Pages (from-to)314-327
Number of pages13
JournalEuropean Journal of Mechanics B. Fluids
Issue number3
Publication statusPublished - Mar 2005


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